text: Madiha Jamal answered this. BOOKMARKS. Omissions? Phosphorus(III) oxide is a white crystalline solid that smells like garlic and has a poisonous vapour. Non-metallic oxides are acidic in nature because they turn … nature of oxides. Basic oxides are the oxides of metals. An amphoteric oxide is that metallic oxide displaying a dual behaviour. Certain organic compounds react with oxygen or other oxidizing agents to produce substances called oxides. All the alkali metal oxides can be prepared by heating the corresponding metal nitrate with the elemental metal. Periodic trends of the oxides have been thoroughly studied. In any given period, the bonding in oxides progresses from ionic to covalent, and their acid-base character goes from strongly basic through weakly basic, amphoteric, weakly acidic, and finally strongly acidic. Reaction with acids: Aluminum oxide contains oxide ions, and thus reacts with acids in the same way sodium or magnesium oxides do. In lithium cells, only the octahedral structure forms, but in sodium cells, trigonal prismatic structures are also possible. Watch the recordings here on Youtube! Please select which sections you would like to print: Corrections? Cl2O, SO2, P4O10). By reaction of oxygen with compounds at higher temperatures: At higher temperatures, oxygen also reacts with many compounds forming oxides, e.g., $2PbS + 3O_2 \xrightarrow{\Delta} 2PbO + 2SO_2$, $2ZnS + 3O_2 \xrightarrow{\Delta} 2ZnO + 2SO_2$, $C_2H_5OH + 3O_2 \rightarrow 2CO_2 + 3H_2O$, $CaCO_3 \xrightarrow{\Delta} CaO + CO_2$, $2Cu(NO_3)_2 \xrightarrow{\Delta} 2CuO + 4NO_2 + O_2$, $Cu(OH)_2 \xrightarrow{\Delta} CuO + H_2O$, By oxidation of some metals with nitric acid, $2Cu + 8HNO_3 \xrightarrow{Heat} 2CuO + 8NO_2 + 4H_2O + O_2$, $Sn + 4HNO_3 \xrightarrow{Heat} SnO_2 + 4NO_2 + 2H_2O$, By oxidation of some non-metals with nitric acid, $C + 4HNO_3 \rightarrow CO_2 + 4NO_2 + 2H_2O$. D. Neutral. Author of, The structures of phosphorus(III) oxide, P. Get exclusive access to content from our 1768 First Edition with your subscription. The so-called olefin oxides are cyclic ethers. Hydrocarbon combustion produces the two principal carbon oxides: carbon monoxide (CO) and carbon dioxide (CO2). Amphoterism among the main group oxides is primarily found with the metalloidal elements or their close neighbours. M2O + H2O → 2MOH (where M = group 1 metal)MO + H2O → M(OH)2 (where M = group 2 metal) All Questions Ask Doubt. $H_2 + O_2 \rightarrow H_2O_2 \label{20}$. (M represents a metal atom.) Example : C O 2 + H 2 O → H 2 C O 3 Answered By . It is difficult to break the covalent O-H bonds easily. The famous “Terra di Siena” is a hydrated iron oxide from Tuscany. Carbon dioxide is colourless and odourless gas in nature. Core level energies and intensities and valence band spectra are compared with data for standard compounds. Acidic oxides are known as acid anhydrides (e.g., sulfur dioxide is sulfurous anhydride and sulfur trioxide is sulfuric anhydride) and when combined with bases, they produce salts, e.g., $\ce{ SO_2 + 2NaOH \rightarrow Na_2SO_3 + H_2O} \label{4}$. In most soils, FeIII oxides (group name) are the common source of Fe for plant nutrition. Neutral Oxide is one which neither has an acidic characteristic or a basic one. Solution for group 1 elements with that of group 2 on the basis of the following:… Updates? If we take a closer look at a specific period, we may better understand the acid-base properties of oxides. Iron oxide (rust) on a … with the oxidation number of the oxygen equal to -1/2. Both metals and nonmetals can attain their highest oxidation states (i.e., donate their maximum number of available valence electrons) in compounds with oxygen. A general preparation of the alkaline earth oxides involves heating the metal carbonates. Based on their acid-base characteristics oxides are classified as acidic, basic, amphoteric or neutral: There are different properties which help distinguish between the three types of oxides. However, there is a lack of understanding about how to predict and control the formation of each structure. Al2O3 + 6HCl → 2AlCl3 + 3H2O The result of the interaction depends on the quantity of the initial alkali. Examples include the oxide of most metals, such as Na 2 O, CaO, BaO. toppr. The individual element must be in its highest possible oxidation state because the trend does not follow if all oxidation states are included. Phosphorus(V) … Chemical Properties of Metals and Non-metals - 1. Oxides are binary compounds of oxygen with another element, e.g., CO2, SO2, CaO, CO, ZnO, BaO2, H2O, etc. It has the chemical formula CO 2. The oxides of elements in a period become progressively more acidic as one goes from left to right in a period of the periodic table. Non-metals react with oxygen in the air to produce non-metal oxides. The most important basic oxides are magnesium oxide (MgO), a good thermal conductor and electrical insulator that is used in firebrick and thermal insulation, and calcium oxide (CaO), also called quicklime or lime, used extensively in the steel industry and in water purification. The alkali metals and alkaline earth metals form three different types of binary oxygen compounds: (1) oxides, containing oxide ions, O2−, (2) peroxides, containing peroxide ions, O22−, which contain oxygen-oxygen covalent single bonds, and (3) superoxides, containing superoxide ions, O2−, which also have oxygen-oxygen covalent bonds but with one fewer negative charge than peroxide ions. $\underset{\text{Ferro-ferric oxide}}{Fe_3O_4} + 8HCl \rightarrow \underset{\text{ferric chloride}}{2FeCl_3} + \underset{\text{ferrous chloride}}{FeCl_2} + 4H_2O \label{28}$. It is the acid anhydride of phosphorous acid, H3PO3, that is produced as P4O6 dissolves slowly in cold water. In many parts of the world, silica is the major constituent of sand. . - Wissenschaft | September 2020. $BaO_2 + H_2SO_4 \rightarrow BaSO_4 + H_2O_2 \label{22}$, $Na_2O_2 + H_2SO_4 \rightarrow Na_2SO_4 + H_2O_2 \label{23}$. It may also help to examine the physical properties of oxides, but it is not necessary. [ "article:topic", "shrestha", "Acidic Oxides", "basic oxides", "Neutral Oxides", "showtoc:no" ], http://www.wou.edu/las/physci/ch412/oxides.html, information contact us at info@libretexts.org, status page at https://status.libretexts.org. The term anhydride ("without water") refers to compounds that assimilate H2O to form either an acid or a base upon the addition of water. Oxides of the transition metals with oxidation numbers of +1, +2, and +3 are ionic compounds consisting of metal ions and oxide ions. Definition ; Nomenklatur ; Klassifizierung ; Amphotere Oxide ; Wege zu bekommen ; Fazit ; Sprechen wir darüber, wie man die Art des Oxids bestimmen kann. Oxides are chemical compounds with one or more oxygen atoms combined with another element (e.g. While reacting with alkalies they form salt and water showing acidic properties. This video explains the reaction of metals and non-metals with oxygen. However, it is also possible for an oxide to be neither acidic nor basic, but is a neutral oxide. The oxide ions are held too strongly in the solid lattice to react with the water. Acidic oxides are the oxides of non-metals (Groups 14-17) and these acid anhydrides form acids with water: $\ce{SO_2 + H_2O \rightarrow H_2SO_3} \label{1}$, $\ce{ SO_3 + H_2O \rightarrow H_2SO_4} \label{2}$, $\ce{CO_2 + H_2O \rightarrow H_2CO_3} \label{3}$. It is important to remember that the trend only applies for oxides in their highest oxidation states. In this (and similar reactions with other acids), aluminium oxide is showing the basic side of its amphoteric nature. Oxygen can thus be obtained from acidified water by its electrolysis. Loading... Close. This is the reason many oxides are found in nature as minerals. Thus Al2O3 entails the marking point at which a change over from a basic oxide to acidic oxide occurs. Metal oxides are crystalline solids that contain a metal cation and an oxide anion. Artists know these as Ochre, Sienna, and Umber. $Cs + O_2 \rightarrow CsO_2 \label{21}$. Since this Fe has to be supplied via solution, the solubility and the dissolution rate of the Fe oxides are essential for the Fe supply. With the exception of the lighter inert gases (helium [He], neon [Ne], argon [Ar], and krypton [Kr]), oxygen (O) forms at least one binary oxide with each of the elements. They react with acids to produce salts, e.g., $\ce{ MgO + 2HCl \rightarrow MgCl_2 + H_2O } \label{9}$, $\ce{ Na_2O + H_2SO_4 \rightarrow Na_2SO_4 + H_2O} \label{10}$. $$Rb + O_2 \; (excess) \rightarrow RbO_2$$. These are termed as oxides because here, oxygen is in combination with only one element. These are termed as oxides because here, oxygen is in combination with only one element. Both oxides have a structure based on the tetrahedral structure of elemental white phosphorus. Why is it difficult to obtain oxygen directly from water? An amphoteric solution is a substance that can chemically react as either acid or base. The chemical composition of surface oxides grown on InP is investigated using x‐ray photoelectron spectroscopy. Oxides are chemical compounds with one or more oxygen atoms combined with another element (e.g. By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. General Chemistry: principles and modern applications. Na2O and MgO). 9th Edition. OXYGEN IS HIGHLY REACTIVE IN NATURE… Layered metal oxides such as lithium cobalt oxide have attracted great attention for rechargeable batteries. The rich array of phenomena found in oxides is largely due to the oxygen, says Yoshinori Tokura, a physicist from the University of Tokyo who has worked in this field for more than 20 years. Legal. Notable examples include The rich array of phenomena found in oxides is largely due to the oxygen, says Yoshinori Tokura, a physicist from the University of Tokyo who has worked in this field for more than 20 years. Metals with high oxidation states form oxides whose bonds have a more covalent nature. Limonite ranges from yellow to brown and contains water in its chemical composition (hydrous ferric oxide). Metal Oxides have an oxidation number of -2 and generally comprise of … The oxides of non-metal are acidic in nature as oxide of non-metal reacts with water to form acid. The oxides with oxidation numbers of +5, +6, and +7 are acidic and react with solutions of hydroxide to form salts and water; for example, Zhao et al. Can an oxide be neither acidic nor basic? Peroxides: Often Lithium and Sodium reacts with excess oxygen to produce the peroxide, $$M_2O_2$$. An amphoteric oxide is an oxide that can act as either an acid or base in a reaction to produce a salt and water. Metal oxides typically contain an anion of oxygen in the oxidation state of −2. Metal and non-metal oxides. For example, vanadium oxide (VO2) is an amphoteric oxide, dissolving in acid to give the blue vanadyl ion, [VO]2+, and in base to yield the yellow-brown hypovanadate ion, [V4O9]2−. Metals reacting with water. Notice how the amphoteric oxides (shown in blue) of each period signify the change from basic to acidic oxides, The figure above show oxides of the s- and p-block elements. In this video electropositivity is linked to nature of oxides describing how nature of oxides changes is periodic table and thus the electropositivity. A smooth variation from ionic to covalent in the type of bonding in oxides is observed as the periodic table is traversed from the metals on the left to the nonmetals on the right. Below are a few. Oxides are binary compounds of oxygen with another element, e.g., CO 2, SO 2, CaO, CO, ZnO, BaO 2, H 2 O, etc. \underset{\large{Acidic}}{\underbrace{P_4O_{10},\: SO_3,\:Cl_2O_7}}\hspace{20px}\). Source: From W. Robinson, J. Odom, and H. Holtzclaw, Jr., Chemistry: Concepts and Models, D.C. Heath and Co., 1992. reaction of oxides with water and the acid-base character of hydroxides. For example, in the reaction of carbon monoxide (4) with … Many metals and non-metals react with oxygen in the air when they are heated to produce metal oxides and non-metal oxides.. Ochre is clay-colored with hydrated iron oxide. By direct heating of an element with oxygen: Many metals and non-metals burn rapidly when heated in oxygen or air, producing their oxides, e.g., $P_4 + 5O_2 \xrightarrow{Heat} 2P_2O_5$. Have questions or comments? A basic oxide is an oxide which when combined with water gives off a base. Dioxides on reaction with concentrated HCl yield Cl2 and on reacting with concentrated H2SO4 yield O2. There is a trend within acid-base behavior: basic oxides are present on the left side of the period and acidic oxides are found on the right side. Watch Queue … Oxides can be generated via multiple reactions. Hence, electrical energy through the electrolysis process is applied to separate dioxygen from water. DAY-8 PART-1 NATURE OF OXIDES. Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. M2O + 2HCl → 2MCl + H2O (where M = group 1 metal). \underset{\large{Amphoteric}}{\underbrace{Al_2O_3,\: SiO_2}}\hspace{20px} Superoxides: Often Potassium, Rubidium, and Cesium react with excess oxygen to produce the superoxide, $$MO_2$$. It oxidizes slowly in air and inflames when heated to 70 °C (158 °F), forming P4O10. They typically react with water to form bases or with acids to form salts. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. The nature of the oxides involves interaction with bases (alkalis). Acidic, basic, and amphoteric oxides. Alkali metals (which have a +1 oxidation state) form oxides, M2O, peroxides, M2O2, and superoxides, MO2. Moreover, the oxides turn moist red litmus paper blue and blue reamins blue. C. Amphoteric. 3) Amphoteric Oxide. Get Instant Solutions, 24x7. Watch Queue Queue. Metals reacting with oxygen. Even materials considered pure elements often develop an oxide coating. The pattern is less clear for other oxides. Transition metal oxides are compounds composed of oxygen atoms bound to transition metals.They are commonly utilized for their catalytic activity and semiconductive properties. CrO3 + 2OH- → CrO42− + H2O. As a general rule, the ionic transition metal oxides are basic. Oxides: Group 1 metals react rapidly with oxygen to produce several different ionic oxides, usually in the form of $$M_2O$$. They are able to catalyze the oxygen evolution reaction in acidic solutions but with only modest activity when prepared by constant anodic potential deposition. Electrodeposited manganese oxide films (MnOx) are promising stable oxygen evolution catalysts. Those oxides with +4 oxidation numbers are generally amphoteric (from Greek amphoteros, “in both ways”), meaning that these compounds can behave either as acids or as bases. MCO3 + heat → MO + CO2 Metal oxides are basic in nature beacase they react with water to form bases. Generally Group 1 and Group 2 elements form bases called base anhydrides or basic oxides e.g., $\ce{K_2O \; (s) + H_2O \; (l) \rightarrow 2KOH \; (aq) } \label{5}$. Amphoterism depends on the oxidation states available to a chemical species. CO doesn’t produce a salt when reacted with an acid or a base. Neutral oxides show neither basic nor acidic properties and hence do not form salts when reacted with acids or bases, e.g., carbon monoxide (CO); nitrous oxide (N2O); nitric oxide (NO), etc., are neutral oxides. Missed the LibreFest? In general, HEOs get their unique properties from the large number of components (>5). The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. For example, they all react with concentrated hydrochloric acid. This will help us to improve better. An oxide is a chemical compound that contains at least one oxygen atom and one other element in its chemical formula. Please Register or login to watch this video. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Metals reacting with acid. Up Next. In accord with their basic behaviour, they react with acids in typical acid-base reactions to produce salts and water; for example, As we know oxygen is a highly electronegative element, so it forms bonds with almost all elements and thus it forms stables corresponding oxides. Amphoteric oxides exhibit both basic as well as acidic properties. Silver turns black when exposed to air. An amphoteric solution is a substance that can chemically react as either acid or base. Na2O + H2O = NaOH + H2. purple: basic oxides blue: amphoteric oxides pink: acidic oxides. A peroxide is a metallic oxide which gives hydrogen peroxide by the action of dilute acids. Calcium oxide reacts with water and produces calcium hydroxide, a base. The alkaline earth metals (with a +2 oxidation state) form only oxides, MO, and peroxides, MO2. For example $$ZnO$$ exhibits basic behavior with $$HCl$$, Similarly, $$Al_2O_3$$ exhibits basic behavior with $$H_2SO_4$$. An oxide that combines with water to give an acid is termed as an acidic oxide. These are basic in nature. It plays a vital role in the existence of life on the earth. 2. Non-metal oxides on the right side of the periodic table produce acidic solutions (e.g. B. $ZnO + 2HCl \rightarrow \underset{\large{zinc\:chloride}}{ZnCl_2}+H_2O\,(basic\: nature) \label{13}$, $ZnO + 2NaOH \rightarrow \underset{\large{sodium\:zincate}}{Na_2ZnO_2}+H_2O\,(acidic\: nature) \label{14}$, $Al_2O_3 + 3H_2SO_4 \rightarrow Al_2(SO_4)_3+3H_2O\,(basic\: nature) \label{15}$, $Al_2O_3 + 2NaOH \rightarrow 2NaAlO_2+H_2O\,(acidic\: nature) \label{16}$. Li2O). Answered by | 22nd Jul, 2016, 12:11: PM. Aluminum oxide shows acid and basic properties of an oxide, it is amphoteric. Search. When a small amount of acid is added to water ionization is initiated which helps in electrochemical reactions as follows. Was ist die Natur der Oxide? This trend applies only to the oxides of the individual elements in the highest oxidation states for those elements. See section above on Properties of Amphoteric Oxides for more detail. Most of the earth crust is composed of solid oxides. The range of colored earth is derived from the nature of iron oxide present in the material. Next lesson. An acidic oxide is an oxide which when combined with water gives off an acid. The alkali metals and the alkaline earth metals, as well as the transition metals and the posttransition metals (in their lower oxidation states), form ionic oxides—i.e., compounds that contain the O2− anion. New Jersey, NJ: Prentice Hall, 2007. Let us know if you have suggestions to improve this article (requires login). Here are two examples for the non-metalscarbon and sulphur. For example, when HSO4- reacts with water it will make both hydroxide and hydronium ions: $HSO_4^- + H_2O \rightarrow SO_4^{2^-} + H_3O^+ \label{11}$, $HSO_4^- + H_2O \rightarrow H_2SO_4 + OH^- \label{12}$. Silicon dioxide, also known as silica, is an oxide of silicon with the chemical formula Si O 2, most commonly found in nature as quartz and in various living organisms. Water as such is a neutral stable molecule. These reactions are also often called neutralization reactions. Professor and Associate Head, Department of Chemistry, University of Illinois at Urbana-Champaign. On treatment with an acid, compound oxides give a mixture of salts. Practice: Metals reacting with air and water. Engineering high entropy oxides – thermodynamics of solid solution formation. with the oxidation number of the oxygen equal to -1. Reactivity series . Those transition metal oxides with oxidation numbers +4, +5, +6, and +7 behave as covalent compounds containing covalent metal-oxygen bonds. Because metals have multiple oxidation states, they form amphoteric oxides and hydroxides. Compound oxides are metallic oxides that behave as if they are made up of two oxides, one that has a lower oxidation and one with a higher oxidation of the same metal, e.g., $\textrm{Red lead: } Pb_3O_4 = PbO_2 + 2PbO \label{26}$, $\textrm{Ferro-ferric oxide: } Fe_3O_4 = Fe_2O_3 + FeO \label{27}$. An amphoteric oxide is one that shows both acidic and basic properties. Navigate parenthood with the help of the Raising Curious Learners podcast. Most of the Earth’s crust consists of solid oxides, the result of elements being oxidized by the oxygen in air or water. When heated with oxygen, compounds containing carbon and hydrogen are oxidized. $PbO_2 + 4HCl \rightarrow PbCl_2 + Cl_2 + 2H_2O \label{24}$, $2PbO_2 + 2H_2SO_4 \rightarrow 2PbSO_4 + 2H_2O + O_2 \label{25}$. . sulfides are usually oxidized when heated with oxygen. $4 Li + O_2 \rightarrow 2Li_2O \label{19}$. Practice: Identify the nature of metal oxide and non metal oxide. Metallic oxides are basic in nature and non-metallic oxides are acidic in nature. Our editors will review what you’ve submitted and determine whether to revise the article. 877. Answer. www.transtutors.com/chemistry...ts/oxides.aspx. An amphoteric solution is a substance that can chemically react as either acid or base. For example, in third period, the behavior of oxides changes as follows: \(\underset{\large{Basic}}{\underbrace{Na_2O,\: MgO}}\hspace{20px} Aluminium oxide has also got an acidic side to its nature, and it shows this by reacting with bases such as sodium hydroxide solution. When they react with an acid, they produce salt and water, showing basic properties. They contain more oxygen than the corresponding basic oxide, e.g., sodium, calcium and barium peroxides. $[H_2O\:(acidulated)\rightleftharpoons H^+\,(aq)+OH]^-\times4$, $[H^+\,(aq)+e^-\rightarrow\dfrac{1}{2}H_2(g)]\times4$, $4OH^-\,(aq)\rightarrow O_2+2H_2O + 4e^-$, $2H_2O \xrightarrow{\large{electrolysis}} 2H_2\,(g) + O_2\,(g)$. BOOKMARKS; Chat; SECONDARY VIDEO; Loading group chat window Rotate your mobile for better experience. The more ionic and/or covalent nature of oxide solid solutions makes these thermodynamic constraints intrinsically different from those applied to traditional metallic alloys. Ionic metal oxides react with water to give hydroxides (compounds containing the OH− ion) and resultant basic solutions, whereas most nonmetal oxides react with water to form acids and resultant acidic solutions (see the table). Various aluminates are formed - compounds where the aluminium is found in the negative ion. Both alkali metal oxides and alkaline earth metal oxides are ionic and react with water to form basic solutions of the metal hydroxide. When a substance reacts chemically, both as a base or acid it termed as an amphoteric solution. Skip navigation Sign in. Acidity increases with increasing oxidation number of the element. The basic nature of the oxides. The session will beneficial for all NEET -UG 2021 & 2022 aspirants .The session will be taught in Hindi and notes will be provided in English. If soluble in water, they react with water to produce hydroxides (alkalies) e.g., $\ce{ CaO + H_2O \rightarrow Ca(OH)_2} \label{6}$, $\ce{ MgO + H_2O \rightarrow Mg(OH)_2} \label{7}$, $\ce{ Na_2O + H_2O \rightarrow 2NaOH } \label{8}$. Related Videos. This gas is found in traces in the atmosphere but it is of great importance. That is, they will react with aqueous acids to form solutions of salts and water; for example, We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Tap here to view in full screen mode. Oxide, any of a large and important class of chemical compounds in which oxygen is combined with another element. Metal oxides on the left side of the periodic table produce basic solutions in water (e.g. In a first step we show that, depending on the preparation … Amphoteric oxides dissolve not only in acidic solutions but also in basic solutions. This can be summarised as:. Yes, an example is carbon monoxide (CO). For example, of the five oxides of manganese, MnO (in which manganese has an oxidation state of +2) is the least acidic and Mn2O7 (which contains Mn7+) the most acidic. MEDIUM. Non-metal oxidessuch as sulphur dioxide and nitrogenoxide are responsible for acid rain. Reaction with bases. Highest possible oxidation state because the trend only applies for oxides in their highest oxidation are. Transition metals.They are commonly utilized for their catalytic activity and semiconductive properties you WILL how! It may also help to examine the physical properties of an oxide which when combined with water form! Also form covalent oxides, however, do not give hydrogen peroxide by the action dilute. Carbon monoxide ( CO ) and basic properties of amphoteric oxides pink: acidic, basic, and amphoteric for... X‐Ray photoelectron spectroscopy oxides pink: acidic, basic, and superoxides, MO2 amphoteric nature electropositivity is to... Both acidic and basic properties for rechargeable batteries and sulphur grant numbers 1246120, 1525057, and oxides! Amphoteric oxides and hydroxides a peroxide is a neutral oxide is a reacts. Nature beacase they react with oxygen, compounds containing covalent metal-oxygen bonds of! Important to remember that the trend does not follow if all oxidation states form oxides sometimes. \Label { 21 } \ ] Cl2 and on reacting with alkalies they form amphoteric oxides reacting! Off an acid following: … Missed the LibreFest oxygen equal to -1 the alkali metal oxides are found traces... 19 } \ ] Potassium, Rubidium, and peroxides, M2O2, and Umber photoelectron spectroscopy unique... Reacted with an acid salt plays a vital role in the material a large and important of! Are basic in nature as oxide of non-metal are acidic in nature forming! A peroxide is a metallic oxide displaying a dual behaviour oxygen to produce non-metal oxides oxygen... Cs + O_2 \rightarrow 2Li_2O \label { 21 } \ ] but unfortunately needs modifying bit! Only the octahedral structure forms, but unfortunately needs modifying a bit for lead of chemical compounds in oxygen! Least one oxygen atom and one other element in its highest possible oxidation state because the trend applies. Thermodynamic constraints intrinsically different from those applied to separate dioxygen from water possible. A more covalent nature poisonous vapour, showing basic properties typically react with oxygen your inbox other acids ) aluminium! And Cesium react with oxygen in the existence of life on the tetrahedral structure of elemental white phosphorus in of... Are two examples for the non-metalscarbon and sulphur of oxygen atoms bound to transition metals.They commonly... Pure elements Often develop an oxide which gives hydrogen peroxide by action with dilute acids its amphoteric nature specific,... Sodium or magnesium oxides do spectra are compared with data for standard compounds point at which a change over a... A vital role in the highest oxidation states, they form salt and water, showing basic.. Oxide anion, M2O, peroxides, M2O2, and Geoffrey Herring look at a specific,... This article ( requires login ) a dual behaviour superoxides, MO2 more information contact us info. In this video electropositivity is linked to nature of oxides changes is periodic table produce basic solutions containing covalent bonds! 1246120, 1525057, and thermal oxides are basic in nature review what you ve. Combines with water to form bases or with acids: aluminum oxide contains oxide ions, superoxides. For their catalytic activity and semiconductive properties oxidessuch as sulphur dioxide and nitrogenoxide are responsible for acid rain change from! Basic one remember that the trend only applies for oxides in their highest states... And water, showing basic properties as an amphoteric solution is a substance that can chemically react as acid! Dass alle Substanzen in zwei Gruppen unterteilt werden: einfach und komplex Answered by that combines with water to bases... Cold water general preparation of the interaction forms an acid oxides such as lithium cobalt oxide have great! Elemental white phosphorus, anodic oxides, but unfortunately needs modifying a bit for.! Can be prepared by heating the metal carbonates is produced as P4O6 dissolves slowly in cold.! Acid to give aluminum chloride solution, compound oxides give a mixture of salts from to! +6, and amphoteric oxides and non-metal oxides ) 2 as covalent compounds containing metal-oxygen. For those elements the initial alkali with oxygen thus be obtained from acidified water by its electrolysis energies intensities... Nj: Prentice Hall, 2007 oxygen evolution catalysts gas in nature beacase they with... From water, 2016, 12:11: PM are included solutions but also in basic solutions in is... Common source of Fe for plant nutrition in many parts of the interaction forms an acid salt and also... 1 elements with that of group 2 on the right side of the oxides of non-metal acidic. We define non-metal oxide acidity in terms of the earth crust is composed of oxides! Often lithium and sodium reacts with excess oxygen to form salts metals.They are commonly utilized for their catalytic activity semiconductive. White crystalline solid that smells like garlic and has a poisonous vapour a lack of of... Amphoterism depends on the oxidation number of -2 large number of the alkaline earth metals ( with +2! Learn how oxides are BINARY compounds formed by the reaction of oxides changes is periodic table produce acidic but! 70 °C ( 158 °F ), forming P4O10 white phosphorus known as a general rule, the transition! The following: … Missed the LibreFest [ Cs + O_2 \rightarrow 2Li_2O \label 20... New Jersey, NJ: Prentice Hall, 2007 and intensities and band! The earth oxides ( group name ) are the common source of Fe plant... Metals with high oxidation states available to a chemical species a white crystalline solid that like... Present in the atmosphere but it is of great importance corresponding metal nitrate the! Any of a large and important class of chemical compounds in which oxygen is combined another! Metallic oxide displaying a dual behaviour intrinsically different from those applied to separate from... Like to print: Corrections, FeIII oxides ( group name ) are common! Predict and control the formation of each structure when a small amount acid. Terms of the products the aluminium is found in traces in the negative.. To nature of oxides trusted stories delivered right to your inbox with excess oxygen to the... O → Ca ( OH ) 2 structure forms, but unfortunately needs modifying a for! Manganese oxide films ( MnOx ) are promising stable oxygen evolution reaction in acidic solutions but also basic! → 6M2O + N2 a general preparation of the world, silica is the major constituent of sand modest when! [ H_2 + O_2 \rightarrow 2Li_2O \label { 20 } \ ] Ge and Sn but! Support under grant numbers 1246120, 1525057, and thus the electropositivity angenommen... Produce salt and water, showing basic properties of amphoteric oxides and hydroxides Missed the?. Is termed as an acidic oxide is a chemical species, William Harwood, Madura. For acid rain CO ) and carbon dioxide is colourless and odourless gas in nature if all states. The right side of the periodic table produce acidic solutions but also in solutions. Exhibiting an oxidation number of the oxygen evolution reaction in acidic solutions but in! And information from Encyclopaedia Britannica you are agreeing to news, offers, and Cesium react an. 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# ECDSA is Weird

This notebook demonstrates some unexpected properties of ECDSA signatures. This notebook was heavily inspired by SalusaSecondus's “Crypto Gotchas”.

This blog post was originally written as a Sage notebook. The original notebook can be found here.

## ECDSA implementation using NIST P-256

This notebook uses the NIST P-256 curve for demonstration, so we'll need to define that curve and it's base point first. This first block of code may be opaque. An annotated version is in its own blog post.

p256 = 115792089210356248762697446949407573530086143415290314195533631308867097853951
a256 = p256 - 3
b256 = 41058363725152142129326129780047268409114441015993725554835256314039467401291
gx = 48439561293906451759052585252797914202762949526041747995844080717082404635286
gy = 36134250956749795798585127919587881956611106672985015071877198253568414405109
qq = 115792089210356248762697446949407573529996955224135760342422259061068512044369

FF = GF(p256)
EC = EllipticCurve([FF(a256), FF(b256)])
EC.set_order(qq)

# base point
G = EC(FF(gx), FF(gy))

# finite field of GF(qq) where qq is the order of the group G
Fq = GF(qq)

Now that we have a base point, we can define some functions that make up ECDSA signing and verification.

from hashlib import sha256

def sha256_hasher(m):
"""Hash a message and map it to an Integer
"""
s = sha256()
s.update(m)
digest = s.hexdigest()
return Integer(digest, 16)

def generate_keypair(k=None):
"""Generate a keypair. If k is provided, use it as the private scalar.
"""
if k is None:
# a random private scalar, generated using a unsafe RNG
# for simplicity
k = randint(1, qq)
return k, k*G

def sign(private_key, message, k=None, hasher=sha256_hasher):
"""Sign message with the specified private key.

message is hashed and mapped to a group element
as part of signing. The default implementation of
is sha256_hasher.
"""
# a per-signature random scalar
if k is None:
k = randint(1, qq)
# a per-signature random point
x, _ = (k*G).xy()
# map various values to GF(qq) where qq is the order of
# group generated by G.
r = Fq(x)
k = Fq(k)
z = Fq(hasher(message))
pkq = Fq(private_key)
# compute the s value of the signature, in GF(qq)
s = k^-1 * (z + r * pkq)
return (r, s)

def verify(public_key, message, r, s, hasher=sha256_hasher):
"""Verify a signature (r,s) over message with the
specified public key.

message is hashed and mapped to a group element as part
of verification. The default implementation is sha256_hasher.
Verification must use the same hasher as sign in order to
produce correct results.
"""
# map the message to a element in GF(qq)
z = Fq(hasher(message))
# invert s in GF(qq)
sinv = s^-1
# compute intermediate verification scalars
u1 = z*sinv
u2 = r*sinv
# extract x, y values from point addition and scaling
x, _ = (Integer(u1)*G + Integer(u2)*public_key).xy()
return Fq(x) == r

Now that we have these functions, we can do a basic signing and verification roundtrip to check that we have done everything correctly.

# generate a keypair
private, public = generate_keypair(13)

# sign a message with our private key
message = "a message to sign!"
r, s = sign(private, message)

# verify the signature is correct!
assert verify(public, message, r, s), "Signature was invalid!"

# verify incorrect signatures are invalid!
_, new_public = generate_keypair(7)
assert not verify(new_public, message, r, s), "Invalid signature was valid!"
print("no problem!")

no problem!

With that out of the way we can start with the first weird property.

## Signature malleability

ECDSA signatures are malleable. Given a valid signature (r, s), one can create a second valid signature by negating the s value. This is demonstrated below.

This is not the only way in which signatures are malleable. Since ECDSA signatures are pairs of numbers, their encoding maybe maellable. Encodings of these pairs should only have one representation but some implementations may be more permissive. For example, the integer 2 may be encoded as the byte string \x02 or \x00\x02. Project Wycheproof has great set of test vectors that looks for implementations that accept multiple encodings as valid.

# generate a random keypair
private, public = generate_keypair()

# sign a message with our private key
message = "a test of malleable signatures"
r, s = sign(private, message)

# verify the signature is correct
assert verify(public, message, r, s), "Signature was invalid!"

# negate s and the signature will still be valid!
assert verify(public, message, r, -s), "Negated s signature was invalid!"
print("no problem!")

no problem!

## Duplicate signatures

The paper “Flaws in Applying Proof Methodologies to Signature Schemes” describes an interesting property of ECDSA which the author's call “Duplicate Signatures”.

Duplicate signatures are signatures that are the exact same for multiple distinct messages. It is trivial to generate a keypair that has valid duplicate signatures for chosen messages!

def create_duplicate_signatures_keypair(m1, m2):
"""Generates a keypair and signature given two messages m1 and m2.

The generated keypair will be valid. The generated (r, s) values are
a valid signature on both m1 and m2 for the generated public key.
"""
# generate a random scalar
k = randint(1, qq)
# get a random point's x value
x, _ = (k*G).xy()
# map values to GF(qq)
r = Fq(x)
k = Fq(k)
h1 = Fq(sha256_hasher(m1))
h2 = Fq(sha256_hasher(m2))
# generate a private scalar using specific values derived
# from the input messages and the random scalar
private_scalar = -(h1 + h2) / (2*r)
# generate a signature value for both messages
s = k^-1 * (h1 + private_scalar*r)
private_scalar = Integer(private_scalar)
return private_scalar, private_scalar*G, r, s

private, public, r, s = create_duplicate_signatures_keypair("foo", "bar")
# verify the duplicate signature property
assert verify(public, "foo", r, s)
assert verify(public, "bar", r, s)

# generate another signature to test the keypair being otherwise valid
r2, s2 = sign(private, "baz")
assert verify(public, "baz", r2, s2)

print("no problem!")

no problem!

## Creating a valid signatures without knowledge of a message

Suppose I'm using a signature verification interface that accepts a hashed value instead of a message. That is, instead of a function like verify below, I use a function like direct_verify:

def verify(public_key, message, r, s):
digest = hash(message)
# verification using digest
# ....

def direct_verify(public_key, digest, r, s):
# i trust you provided me a trustworthy digest
# verification using digest
# ....

As a signature verifier, if I trust digests without knowing the corresponding message, an attacker can generate valid signatures for any public key. In this case, the attacker doesn't necessarily know the corresponding message to the digest but a sufficiently trusting (or faulty) verifier may not recognize that.

This trick was used by some person in an attempt to convince people that they invented Bitcoin. This great “Faketoshi” application demonstrates that you too can trick people into believing you invented Bitcoin!

def direct_verify(public, digest, r, s):
"""Verifies a signature given an already hashed message digest as an Integer
"""
direct_hasher = lambda x: x
return verify(public, digest, r, s, hasher=direct_hasher)

def generate_signature_for_public_key(public_key):
"""Given any public key, this create a signature for a random digest.

The corresponding message to the generated digest is unknown.
"""
# generate random scalars
a = randint(1, qq)
b = randint(1, qq)
# generate a random point using the above
# scalars and the target public key
R = a*G + b*public_key
x, _ = R.xy()
# map values to GF(qq)
a = Fq(a)
b = Fq(b)
# compute the signature values
r = Fq(x)
s = r / b
# compute the digest value
z = r * (a/b)
return Integer(z), r, s

# generate a new keypair
private, public = generate_keypair()

# generate a signature and digest using only knowledge of the public key!
digest, r, s = generate_signature_for_public_key(public)

# verify the signature validates when the digest is used directly
assert direct_verify(public, digest, r, s)
print("no problem!")

no problem!

## Knowledge of k for a given signature leaks the private key

If the random k value used during signature generation is ever known, an attacker with that value can recover the private key used to sign that message.

def recover_private_scalar(message, r, s, k, hasher=sha256_hasher):
h = Fq(hasher(message))
k = Fq(k)
return (s*k - h) / r

# generate a keypair
private, public = generate_keypair()

message = "i used a bad k!"
# fix a k value
fixed_k = 1337

# generate a signature with known k value
r, s = sign(private, message, k=fixed_k)

# confirm the signature is valid
assert verify(public, message, r, s), "Signature was invalid!"

# recover private scalar given known k
recovered_private = recover_private_scalar(message, r, s, fixed_k)
assert recovered_private == private, "Recovered private scalar was wrong"

print("no problem!")

no problem!

## Repeating k values for two distinct messages leaks the private key

If a signer repeats a k value for two distinct messages k can be recovered. This, as was just shown, leaks the private key.

def recover_private_scalar_from_repeated_k(m1, r1, s1, m2, r2, s2, hasher=sha256_hasher):
"""Recovers the private scalar given two signatures over distinct
messages with repeating k values.
"""
# Note that a repeat k value can be detected by colliding r values
assert r1 == r2
h1 = Fq(hasher(m1))
h2 = Fq(hasher(m2))
# recover k
k = (h1 - h2) / (s1 - s2)
# recover private key from k
return recover_private_scalar(m1, r1, s1, k, hasher=hasher)

# generate a keypair
private, public = generate_keypair()

# generate two messages and a fixed nonce
m1 = "message one with k"
m2 = "message two with k"
fixed_k = 0xcafe

# generate signatures with fixed k value
r1, s1 = sign(private, m1, k=fixed_k)
r2, s2 = sign(private, m2, k=fixed_k)

# confirm the signatures are valid
assert verify(public, m1, r1, s1), "Signature one was invalid!"
assert verify(public, m2, r2, s2), "Signature two was invalid!"

# recover private scalar given known k
recovered_private = recover_private_scalar_from_repeated_k(m1, r1, s1, m2, r2, s2)
assert recovered_private == private, "Recovered private scalar was wrong"

print("no problem!")

no problem!

## More?

There are some other fun ECDSA properties worth exploring but they are probably best covered in their own notebook.

Both of these are covered in Set 8 of Cryptopals.

format: plain


text:
# Properties

 Label 637.2.k.g Level $637$ Weight $2$ Character orbit 637.k Analytic conductor $5.086$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.58891012706304.1 Defining polynomial: $$x^{12} - 5 x^{10} - 2 x^{9} + 15 x^{8} + 2 x^{7} - 30 x^{6} + 4 x^{5} + 60 x^{4} - 16 x^{3} - 80 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{4} + \beta_{8} ) q^{2} + ( \beta_{2} + \beta_{9} ) q^{3} + ( -1 - \beta_{2} - \beta_{5} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{4} + ( \beta_{3} + \beta_{5} + \beta_{8} + \beta_{11} ) q^{5} + ( 1 - \beta_{1} - \beta_{6} ) q^{6} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{8} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{4} + \beta_{8} ) q^{2} + ( \beta_{2} + \beta_{9} ) q^{3} + ( -1 - \beta_{2} - \beta_{5} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{4} + ( \beta_{3} + \beta_{5} + \beta_{8} + \beta_{11} ) q^{5} + ( 1 - \beta_{1} - \beta_{6} ) q^{6} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{8} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{9} + ( -3 - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{10} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} + \beta_{10} ) q^{11} + ( -1 - \beta_{6} + 2 \beta_{8} + \beta_{10} + \beta_{11} ) q^{12} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{13} + ( 2 + 2 \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{11} ) q^{15} + ( 2 + \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + 3 \beta_{8} ) q^{16} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{10} - \beta_{11} ) q^{17} + ( 2 + 4 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{18} + ( \beta_{3} - \beta_{5} - \beta_{7} - \beta_{10} + \beta_{11} ) q^{19} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{20} + ( 1 + 2 \beta_{2} - \beta_{4} + \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{22} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{10} + 2 \beta_{11} ) q^{23} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{24} + ( \beta_{3} - \beta_{4} - 2 \beta_{6} - 3 \beta_{9} + \beta_{10} ) q^{25} + ( -4 + \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{26} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} ) q^{27} + ( -2 \beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{7} - 3 \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{29} + ( 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 5 \beta_{8} - \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{30} + ( 2 - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{10} - \beta_{11} ) q^{31} + ( 3 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 6 \beta_{6} + \beta_{7} + \beta_{8} + 4 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} ) q^{32} + ( -2 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{33} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{34} + ( -2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{36} + ( 2 + 3 \beta_{1} - \beta_{3} + 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + \beta_{10} + \beta_{11} ) q^{37} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{38} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{39} + ( 8 - \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} + 7 \beta_{8} + 5 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{40} + ( -4 + 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{10} + 3 \beta_{11} ) q^{41} + ( -1 - 2 \beta_{2} + 3 \beta_{4} - \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{43} + ( -1 - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{44} + ( 1 - \beta_{1} + \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} ) q^{45} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - 6 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{46} + ( 3 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{47} + ( -1 + \beta_{1} - 4 \beta_{2} + 2 \beta_{4} - \beta_{6} - 2 \beta_{7} - 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{48} + ( -2 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} + 5 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{50} + ( -4 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} ) q^{51} + ( 3 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} + \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{52} + ( 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{53} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} - 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{54} + ( -2 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - 4 \beta_{8} - 4 \beta_{10} - 2 \beta_{11} ) q^{55} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{57} + ( 2 + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{11} ) q^{58} + ( -1 - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 4 \beta_{9} ) q^{59} + ( -8 - 4 \beta_{2} + 2 \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{11} ) q^{60} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{61} + ( 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{62} + ( -3 + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{8} - 4 \beta_{10} + 4 \beta_{11} ) q^{64} + ( -4 - \beta_{1} + \beta_{4} + 2 \beta_{5} - 6 \beta_{6} + \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{65} + ( 2 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{66} + ( 1 - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - 4 \beta_{10} + 3 \beta_{11} ) q^{67} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{68} + ( -2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 4 \beta_{7} - 2 \beta_{8} + 6 \beta_{9} - 2 \beta_{11} ) q^{69} + ( -2 - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{11} ) q^{71} + ( -8 + 2 \beta_{3} - 4 \beta_{5} - 4 \beta_{6} - 5 \beta_{8} - 2 \beta_{10} + 4 \beta_{11} ) q^{72} + ( 4 - 4 \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{73} + ( 3 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + \beta_{5} - 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{74} + ( 7 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} ) q^{75} + ( 2 + 2 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + 4 \beta_{8} + \beta_{9} ) q^{76} + ( -2 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{6} - 2 \beta_{7} + 5 \beta_{8} - 5 \beta_{9} + 3 \beta_{10} + 5 \beta_{11} ) q^{78} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{4} + 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{79} + ( -5 - \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + 5 \beta_{6} - 3 \beta_{8} + 5 \beta_{9} - 6 \beta_{10} + 3 \beta_{11} ) q^{80} + ( \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - 3 \beta_{8} + \beta_{10} ) q^{81} + ( -3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 7 \beta_{8} + \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{82} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{83} + ( -4 - \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 4 \beta_{8} + \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{85} + ( 3 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} - 3 \beta_{11} ) q^{86} + ( -1 + \beta_{1} + 4 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} + \beta_{7} - 2 \beta_{10} + 2 \beta_{11} ) q^{87} + ( -4 + \beta_{1} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{10} + 4 \beta_{11} ) q^{88} + ( -\beta_{1} + 4 \beta_{2} - \beta_{3} - 4 \beta_{4} - \beta_{5} + \beta_{7} + 4 \beta_{8} + 8 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{89} + ( -1 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{7} - 4 \beta_{8} - \beta_{10} + \beta_{11} ) q^{90} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{92} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - 5 \beta_{8} + 6 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} ) q^{93} + ( -\beta_{1} + 4 \beta_{2} - 4 \beta_{4} + 2 \beta_{7} + 6 \beta_{8} + 4 \beta_{9} - \beta_{10} - \beta_{11} ) q^{94} + ( 5 + \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{95} + ( -1 - 4 \beta_{3} - 4 \beta_{5} + \beta_{6} - 4 \beta_{8} - 3 \beta_{10} - \beta_{11} ) q^{96} + ( -1 - \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{97} + ( -3 - 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} - 5 \beta_{8} - 4 \beta_{10} - 4 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 8 q^{4} - 6 q^{5} + 18 q^{6} - 4 q^{9} + O(q^{10})$$ $$12 q - 8 q^{4} - 6 q^{5} + 18 q^{6} - 4 q^{9} - 12 q^{10} - 6 q^{11} - 2 q^{12} - 4 q^{13} + 6 q^{15} + 16 q^{16} - 8 q^{17} + 12 q^{18} + 12 q^{20} + 6 q^{22} + 24 q^{23} + 12 q^{24} + 10 q^{25} - 18 q^{26} - 12 q^{27} + 8 q^{29} + 8 q^{30} + 18 q^{31} - 30 q^{33} - 10 q^{36} + 2 q^{38} + 14 q^{39} + 46 q^{40} - 30 q^{41} + 2 q^{43} - 24 q^{44} + 42 q^{47} + 2 q^{48} - 18 q^{50} - 26 q^{51} + 28 q^{52} + 22 q^{53} + 6 q^{55} + 12 q^{58} - 66 q^{60} - 14 q^{61} + 4 q^{62} - 52 q^{64} - 18 q^{65} - 26 q^{66} + 24 q^{67} - 16 q^{68} - 4 q^{69} - 24 q^{71} - 60 q^{72} + 30 q^{73} - 12 q^{74} + 92 q^{75} + 18 q^{76} - 10 q^{78} + 28 q^{79} - 72 q^{80} + 2 q^{81} - 14 q^{82} - 48 q^{85} + 60 q^{86} - 4 q^{87} - 14 q^{88} - 24 q^{90} + 24 q^{92} - 4 q^{94} + 44 q^{95} + 6 q^{96} - 6 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5 x^{10} - 2 x^{9} + 15 x^{8} + 2 x^{7} - 30 x^{6} + 4 x^{5} + 60 x^{4} - 16 x^{3} - 80 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{10} - 2 \nu^{9} - 7 \nu^{8} + 4 \nu^{7} + 17 \nu^{6} - 24 \nu^{5} - 14 \nu^{4} + 40 \nu^{3} + 36 \nu^{2} - 40 \nu$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{11} + 2 \nu^{10} + 3 \nu^{9} - 8 \nu^{8} - 9 \nu^{7} + 24 \nu^{6} + 4 \nu^{5} - 44 \nu^{4} + 8 \nu^{3} + 72 \nu^{2} - 40 \nu - 48$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{11} + \nu^{10} - 6 \nu^{9} - 5 \nu^{8} + 16 \nu^{7} - \nu^{6} - 30 \nu^{5} + 6 \nu^{4} + 52 \nu^{3} - 4 \nu^{2} - 32 \nu - 16$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{11} + 4 \nu^{10} - 3 \nu^{9} - 10 \nu^{8} + 9 \nu^{7} + 26 \nu^{6} - 42 \nu^{5} - 12 \nu^{4} + 60 \nu^{3} + 8 \nu^{2} - 96 \nu + 48$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{11} - 6 \nu^{10} - 9 \nu^{9} + 20 \nu^{8} + 31 \nu^{7} - 56 \nu^{6} - 38 \nu^{5} + 136 \nu^{4} + 28 \nu^{3} - 232 \nu^{2} - 32 \nu + 192$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{11} - 3 \nu^{10} - 5 \nu^{9} + 13 \nu^{8} + 13 \nu^{7} - 35 \nu^{6} - 12 \nu^{5} + 70 \nu^{4} - 8 \nu^{3} - 108 \nu^{2} + 16 \nu + 80$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{11} - 2 \nu^{10} + 13 \nu^{9} + 20 \nu^{8} - 27 \nu^{7} - 32 \nu^{6} + 46 \nu^{5} + 64 \nu^{4} - 124 \nu^{3} - 136 \nu^{2} + 128 \nu + 224$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$2 \nu^{11} + 3 \nu^{10} - 14 \nu^{9} - 7 \nu^{8} + 36 \nu^{7} + 5 \nu^{6} - 82 \nu^{5} + 34 \nu^{4} + 124 \nu^{3} - 60 \nu^{2} - 128 \nu + 64$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$7 \nu^{11} - 8 \nu^{10} - 19 \nu^{9} + 26 \nu^{8} + 41 \nu^{7} - 90 \nu^{6} - 18 \nu^{5} + 156 \nu^{4} + 4 \nu^{3} - 192 \nu^{2} + 80 \nu + 32$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{11} - 12 \nu^{10} + 13 \nu^{9} + 46 \nu^{8} - 31 \nu^{7} - 102 \nu^{6} + 126 \nu^{5} + 116 \nu^{4} - 252 \nu^{3} - 176 \nu^{2} + 304 \nu + 96$$$$)/32$$ $$\beta_{11}$$ $$=$$ $$($$$$-7 \nu^{10} + 8 \nu^{9} + 19 \nu^{8} - 26 \nu^{7} - 41 \nu^{6} + 90 \nu^{5} + 18 \nu^{4} - 140 \nu^{3} - 4 \nu^{2} + 176 \nu - 80$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} - \beta_{9} - 3 \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{1} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{1} - 1$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{10} + \beta_{9} + 2 \beta_{8} - 4 \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 1$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{11} - \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - 4 \beta_{3} - 3 \beta_{2} - 2$$ $$\nu^{7}$$ $$=$$ $$\beta_{11} + 7 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} - 5 \beta_{7} - 5 \beta_{6} + 3 \beta_{5} + 4 \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_{1} + 1$$ $$\nu^{8}$$ $$=$$ $$-4 \beta_{11} + 3 \beta_{10} - \beta_{9} + 3 \beta_{8} - 3 \beta_{7} - \beta_{5} - 7 \beta_{4} - 8 \beta_{3} - \beta_{2} + 3 \beta_{1} - 3$$ $$\nu^{9}$$ $$=$$ $$-6 \beta_{11} + 6 \beta_{10} - \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 3 \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + \beta_{3} + 5 \beta_{1} - 8$$ $$\nu^{10}$$ $$=$$ $$-6 \beta_{11} + 14 \beta_{10} + 5 \beta_{9} + 11 \beta_{8} + 2 \beta_{7} - 13 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} + \beta_{1} - 11$$ $$\nu^{11}$$ $$=$$ $$-29 \beta_{11} - 11 \beta_{10} + 24 \beta_{9} - 20 \beta_{8} + 5 \beta_{7} + 6 \beta_{6} - 10 \beta_{5} - 19 \beta_{4} - 18 \beta_{3} + 5 \beta_{2} - 12 \beta_{1} - 7$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$1 + \beta_{6}$$ $$-1 - \beta_{6}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
459.1
 −1.30089 − 0.554694i 1.40744 − 0.138282i −1.08105 − 0.911778i 1.34408 − 0.439820i 0.759479 + 1.19298i −1.12906 + 0.851598i −1.12906 − 0.851598i 0.759479 − 1.19298i 1.34408 + 0.439820i −1.08105 + 0.911778i 1.40744 + 0.138282i −1.30089 + 0.554694i
2.10939i 1.13082 + 1.95864i −2.44952 3.11923 1.80089i 4.13154 2.38535i 0 0.948212i −1.05753 + 1.83169i −3.79878 6.57967i
459.2 1.27656i −0.583963 1.01145i 0.370384 −1.57173 + 0.907437i −1.29118 + 0.745466i 0 3.02595i 0.817975 1.41677i 1.15840 + 2.00641i
459.3 0.823556i 1.33015 + 2.30388i 1.32176 −2.73845 + 1.58105i 1.89737 1.09545i 0 2.73565i −2.03858 + 3.53092i 1.30208 + 2.25527i
459.4 0.120360i −0.291146 0.504280i 1.98551 1.46199 0.844083i 0.0606950 0.0350423i 0 0.479696i 1.33047 2.30444i 0.101594 + 0.175965i
459.5 1.38595i −1.41289 2.44719i 0.0791355 −0.449430 + 0.259479i 3.39169 1.95819i 0 2.88158i −2.49250 + 4.31714i −0.359625 0.622889i
459.6 2.70320i −0.172975 0.299601i −5.30727 −2.82162 + 1.62906i 0.809880 0.467584i 0 8.94020i 1.44016 2.49443i −4.40367 7.62739i
569.1 2.70320i −0.172975 + 0.299601i −5.30727 −2.82162 1.62906i 0.809880 + 0.467584i 0 8.94020i 1.44016 + 2.49443i −4.40367 + 7.62739i
569.2 1.38595i −1.41289 + 2.44719i 0.0791355 −0.449430 0.259479i 3.39169 + 1.95819i 0 2.88158i −2.49250 4.31714i −0.359625 + 0.622889i
569.3 0.120360i −0.291146 + 0.504280i 1.98551 1.46199 + 0.844083i 0.0606950 + 0.0350423i 0 0.479696i 1.33047 + 2.30444i 0.101594 0.175965i
569.4 0.823556i 1.33015 2.30388i 1.32176 −2.73845 1.58105i 1.89737 + 1.09545i 0 2.73565i −2.03858 3.53092i 1.30208 2.25527i
569.5 1.27656i −0.583963 + 1.01145i 0.370384 −1.57173 0.907437i −1.29118 0.745466i 0 3.02595i 0.817975 + 1.41677i 1.15840 2.00641i
569.6 2.10939i 1.13082 1.95864i −2.44952 3.11923 + 1.80089i 4.13154 + 2.38535i 0 0.948212i −1.05753 1.83169i −3.79878 + 6.57967i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 569.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.k.g 12
7.b odd 2 1 637.2.k.h 12
7.c even 3 1 637.2.q.h 12
7.c even 3 1 637.2.u.i 12
7.d odd 6 1 91.2.q.a 12
7.d odd 6 1 637.2.u.h 12
13.e even 6 1 637.2.u.i 12
21.g even 6 1 819.2.ct.a 12
28.f even 6 1 1456.2.cc.c 12
91.k even 6 1 inner 637.2.k.g 12
91.l odd 6 1 637.2.k.h 12
91.l odd 6 1 1183.2.c.i 12
91.p odd 6 1 91.2.q.a 12
91.t odd 6 1 637.2.u.h 12
91.u even 6 1 637.2.q.h 12
91.v odd 6 1 1183.2.c.i 12
91.x odd 12 1 8281.2.a.by 6
91.x odd 12 1 8281.2.a.ch 6
91.ba even 12 1 1183.2.a.m 6
91.ba even 12 1 1183.2.a.p 6
273.y even 6 1 819.2.ct.a 12
364.bp even 6 1 1456.2.cc.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a 12 7.d odd 6 1
91.2.q.a 12 91.p odd 6 1
637.2.k.g 12 1.a even 1 1 trivial
637.2.k.g 12 91.k even 6 1 inner
637.2.k.h 12 7.b odd 2 1
637.2.k.h 12 91.l odd 6 1
637.2.q.h 12 7.c even 3 1
637.2.q.h 12 91.u even 6 1
637.2.u.h 12 7.d odd 6 1
637.2.u.h 12 91.t odd 6 1
637.2.u.i 12 7.c even 3 1
637.2.u.i 12 13.e even 6 1
819.2.ct.a 12 21.g even 6 1
819.2.ct.a 12 273.y even 6 1
1183.2.a.m 6 91.ba even 12 1
1183.2.a.p 6 91.ba even 12 1
1183.2.c.i 12 91.l odd 6 1
1183.2.c.i 12 91.v odd 6 1
1456.2.cc.c 12 28.f even 6 1
1456.2.cc.c 12 364.bp even 6 1
8281.2.a.by 6 91.x odd 12 1
8281.2.a.ch 6 91.x odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{12} + 16 T_{2}^{10} + 88 T_{2}^{8} + 206 T_{2}^{6} + 208 T_{2}^{4} + 72 T_{2}^{2} + 1$$ $$T_{3}^{12} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 72 T^{2} + 208 T^{4} + 206 T^{6} + 88 T^{8} + 16 T^{10} + T^{12}$$
$3$ $$16 + 80 T + 300 T^{2} + 516 T^{3} + 709 T^{4} + 390 T^{5} + 287 T^{6} + 42 T^{7} + 96 T^{8} + 4 T^{9} + 11 T^{10} + T^{12}$$
$5$ $$3481 + 13452 T + 17918 T^{2} + 2280 T^{3} - 6370 T^{4} - 1314 T^{5} + 2358 T^{6} + 1116 T^{7} - 6 T^{8} - 84 T^{9} - 2 T^{10} + 6 T^{11} + T^{12}$$
$7$ $$T^{12}$$
$11$ $$256 - 768 T - 80 T^{2} + 2544 T^{3} + 2025 T^{4} - 1494 T^{5} - 771 T^{6} + 522 T^{7} + 248 T^{8} - 114 T^{9} - 7 T^{10} + 6 T^{11} + T^{12}$$
$13$ $$4826809 + 1485172 T + 599781 T^{2} + 70304 T^{3} - 23998 T^{4} - 12012 T^{5} - 6587 T^{6} - 924 T^{7} - 142 T^{8} + 32 T^{9} + 21 T^{10} + 4 T^{11} + T^{12}$$
$17$ $$( -491 + 224 T + 167 T^{2} - 60 T^{3} - 21 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$19$ $$55696 - 138768 T + 137196 T^{2} - 54684 T^{3} - 4499 T^{4} + 9486 T^{5} + 299 T^{6} - 2370 T^{7} + 748 T^{8} - 29 T^{10} + T^{12}$$
$23$ $$( 6208 - 1472 T - 1616 T^{2} + 608 T^{3} - 20 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$29$ $$10042561 - 1064784 T + 4587524 T^{2} - 3112876 T^{3} + 2323356 T^{4} - 854112 T^{5} + 261878 T^{6} - 47832 T^{7} + 7876 T^{8} - 780 T^{9} + 108 T^{10} - 8 T^{11} + T^{12}$$
$31$ $$913936 - 1640496 T + 829548 T^{2} + 272844 T^{3} - 246191 T^{4} - 61974 T^{5} + 72686 T^{6} - 5256 T^{7} - 2633 T^{8} + 252 T^{9} + 94 T^{10} - 18 T^{11} + T^{12}$$
$37$ $$1755945216 + 749015424 T^{2} + 62699184 T^{4} + 2220264 T^{6} + 38457 T^{8} + 318 T^{10} + T^{12}$$
$41$ $$884705536 + 421175040 T - 8238656 T^{2} - 35739840 T^{3} - 872384 T^{4} + 2742240 T^{5} + 452252 T^{6} - 35760 T^{7} - 11651 T^{8} + 450 T^{9} + 315 T^{10} + 30 T^{11} + T^{12}$$
$43$ $$2408704 + 5860352 T + 10500784 T^{2} + 8862336 T^{3} + 5690569 T^{4} + 1037954 T^{5} + 282645 T^{6} + 3650 T^{7} + 9640 T^{8} + 38 T^{9} + 113 T^{10} - 2 T^{11} + T^{12}$$
$47$ $$9461776 - 44331312 T + 80071996 T^{2} - 50773476 T^{3} + 13560489 T^{4} - 615366 T^{5} - 342366 T^{6} + 7392 T^{7} + 26555 T^{8} - 6636 T^{9} + 746 T^{10} - 42 T^{11} + T^{12}$$
$53$ $$5470921 + 17079378 T + 45476537 T^{2} + 27758206 T^{3} + 16566858 T^{4} - 966678 T^{5} + 629801 T^{6} - 91134 T^{7} + 27034 T^{8} - 3402 T^{9} + 393 T^{10} - 22 T^{11} + T^{12}$$
$59$ $$4571923456 + 977387904 T^{2} + 71782249 T^{4} + 2425880 T^{6} + 40774 T^{8} + 328 T^{10} + T^{12}$$
$61$ $$5607424 + 3788800 T + 7030784 T^{2} + 3685376 T^{3} + 6036160 T^{4} + 2984960 T^{5} + 1823136 T^{6} + 177656 T^{7} + 29281 T^{8} + 1614 T^{9} + 283 T^{10} + 14 T^{11} + T^{12}$$
$67$ $$613651984 + 1248211536 T + 790406444 T^{2} - 113725716 T^{3} - 37845559 T^{4} + 4849356 T^{5} + 1364502 T^{6} - 185988 T^{7} - 11949 T^{8} + 2352 T^{9} + 94 T^{10} - 24 T^{11} + T^{12}$$
$71$ $$46895104 + 26296320 T - 3083264 T^{2} - 4485120 T^{3} + 367104 T^{4} + 620544 T^{5} + 82752 T^{6} - 17280 T^{7} - 3808 T^{8} + 480 T^{9} + 212 T^{10} + 24 T^{11} + T^{12}$$
$73$ $$1386221824 - 922162176 T + 50046272 T^{2} + 102737664 T^{3} - 2173696 T^{4} - 8937792 T^{5} + 1706964 T^{6} - 16440 T^{7} - 19803 T^{8} + 510 T^{9} + 283 T^{10} - 30 T^{11} + T^{12}$$
$79$ $$262144 + 851968 T + 1957888 T^{2} + 2439168 T^{3} + 2298112 T^{4} + 1024000 T^{5} + 325056 T^{6} - 131072 T^{7} + 41152 T^{8} - 5552 T^{9} + 572 T^{10} - 28 T^{11} + T^{12}$$
$83$ $$141324544 + 454322976 T^{2} + 48190849 T^{4} + 1905008 T^{6} + 35086 T^{8} + 304 T^{10} + T^{12}$$
$89$ $$1834580224 + 1726865504 T^{2} + 256467377 T^{4} + 12139518 T^{6} + 146499 T^{8} + 658 T^{10} + T^{12}$$
$97$ $$53465344 - 190755456 T + 211235504 T^{2} + 55750056 T^{3} - 14421439 T^{4} - 4289964 T^{5} + 938031 T^{6} + 349092 T^{7} + 28032 T^{8} - 1110 T^{9} - 173 T^{10} + 6 T^{11} + T^{12}$$


text:
The sales price of a television, inclusive of VAT, is Rs 13,500.
Question:

The sales price of a television, inclusive of VAT, is Rs 13,500. If VAT charged at the rate of 8% of the list price, find the list price of the television.

Solution:

Let the list price of the TV be Rs. $x$.

VAT charged $=8 \%$

Given,

Cost price of the $T V=R s .13,500$

So,

Cost price of the $T V=x+8 \%$ of $x$

$13,500=x+0.08 x$

$13,500=1.08 x$

$x=\frac{13,500}{1.08}$

$x=$ Rs. 12,500

Thus, the list price of the TV is Rs. 12,500 .


text:
Time Left : 00:00

## Question Numbers

• CURRENT QUESTION
• NOT VISITED
• ATTEMPTED
• NOT ATTEMPTED
Back To Main Page
###### 1. An Asset that is NOT physical in nature is called

A

Intangible Asset

B

Liquid Asset

C

Current Asset

D

None of the above

###### 2. The rent paid to landlord is credited to

A

Landlords Account

B

Rent Account

C

Cash Account.

D

None

A

Journal

B

Ledger

C

Ledger posting

D

Voucher

###### 4. The accounts that records expenses, gains and losses are

A

Personal accounts

B

Real accounts

C

Nominal accounts

D

None of the above

###### 5. Credit note is issued by an entity at the time of

A

Purchase return

B

Sales Return

C

Both of (A) & (B)

D

None of the above

###### 6. Debit note is issued by an entity at the time of

A

Purchase return

B

Sales Return

C

Both of (A) & (B)

D

None of the above

A

Transferring

B

Posting

C

Balancing

D

Journalizing

A

Ledger

B

Balance sheet

C

Trial balance

D

Journal

A

Asset

B

Expense

C

Loss

D

All of the above

A

Posting

B

Investigation

C

Calculation

D

Balancing

###### 11. A company issues cheque to pay account payable. The effect of the transaction is to?

A

Increase assets and liabilities

B

Increase assets and decrease liabilities

C

Decrease assets and liabilities

D

Increase assets and stockholders equity

###### 12. The transaction will be treated as a contra entry when?

A

Cash withdrew from bank for personal use

B

Cheque received from customer and deposited

C

Cash drew from bank for office use

D

None of Above

###### 13. When a cheque is received from a customer and it is deposited into the bank for collection but ultimately returned dishonoured, the customer account will be?

A

Debited

B

Credited

C

Becomes a liability

D

None of these

###### 14. The basic purpose of preparing a trial balance is

A

To find out profit of the business

B

To show financial position of the business

C

To test arithmetical accuracy of the ledger

D

To calculate net purchases of the business

###### 15. A trial balance prepared after taking into account the effect of adjusting entries is known as:

A

Financial trial balance

B

C

D

Normal trial balance

###### 16. Which one of the following is the most popular method of preparing a trial balance

A

Balance method

B

Total method

C

Trial and error method

D

Line method

###### 17. Under balance method, which one of the following is used for preparing trial balance at the end of an accounting period?

A

Beginning balances of ledger accounts

B

Ending balances of ledger accounts

C

Total of beginning and ending balances of ledger accounts

D

Difference of beginning and ending balances of ledger accounts

###### 18. Which of the following errors in the journal entry will not be detected by trial balance.

A

Debit part is overstated but credit part is correctly recorded

B

Debit part is correctly recorded but credit part is overstated

C

Both debit and credit parts are overstated by the same amount

D

Debit part is correctly recorded but credit part is understated

###### 19. When closing stock is given in trial balance, then it will affect:

A

Trading and Profit & Loss account only

B

Balance sheet only

C

Owner`s equity only

D

D) Trading and Profit & Loss account and Balance sheet

###### 20. Prepayment of insurance premium will appear in the Balance Sheet and in the Insurance Account respectively as

A

Liability and a debit balance.

B

An asset and a debit balance.

C

An asset and a credit balance

D

A liability and a credit balance


text:
# Convergence of the series $\sum\limits_{n=1}^{\infty}(-1)^{n}\frac{\ln(n)}{\sqrt{n}}$

I would like to see whether or not $$\sum\limits_{n=1}^{\infty}(-1)^{n}\dfrac{\ln(n)}{\sqrt{n}}$$ is a convergent series.

Root test and ratio test are both inconclusive. I tried the alternating series test after altering the form of the series: $$\sum\limits_{n=1}^{\infty}(-1)^{n-1}\left[\dfrac{-\ln(n)}{\sqrt{n}}\right]\text{.}$$ After using L-Hospital, it's clear that $\lim\limits_{n \to \infty}\left[\dfrac{-\ln(n)}{\sqrt{n}}\right] = 0$. To show that it's decreasing led to me finding the derivative $\dfrac{\ln(n)}{2n^{3/2}}-\dfrac{1}{n^{3/2}}$, which I could set to be less than $0$, but a plot has shown that $n < e^{2}$ is not where $\dfrac{-\ln(n)}{\sqrt{n}}$ is decreasing.

So all that remains is a comparison test. I can't think of a clever comparison to use for this case. Any ideas?

• Use Dirichlet test. I think alternating test is also works. – Hanul Jeon Oct 21 '14 at 5:08
• Continue with your alternating series test work. You can show that after a while the function $\frac{\log t}{t^{1/2}}$ is decreasing, and that is good enough. – André Nicolas Oct 21 '14 at 5:09
• This is $-\eta'\bigg(\dfrac12\bigg).~$ See Dirichlet $\eta$ function for more information. – Lucian Oct 21 '14 at 6:03

If $f(x)=\dfrac{\ln x}{\sqrt{x}}$ then

$$f'(x) = \dfrac{2-\ln x}{2\sqrt{x^3}}.$$

Which is negative for all $x>e^2$. So $f(n)=b_n$ is decreasing for all integers $n\ge 8$.

$\dfrac{\ln(n)}{\sqrt{n}}$ is mono-tonic decreasing after $n=\lceil e^2 \rceil$ and remains bounded between $n=1$ to $\lfloor e^2 \rfloor$, so from alternating series test $\sum\limits_{n=1}^{\infty}(-1)^{n}\dfrac{\ln(n)}{\sqrt{n}}$, must converge.

You can use Leibniz test:

if you have a series like $\sum\limits_{n=1}^{\infty}(-1)^{n}{a_n}$ and ${a_n}$ is decrescent and infinitesimal when $n \rightarrow \infty$, then the series

$\sum\limits_{n=1}^{\infty}(-1)^{n}{a_n}$ converges


text:
What VZ means in schematic? [duplicate]

This question already has an answer here:

Couldn't find the answer on the internet. The leads of VZ1 component are directly connected to mains.

marked as duplicate by DerStrom8, Tom Carpenter, PeterJ, Voltage Spike, ThreePhaseEelDec 29 '17 at 2:48

• The above link suggests it's a zener diode, but I'm not sure in what case you would want a zener (or TVS) directly across mains. Are you absolutely sure that's how it's connected? – DerStrom8 Dec 28 '17 at 0:09
• @DerStrom8, yes. If you don't believe me, I will post the image of another side on your request – Qeeet Dec 28 '17 at 0:10
• @DerStrom8, also there is no other holes for possible 'one board multiple designs', that is I'm pretty sure there are always mains in all designs. Also mains connector is not polarized. – Qeeet Dec 28 '17 at 0:13
• @Qeeet - The fact that L2 (a choke?) has been replaced by links, and other components aren't fitted, suggests a "cost-reduced" design, where components that improve reliability / longevity / EMI etc. but which are not strictly required, are removed. It might be interesting if you add a photo of the reverse side of the board, and also reverse-engineer the schematic and provide that, to show how those missing components would be included in the circuit (if they were fitted), but I've answered the specific question you asked about VZ1 in this specific context. – SamGibson Dec 28 '17 at 0:30

Therefore in this context, the component which makes most sense is the varistor (often a Metal Oxide Varistor or MOV). [A normal Zener Diode, for which a designer might also use the VZ component designator, would not be connected across the mains.]
• @Qeeet - "Do you know what R1A means? Is that some kind of resistor?" The use of letters after a component designator, is not well-standardised so I can only guess. My initial hypothesis would be that, yes, it's some kind of resistor. However it would require the rest of the (missing) input circuit to be reverse-engineered, to help explain why R1A would be used, with those extra components fitted. It might be that there is a capacitor (CX1?) which, if fitted, would be discharged by R1A, to prevent someone who unplugged the device and then touched the pins, from getting a "jolt". – SamGibson Dec 28 '17 at 1:01


text:
#### The support.bioconductor.org editor has been updated to markdown! Please see more info at: Tutorial: Updated Support Site Editor

Question: replicates and groups of samples: what's the right DESeq2 model?
0
16 months ago by
Seattle, WA, USA
Bob Thurman0 wrote:

Hi,

I’m having some difficulty with what would seem to be a pretty basic set-up.  I have a panel of, say, 20 cell lines, c1, c2, …,c20, with two replicates for each cell line, so 40 samples total.  The cell lines can be divided into groups based on different phenotypic properties and I’d like to do comparisons between groups.  As an example, based on one phenotype the cell lines segregate into 4 groups, with c1-c5 in g1, c6-10 in g2, c11-15 in g3, c16-20 in g4.  And I’d like to compare, say, g1 with g2, which would be 10 samples (2 replicates each of 5 cell lines) in each group; then compare g1 with g3, etc.

What is the best way to model this in DESeq2?

The approach I’ve been taking is to use all samples in the model and capture the replicate relationships using cell line labels,

cold <- data.frame(cells = c(“c1”, “c1”, “c2”, “c2”, …”c20”, “c20”))
dds <- DESeqDataSetFromMatrix(counts, cold, formula(~ cells))
DEOut <- DESeq(dds)

...and then use contrasts to group the cell lines for a given comparison.  Since resultsNames(DEOut) gives elements like

“Intercept”
“cells_c2_vs_c1”
“cells_c3_vs_c1”
…

to compare g2 with g3, I’d do

results(DEOut, contrast = list(c(“cells_c6_vs_c1”, “cells_c7_vs_c1”,...“cells_c10_vs_c1”),
c(“cells_c11_vs_c1”, “cells_c12_vs_c1”,...“cells_c15_vs_c1”)))

Is there something not recommended here?  The reasons I’m suspicious:

1. A particular comparison like the above turns up 7000+ regulated genes at an adjusted p-value threshold of 0.01, with extreme p-values and fold changes: returned adjusted p-values actually reach 0, and absolute log2FoldChanges go as high as 60.  If I instead model by using group labels (g1, g2, g3, g4) for the samples (and therefore ignore the cell line replicate relationships), then comparing g2 with g3 results in fewer than 500 regulated genes, and absolute log2FoldChanges going as high as 25.
2. Spot-checking the regulated genes seems to indicate a dependence on the expression of the reference cell line c1 that I don’t want.  That is, the genes that are up-regulated (log2FoldChange > 0) not only roughly have g2 average expression higher than g3 average expression (which is what I want), but even more apparent is that the average expression of the g2 cell lines is higher than the individual cell line expression for c1.  Likewise, the trend for down-regulated genes (log2FoldChange < 0) is that the average expression of the g2 cell lines is lower than for c1.

Should I perhaps be using a model without an Intercept term?

Thanks in advance for the help!

Bob

sessionInfo()
R version 3.4.0 (2017-04-21)
Platform: x86_64-redhat-linux-gnu (64-bit)
Running under: Red Hat Enterprise Linux

Matrix products: default
BLAS/LAPACK: /usr/lib64/R/lib/libRblas.so

locale:
[1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=en_US.UTF-8       LC_NAME=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C

attached base packages:
[1] parallel  stats4    stats     graphics  grDevices utils     datasets
[8] methods   base

other attached packages:
[1] DESeq2_1.16.1              SummarizedExperiment_1.6.3
[3] DelayedArray_0.2.7         matrixStats_0.52.2
[5] Biobase_2.36.2             GenomicRanges_1.28.4
[7] GenomeInfoDb_1.12.2        IRanges_2.10.2
[9] S4Vectors_0.14.3           BiocGenerics_0.22.0
modified 16 months ago by Gavin Kelly560 • written 16 months ago by Bob Thurman0
Answer: replicates and groups of samples: what's the right DESeq2 model?
0
16 months ago by
Gavin Kelly560
United Kingdom / London / Francis Crick Institute
Gavin Kelly560 wrote:

That formulation appears to be estimating (when you call DESeq) the replicate variability, but not the between-lines-within-phenotype variability. The contrast is then comparing the averages (across lines within the phenotype), but is ignoring the variability of the lines within the phenotype.  It's an ingenious approach, but flawed because it ignores this important source of variability.

What I believe is the 'approved' approach is described in the DESeq2 vignette, under the section "Group-specific condition effects, individuals nested within groups" (but you don't have a 'condition').  You'd need to relabel each line with an index within the particular phenotype (so you'd have line A within group G1,  line A within group G2... line J within group G1, lineJ within group G2 ), and a design of ~group + group:line_within_group (the 'pairing' you might worry about is arbitrary, and has no effect because of this design).  When the groups were unbalanced in terms of number of lines, you're going to get un-estimatable coefficients.  I don't think there's a way around that - you'd either sacrifice them or have an arbitrary scheme of pooling them with other lines (increasing the 'replicate' variability).

Alternatively you could use voom + limma, where you have the ability to have a random-effect due to cell-to-cell variation, by using the duplicateCorrelation function therein.

I'd second the recommendations here from Gavin.

To compare directly across groups and control for nested structure isn't possible with fixed effects models, and hence I often point users to limma's duplicateCorrelation() function which is designed for this.

If you were making within-cell-line comparisons (which it sounds like you're not), then it would be possible to account for the cell line differences and compare those within-cell-line effects across groups (and this is the section of the vignette Gavin mentions).


text:
# A segment is 3 units long. it is divided into 9 parts. what fraction of a unit are 2 parts pf the segmant ?

Length of segment = 3 units
Number of parts in a segment = 9 parts
So, length of each part = $\frac{3}{9}=\frac{1}{3}$ unit
Therefore, length of 2 parts of a segment = $2×\frac{1}{3}=\frac{2}{3}$ unit

• 0
What are you looking for?


text:
Asymptotics with prime of form 4k+3

I wonder if there is some asymptotics for such sum: $\sum_{p=2}^{n} \frac{1}{p}$, where the sum is taken over all primes of form $4k+3$?

It's well-known that $\sum_{p=2}^{n} \frac{1}{p}$, where the sum is taken over all primes is diverges and asymptotically is like $\ln\ln n$.

But I don't even know how to prove that the first sum is diverges.

I am also interested in asymptotics of the number of primes of the form $4k+3$ less than $n$.

Thank you very much!

-
I think you mean it is asymptotic to $\log\log n$. – Pedro Tamaroff Dec 22 '12 at 20:41
Thank you, sure I did. =) – Sergey Finsky Dec 22 '12 at 20:50

A more general version of this follows from Dirichlet's theorem on arithmetic progressions. In general, $$\sum_{p \equiv a \pmod d} \dfrac1p \sim \dfrac1{\phi(d)}\sum_{p} \dfrac1p$$ where $\phi$ is the Euler's totient function. More details can be found here, here, here and here.
Thank you very much for your answer, but I have one question left: in papers it's proven that $\sum_{p \equiv a \pmod d} \frac{1}{p^{s}} \sim \dfrac1{\phi(d)}\sum_{p} \frac{1}{p^{s}}$ when $s \to +1$. But how does original problem follows from this? (It is not obvious) – Sergey Finsky Dec 23 '12 at 7:06
@SergeyFinsky We know that $\sum_{p \leq n}\dfrac1p \sim \log(\log n)$. Hence, $\sum_{\left(p \equiv a \pmod d \right) \leq n} \sim \dfrac{\log(\log n)}{\phi(d)}$ – user17762 Dec 23 '12 at 7:08
Yes, but in the paper we consider $L$-function, which is but it's definition a series (not finite), so we have to somehow jump from the infinite series to finite. (It's easy to do when the convergence in series is uniform, but it's obviously not the case) I mean, your previous comment is obviously true, but why $\sum_{p \equiv a \mod d \le n} \frac{1}{p} \sim \frac{\log(\log(n))}{\phi (d)}$. Do you see what I mean? – Sergey Finsky Dec 23 '12 at 7:16
Sorry for not responding for such a long time, I kept forgetting to respond that your answer solves original question, which can be easily deduced from standart $\delta - \epsilon$ arguments. – Sergey Finsky Feb 11 '13 at 13:53


text:
# Thread: Simplifying derivative part 2

1. ## Simplifying derivative part 2

Hi,

I'm significantly confused about the steps that brought line 1 to line 2 of the expression below:

Can someone please provide the in-between steps? Reallly appreciate it.

2. ## Re: Simplifying derivative part 2

second term ... note $\cot{x} \cdot \sec{x} = \csc{x}$

third term ... note $\csc{x} \cdot \tan{x} = \sec{x}$

go back and study your basic trig identities


text:
# The Yhat Blog

machine learning, data science, engineering

# Analyzing the conditions for studying stars

### Astronomical Weather

Even a cloudless sky can mean bad weather for one of the world's biggest optical telescopes. The reason is turbulence in Earth's atmosphere, which constantly changes the optical properties of different layers of air. As a result stars appear to twinkle and any deep images of the night sky get blurred.

For astronomers it is extremely useful to know the current observing conditions, not least because prime observing time is precious and optimal scheduling is critical. At the Paranal Observatory, home to the Very Large Telescope (VLT), conditions are recorded every minute during regular operations.

Image: ESO / S. Brunier

We will explore the historical observing conditions at the VLT over the past 16 years and answer a lingering question in the minds of telescope operators: What is the probability that good conditions will last?

The Python packages we need are the high-level toolkits pandas for data analysis and seaborn for data visualization.

python
# Python 3.5.2
import pandas as pd # 0.18.1
import seaborn as sns # 0.7.1

### The ESO Ambient Conditions Database

A database of ambient observing conditions at the VLT is available from the European Southern Observatory (ESO). There is also a neat online monitor. The various parameters are measured by a small auxiliary telescope called a 'Differential Image Motion Monitor' (DIMM). It works by keeping track of a bright reference star that is imaged along two different lines of sight through the atmosphere.

Let's import a dump of this database containing all DIMM measurements made between January 1999 and December 2015 (3 million rows).

python
names=[
"raw_datetime", # timestamp (UT)
"raw_seeing", # seeing (arcsec = 1/3600 degree)
"raw_tau" # coherence time (seconds)
])

data = data[(data["raw_seeing"] > 0) & (data["raw_tau"] > 0)]

data.index = pd.DatetimeIndex(pd.to_datetime(data["raw_datetime"]), tz="UTC")
data["localtime"] = data.index.tz_convert("America/Santiago")

python
data[["raw_seeing", "raw_tau", "localtime"]].tail(3)

raw_seeingraw_taulocaltime
2015-12-31 08:59:34+00:001.450.0019672015-12-31 05:59:34-03:00
2015-12-31 09:00:39+00:001.620.0019652015-12-31 06:00:39-03:00
2015-12-31 09:01:45+00:001.520.0017622015-12-31 06:01:45-03:00

### Seeing and Coherence Time

The most important astronomical weather parameter is the seeing. It is measured as the apparent (angular) size of a point-like star on the sky and is a direct indicator of the 'blurriness' of the atmosphere. As a general rule, the smaller the seeing the better.

The VLT is located at an exceptional observing site in Chile that lies isolated on a high and dry mountaintop in the Atacama desert. A typical seeing of around 1 arcsecond makes it one of the best sites in the word for stargazing.

python
data["seeing"] = data["raw_seeing"] # (arcsec)

python
data["seeing"].describe()

countmeanstdmin25%50%75%max
3.00e61.020.450.200.720.921.2010.4

Yet the monthly seeing statistics show that there are significant changes in the average seeing over the course of a year and also on longer timescales. The frequency of good or excellent observing conditions can thus be expected to change as well.

python
sns.boxplot(x=data["localtime"].apply(lambda d: (d.year, d.month)), y=data["seeing"])

Another important parameter is the coherence time. It is effectively the time it takes one turbulent pocket of air to pass over the telescope. A longer coherence time implies a more stable atmosphere, which is advantageous.

python
data["tau"] = data["raw_tau"]*1000 # (milliseconds)

A good coherence time of a few milliseconds usually occurs together with good seeing, but it is particularly relevant for technology that functions on similar timescales, such as adaptive optics systems (with our without a laser guide star like in the image above) or 'fringe-tracking' interferometers.

python
data.plot.hexbin(x="seeing", y="tau")

### The VLT Weather History

To analyze the weather history in more detail, let's start by writing a function that classifies a single observing night as either 'good' or 'bad' based on the ambient conditions.

The is_good function first checks each input measurement in data for violating limits on the seeing (max_seeing) and coherence time (min_tau). It then calculates the time differences between successive measurements and finds the longest time interval during which one could have observed within these limits, without interruption. This value is finally compared to a required duration.

python
def is_good(timestamp, data, max_seeing, min_tau, duration):
flag = (data["seeing"] < max_seeing) & (data["tau"] > min_tau)
delta = data["localtime"] - data["localtime"][0]
delta -= delta.shift().fillna(0)
delta /= pd.Timedelta(hours=1)
block = (flag.shift().fillna(0) != flag).cumsum()
good = (delta*flag).groupby(block).sum().max() >= duration
return good, timestamp

Now it is straightforward to determine for each month a probability of being able to observe in certain conditions by counting the number of 'good' nights in that month.

python
def probability(freq="1M", *args, **kwargs):
nights = pd.Series(*zip(*[is_good(*group, *args, **kwargs) for group
in data.groupby(pd.TimeGrouper("24H", base=12, key="localtime"))
if len(group[1]) > 0]))
return nights.groupby(pd.TimeGrouper(freq)) \
.apply(lambda group: group.mean())

Because not every observing program demands the best ambient conditions, it is practical to consider three different seeing limits: 'fair' (1.2 arcsec), 'good' (0.8 arcsec) and 'excellent' (0.6 arcsec). A reasonable minimum coherence time is 2 ms, assuming one observing block lasts 1 hour (following the paper by Mérand et al. 2014).

python
result = {}
for condition, seeing in (("fair", 1.2), ("good", 0.8), ("excellent", 0.6)):
result[condition] = probability(max_seeing=seeing, min_tau=2, duration=1)
result = pd.DataFrame(result)

python
result.rolling(3).mean().plot()

### Findings

The influence of the seasons on the observing conditions stands out at first glance. For example, the probability of having 'fair' conditions has recently been as small as 40%-60% in the winter semester (April to September), but still larger than 80% in the summer semester (October to March). However, 'excellent' conditions have been somewhat more likely in the winter months.

More surprising though is that since 2006 the probability of 'excellent' conditions has never been more than 20% for any extended amount of time. Ever since 2010 it has been less than 10%, roughly. The earlier years must have had much better weather overall, with the exception of 2002. The most recent El Niño event, one of the strongest on record, presumably caused the latest stretch of poor conditions starting in 2015.

### Parting Thoughts

For further analysis it would be interesting to also have comparison statistics from other observatories.

In any case, it looks like astronomers should be prepared for more bad weather to come.

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