Base field 4.4.8768.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 5x^{2} + 6x + 7\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[31, 31, w^{2} - 5]$ |
| Dimension: | $13$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $23$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{13} - x^{12} - 29x^{11} + 25x^{10} + 278x^{9} - 165x^{8} - 1084x^{7} + 389x^{6} + 1612x^{5} - 566x^{4} - 948x^{3} + 359x^{2} + 190x - 77\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -w^{2} + w + 3]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w]$ | $\phantom{-}\frac{35364787}{6415733}e^{12} - \frac{9832009}{6415733}e^{11} - \frac{1031546680}{6415733}e^{10} + \frac{138287661}{6415733}e^{9} + \frac{9899210791}{6415733}e^{8} + \frac{1337858267}{6415733}e^{7} - \frac{37076011898}{6415733}e^{6} - \frac{13166364668}{6415733}e^{5} + \frac{46464761625}{6415733}e^{4} + \frac{13811321454}{6415733}e^{3} - \frac{22384144666}{6415733}e^{2} - \frac{3827483174}{6415733}e + \frac{3585653102}{6415733}$ |
| 7 | $[7, 7, -w^{2} + 2w + 1]$ | $\phantom{-}\frac{89314850}{6415733}e^{12} - \frac{22158846}{6415733}e^{11} - \frac{2608733805}{6415733}e^{10} + \frac{273322475}{6415733}e^{9} + \frac{25090279642}{6415733}e^{8} + \frac{4080235267}{6415733}e^{7} - \frac{94258628390}{6415733}e^{6} - \frac{35826788721}{6415733}e^{5} + \frac{118860344411}{6415733}e^{4} + \frac{38193734366}{6415733}e^{3} - \frac{58074532009}{6415733}e^{2} - \frac{10811925193}{6415733}e + \frac{9513987953}{6415733}$ |
| 7 | $[7, 7, w^{2} - 2]$ | $-\frac{29072603}{12831466}e^{12} + \frac{4189884}{6415733}e^{11} + \frac{849461103}{12831466}e^{10} - \frac{61188204}{6415733}e^{9} - \frac{4090495535}{6415733}e^{8} - \frac{1021815979}{12831466}e^{7} + \frac{30904072611}{12831466}e^{6} + \frac{5333374616}{6415733}e^{5} - \frac{19867397061}{6415733}e^{4} - \frac{5893603709}{6415733}e^{3} + \frac{9903079911}{6415733}e^{2} + \frac{3471002759}{12831466}e - \frac{3303318751}{12831466}$ |
| 7 | $[7, 7, w - 1]$ | $-\frac{54415157}{12831466}e^{12} + \frac{6116633}{6415733}e^{11} + \frac{1591687133}{12831466}e^{10} - \frac{65340969}{6415733}e^{9} - \frac{7673596865}{6415733}e^{8} - \frac{2817440113}{12831466}e^{7} + \frac{57895218467}{12831466}e^{6} + \frac{11547820863}{6415733}e^{5} - \frac{36872322277}{6415733}e^{4} - \frac{12580789683}{6415733}e^{3} + \frac{18366465988}{6415733}e^{2} + \frac{7335061759}{12831466}e - \frac{6197451149}{12831466}$ |
| 23 | $[23, 23, -w^{3} + w^{2} + 3w - 1]$ | $\phantom{-}\frac{113791765}{25662932}e^{12} - \frac{3141949}{6415733}e^{11} - \frac{3337978105}{25662932}e^{10} - \frac{24309244}{6415733}e^{9} + \frac{16155394795}{12831466}e^{8} + \frac{9298199113}{25662932}e^{7} - \frac{122112395599}{25662932}e^{6} - \frac{30094220883}{12831466}e^{5} + \frac{77544389269}{12831466}e^{4} + \frac{16255430105}{6415733}e^{3} - \frac{19888435724}{6415733}e^{2} - \frac{18893855557}{25662932}e + \frac{14363492757}{25662932}$ |
| 23 | $[23, 23, w^{3} - 2w^{2} - 2w + 2]$ | $-\frac{13193133}{6415733}e^{12} + \frac{1068662}{6415733}e^{11} + \frac{386906506}{6415733}e^{10} + \frac{22767412}{6415733}e^{9} - \frac{3741788669}{6415733}e^{8} - \frac{1190174251}{6415733}e^{7} + \frac{14091394654}{6415733}e^{6} + \frac{7395671818}{6415733}e^{5} - \frac{17674408137}{6415733}e^{4} - \frac{8008354664}{6415733}e^{3} + \frac{8957850137}{6415733}e^{2} + \frac{2357171481}{6415733}e - \frac{1605574912}{6415733}$ |
| 31 | $[31, 31, w^{2} - 5]$ | $-1$ |
| 31 | $[31, 31, -w^{2} + 2w + 4]$ | $-\frac{479594345}{25662932}e^{12} + \frac{30563740}{6415733}e^{11} + \frac{14004334985}{25662932}e^{10} - \frac{389843481}{6415733}e^{9} - \frac{67317525721}{12831466}e^{8} - \frac{21083380629}{25662932}e^{7} + \frac{505569399419}{25662932}e^{6} + \frac{94755715605}{12831466}e^{5} - \frac{318581459185}{12831466}e^{4} - \frac{50481273552}{6415733}e^{3} + \frac{77712301032}{6415733}e^{2} + \frac{57157738169}{25662932}e - \frac{50837465249}{25662932}$ |
| 41 | $[41, 41, -w^{3} + 2w^{2} + 2w - 6]$ | $-\frac{595433607}{25662932}e^{12} + \frac{37271595}{6415733}e^{11} + \frac{17391074687}{25662932}e^{10} - \frac{465003276}{6415733}e^{9} - \frac{83629934933}{12831466}e^{8} - \frac{26873854767}{25662932}e^{7} + \frac{628410410213}{25662932}e^{6} + \frac{118954120509}{12831466}e^{5} - \frac{396357216737}{12831466}e^{4} - \frac{63604007590}{6415733}e^{3} + \frac{96769858247}{6415733}e^{2} + \frac{72242639023}{25662932}e - \frac{63338487923}{25662932}$ |
| 41 | $[41, 41, w^{3} - w^{2} - 3w - 3]$ | $-\frac{49593019}{12831466}e^{12} + \frac{6263510}{6415733}e^{11} + \frac{1447365969}{12831466}e^{10} - \frac{78704262}{6415733}e^{9} - \frac{6949766383}{6415733}e^{8} - \frac{2223651843}{12831466}e^{7} + \frac{52060346925}{12831466}e^{6} + \frac{9884817398}{6415733}e^{5} - \frac{32545207522}{6415733}e^{4} - \frac{10524070493}{6415733}e^{3} + \frac{15657712109}{6415733}e^{2} + \frac{6033310283}{12831466}e - \frac{5021760305}{12831466}$ |
| 47 | $[47, 47, w^{2} - 2w - 5]$ | $-\frac{353597809}{25662932}e^{12} + \frac{24563673}{6415733}e^{11} + \frac{10314869121}{25662932}e^{10} - \frac{345424381}{6415733}e^{9} - \frac{49500662445}{12831466}e^{8} - \frac{13377430345}{25662932}e^{7} + \frac{370904009863}{25662932}e^{6} + \frac{65807697741}{12831466}e^{5} - \frac{232597834623}{12831466}e^{4} - \frac{34505847836}{6415733}e^{3} + \frac{56093361174}{6415733}e^{2} + \frac{38280534393}{25662932}e - \frac{36018733409}{25662932}$ |
| 47 | $[47, 47, w^{2} - 6]$ | $-\frac{312412263}{12831466}e^{12} + \frac{38995029}{6415733}e^{11} + \frac{9122995731}{12831466}e^{10} - \frac{484584358}{6415733}e^{9} - \frac{43853247023}{6415733}e^{8} - \frac{14156628099}{12831466}e^{7} + \frac{329219679805}{12831466}e^{6} + \frac{62429735539}{6415733}e^{5} - \frac{207088211210}{6415733}e^{4} - \frac{66373561366}{6415733}e^{3} + \frac{100780493382}{6415733}e^{2} + \frac{37538105497}{12831466}e - \frac{32864982945}{12831466}$ |
| 71 | $[71, 71, -w^{3} + 2w^{2} + 3w - 1]$ | $\phantom{-}\frac{54716007}{12831466}e^{12} - \frac{7316879}{6415733}e^{11} - \frac{1595357931}{12831466}e^{10} + \frac{99053802}{6415733}e^{9} + \frac{7646828979}{6415733}e^{8} + \frac{2202147653}{12831466}e^{7} - \frac{57092020033}{12831466}e^{6} - \frac{10341945591}{6415733}e^{5} + \frac{35370288745}{6415733}e^{4} + \frac{10557461551}{6415733}e^{3} - \frac{16730430631}{6415733}e^{2} - \frac{5682813613}{12831466}e + \frac{5190399629}{12831466}$ |
| 71 | $[71, 71, -w^{3} + w^{2} + 4w - 3]$ | $-\frac{241645005}{6415733}e^{12} + \frac{58816870}{6415733}e^{11} + \frac{7060410829}{6415733}e^{10} - \frac{707602836}{6415733}e^{9} - \frac{67945697775}{6415733}e^{8} - \frac{11331612854}{6415733}e^{7} + \frac{255511829099}{6415733}e^{6} + \frac{98051914654}{6415733}e^{5} - \frac{322982374121}{6415733}e^{4} - \frac{105059460215}{6415733}e^{3} + \frac{158406750488}{6415733}e^{2} + \frac{29904042870}{6415733}e - \frac{26068241548}{6415733}$ |
| 79 | $[79, 79, -w - 3]$ | $\phantom{-}\frac{75876584}{6415733}e^{12} - \frac{12423000}{6415733}e^{11} - \frac{2221600051}{6415733}e^{10} + \frac{49255496}{6415733}e^{9} + \frac{21442769906}{6415733}e^{8} + \frac{5164330115}{6415733}e^{7} - \frac{80784447213}{6415733}e^{6} - \frac{36539264694}{6415733}e^{5} + \frac{102036992709}{6415733}e^{4} + \frac{39500758589}{6415733}e^{3} - \frac{50989101496}{6415733}e^{2} - \frac{11456325652}{6415733}e + \frac{8767972203}{6415733}$ |
| 79 | $[79, 79, w - 4]$ | $\phantom{-}\frac{54340085}{12831466}e^{12} - \frac{8547076}{6415733}e^{11} - \frac{1587453605}{12831466}e^{10} + \frac{135231412}{6415733}e^{9} + \frac{7644211450}{6415733}e^{8} + \frac{1511062561}{12831466}e^{7} - \frac{57834446701}{12831466}e^{6} - \frac{9236325818}{6415733}e^{5} + \frac{37384157651}{6415733}e^{4} + \frac{10184554461}{6415733}e^{3} - \frac{18591971940}{6415733}e^{2} - \frac{5958906329}{12831466}e + \frac{6074874429}{12831466}$ |
| 81 | $[81, 3, -3]$ | $-\frac{642239571}{25662932}e^{12} + \frac{44183395}{6415733}e^{11} + \frac{18739223391}{25662932}e^{10} - \frac{614657414}{6415733}e^{9} - \frac{89966959095}{12831466}e^{8} - \frac{24812991291}{25662932}e^{7} + \frac{674635917089}{25662932}e^{6} + \frac{120714917855}{12831466}e^{5} - \frac{423832613951}{12831466}e^{4} - \frac{63886538120}{6415733}e^{3} + \frac{102274639423}{6415733}e^{2} + \frac{71588623775}{25662932}e - \frac{65650146219}{25662932}$ |
| 89 | $[89, 89, w^{2} - 3w - 2]$ | $\phantom{-}\frac{138934829}{12831466}e^{12} - \frac{16271838}{6415733}e^{11} - \frac{4064265579}{12831466}e^{10} + \frac{185525614}{6415733}e^{9} + \frac{19600192398}{6415733}e^{8} + \frac{6848892519}{12831466}e^{7} - \frac{148076824393}{12831466}e^{6} - \frac{28900790656}{6415733}e^{5} + \frac{94773823191}{6415733}e^{4} + \frac{31622732131}{6415733}e^{3} - \frac{47531711285}{6415733}e^{2} - \frac{18477240199}{12831466}e + \frac{16145704821}{12831466}$ |
| 89 | $[89, 89, w^{2} + w - 4]$ | $\phantom{-}\frac{366093485}{12831466}e^{12} - \frac{48986831}{6415733}e^{11} - \frac{10690980477}{12831466}e^{10} + \frac{662918093}{6415733}e^{9} + \frac{51407490621}{6415733}e^{8} + \frac{14809378865}{12831466}e^{7} - \frac{386619717331}{12831466}e^{6} - \frac{70082249377}{6415733}e^{5} + \frac{244793020470}{6415733}e^{4} + \frac{75009797175}{6415733}e^{3} - \frac{119697874144}{6415733}e^{2} - \frac{42574670759}{12831466}e + \frac{38999350335}{12831466}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31, 31, w^{2} - 5]$ | $1$ |