Base field 4.4.9792.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 7x^{2} + 2x + 7\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[31, 31, -2w^{3} + 7w^{2} + 5w - 12]$ |
| Dimension: | $18$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $26$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{18} - 90x^{16} + 3326x^{14} - 65222x^{12} + 732231x^{10} - 4730352x^{8} + 16725808x^{6} - 28575568x^{4} + 17568016x^{2} - 846400\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, w^{3} - 3w^{2} - 3w + 4]$ | $-\frac{91792394648359}{384553142260554276320}e^{16} + \frac{3111442760886223}{192276571130277138160}e^{14} - \frac{75048181498640359}{192276571130277138160}e^{12} + \frac{139167066533865901}{38455314226055427632}e^{10} - \frac{1208525104986046329}{384553142260554276320}e^{8} - \frac{1925997461454556421}{24034571391284642270}e^{6} - \frac{27038353770084329937}{96138285565138569080}e^{4} + \frac{33087487114364380432}{12017285695642321135}e^{2} - \frac{688420996828065511}{2403457139128464227}$ |
| 7 | $[7, 7, w]$ | $-\frac{156460654095889}{384553142260554276320}e^{16} + \frac{8438812845758373}{192276571130277138160}e^{14} - \frac{367598750147565449}{192276571130277138160}e^{12} + \frac{1665531003824143595}{38455314226055427632}e^{10} - \frac{209885564483463337759}{384553142260554276320}e^{8} + \frac{181544220071021875623}{48069142782569284540}e^{6} - \frac{1263936442980768075167}{96138285565138569080}e^{4} + \frac{450410218051093080109}{24034571391284642270}e^{2} - \frac{11455003424701090756}{2403457139128464227}$ |
| 7 | $[7, 7, w^{3} - 3w^{2} - 4w + 5]$ | $-\frac{59053784790677}{384553142260554276320}e^{16} + \frac{192673900567267}{96138285565138569080}e^{14} + \frac{69203723979120313}{192276571130277138160}e^{12} - \frac{555268713390087159}{38455314226055427632}e^{10} + \frac{81230876819732952473}{384553142260554276320}e^{8} - \frac{243967447656625096929}{192276571130277138160}e^{6} + \frac{199462909466855110749}{96138285565138569080}e^{4} + \frac{136729698042168029469}{48069142782569284540}e^{2} - \frac{5190493972674171603}{2403457139128464227}$ |
| 9 | $[9, 3, w^{3} - 4w^{2} - w + 9]$ | $\phantom{-}e$ |
| 17 | $[17, 17, 2w^{3} - 6w^{2} - 7w + 8]$ | $\phantom{-}\frac{31477381899241027}{17689444543985496710720}e^{17} - \frac{69291858792870313}{442236113599637417768}e^{15} + \frac{49937888943538015321}{8844722271992748355360}e^{13} - \frac{952477186285165467727}{8844722271992748355360}e^{11} + \frac{20811020370408942995817}{17689444543985496710720}e^{9} - \frac{65907823423873282796167}{8844722271992748355360}e^{7} + \frac{116706429276921317606539}{4422361135996374177680}e^{5} - \frac{4506570013882193913129}{96138285565138569080}e^{3} + \frac{15787747223323754725091}{552795141999546772210}e$ |
| 17 | $[17, 17, -w^{3} + 3w^{2} + 4w - 3]$ | $\phantom{-}\frac{10138212127302017}{17689444543985496710720}e^{17} - \frac{259510939461462547}{4422361135996374177680}e^{15} + \frac{22017857978587302807}{8844722271992748355360}e^{13} - \frac{99997375418999741653}{1768944454398549671072}e^{11} + \frac{13110899333053435231547}{17689444543985496710720}e^{9} - \frac{49921680998562523327651}{8844722271992748355360}e^{7} + \frac{105349354453101809082721}{4422361135996374177680}e^{5} - \frac{4714863957679187010513}{96138285565138569080}e^{3} + \frac{3606778878283167662723}{110559028399909354442}e$ |
| 17 | $[17, 17, -w + 2]$ | $-\frac{10099072620861841}{8844722271992748355360}e^{17} + \frac{432804234798525439}{4422361135996374177680}e^{15} - \frac{15065990435088182229}{4422361135996374177680}e^{13} + \frac{274895242125128066099}{4422361135996374177680}e^{11} - \frac{5690414339190683499191}{8844722271992748355360}e^{9} + \frac{1708337731541462831693}{442236113599637417768}e^{7} - \frac{29853609667671088492419}{2211180567998187088840}e^{5} + \frac{641694248795148851533}{24034571391284642270}e^{3} - \frac{6036136693306699982147}{276397570999773386105}e$ |
| 23 | $[23, 23, 2w^{3} - 7w^{2} - 4w + 12]$ | $-\frac{3840320806655949}{4422361135996374177680}e^{17} + \frac{326136010807912803}{4422361135996374177680}e^{15} - \frac{709446051247977846}{276397570999773386105}e^{13} + \frac{26393829574391602763}{552795141999546772210}e^{11} - \frac{2301411854453959614489}{4422361135996374177680}e^{9} + \frac{15039000546510013604991}{4422361135996374177680}e^{7} - \frac{1396433587651073744643}{110559028399909354442}e^{5} + \frac{1038972582599412346779}{48069142782569284540}e^{3} - \frac{2398148442869729323857}{276397570999773386105}e$ |
| 23 | $[23, 23, -w^{2} + 2w + 3]$ | $\phantom{-}\frac{33610515975123347}{8844722271992748355360}e^{17} - \frac{76578839287460867}{221118056799818708884}e^{15} + \frac{57336577515146351691}{4422361135996374177680}e^{13} - \frac{1137659432637525923327}{4422361135996374177680}e^{11} + \frac{25724674934250387253217}{8844722271992748355360}e^{9} - \frac{82622857918093650280227}{4422361135996374177680}e^{7} + \frac{70385122745110271674687}{1105590283999093544420}e^{5} - \frac{1162128830860442896706}{12017285695642321135}e^{3} + \frac{11652190102155027530376}{276397570999773386105}e$ |
| 31 | $[31, 31, -2w^{3} + 7w^{2} + 5w - 12]$ | $-1$ |
| 31 | $[31, 31, -w^{3} + 4w^{2} + 2w - 8]$ | $-\frac{21276071684109}{76910628452110855264}e^{16} + \frac{751006336966791}{38455314226055427632}e^{14} - \frac{18743702850193149}{38455314226055427632}e^{12} + \frac{170650392491208375}{38455314226055427632}e^{10} + \frac{800529071756693029}{76910628452110855264}e^{8} - \frac{9113625867366552913}{19227657113027713816}e^{6} + \frac{68133495916006386947}{19227657113027713816}e^{4} - \frac{55649867126825825659}{4806914278256928454}e^{2} + \frac{27166499856756942296}{2403457139128464227}$ |
| 41 | $[41, 41, 3w^{3} - 10w^{2} - 7w + 16]$ | $-\frac{13963906669998049}{8844722271992748355360}e^{17} + \frac{557191378065331669}{4422361135996374177680}e^{15} - \frac{17342091747955937693}{4422361135996374177680}e^{13} + \frac{261065540260255025327}{4422361135996374177680}e^{11} - \frac{3653472590806944632479}{8844722271992748355360}e^{9} + \frac{712737176493627357157}{1105590283999093544420}e^{7} + \frac{3010150872330020651471}{442236113599637417768}e^{5} - \frac{745129996466116321679}{24034571391284642270}e^{3} + \frac{8351162761226756190309}{276397570999773386105}e$ |
| 41 | $[41, 41, 2w^{3} - 7w^{2} - 5w + 10]$ | $-\frac{19832113791839327}{17689444543985496710720}e^{17} + \frac{446636885786083173}{4422361135996374177680}e^{15} - \frac{6578821952328743393}{1768944454398549671072}e^{13} + \frac{638676064722389815959}{8844722271992748355360}e^{11} - \frac{14076537224926149924117}{17689444543985496710720}e^{9} + \frac{44040889061139434038853}{8844722271992748355360}e^{7} - \frac{72757953862355374989577}{4422361135996374177680}e^{5} + \frac{421439219076924993453}{19227657113027713816}e^{3} + \frac{780335122511936794133}{552795141999546772210}e$ |
| 49 | $[49, 7, 2w^{3} - 6w^{2} - 6w + 9]$ | $\phantom{-}\frac{156656752645287}{76910628452110855264}e^{16} - \frac{5677881349353071}{38455314226055427632}e^{14} + \frac{149289642599457747}{38455314226055427632}e^{12} - \frac{1556183306592640569}{38455314226055427632}e^{10} + \frac{1678312137990094761}{76910628452110855264}e^{8} + \frac{5940621917499042320}{2403457139128464227}e^{6} - \frac{275667867641423132453}{19227657113027713816}e^{4} + \frac{45403034280264152852}{2403457139128464227}e^{2} + \frac{6626302226762951768}{2403457139128464227}$ |
| 71 | $[71, 71, w^{2} - 2w - 2]$ | $\phantom{-}\frac{6733355744697389}{4422361135996374177680}e^{17} - \frac{152066340678994067}{1105590283999093544420}e^{15} + \frac{5615245774532823419}{1105590283999093544420}e^{13} - \frac{217551300682105504697}{2211180567998187088840}e^{11} + \frac{4695254290875025959819}{4422361135996374177680}e^{9} - \frac{13627663604888157974973}{2211180567998187088840}e^{7} + \frac{7148559993268915150903}{442236113599637417768}e^{5} - \frac{354955191663268996989}{48069142782569284540}e^{3} - \frac{5133750417711458416423}{276397570999773386105}e$ |
| 71 | $[71, 71, 2w^{3} - 7w^{2} - 4w + 13]$ | $\phantom{-}\frac{122570752035460793}{17689444543985496710720}e^{17} - \frac{531368855886902415}{884472227199274835536}e^{15} + \frac{187362291317778918219}{8844722271992748355360}e^{13} - \frac{3463953031203799461233}{8844722271992748355360}e^{11} + \frac{72296814884087400873363}{17689444543985496710720}e^{9} - \frac{213632030831123932794403}{8844722271992748355360}e^{7} + \frac{339642090472684339572731}{4422361135996374177680}e^{5} - \frac{11169500332209625868351}{96138285565138569080}e^{3} + \frac{35239011005680319911939}{552795141999546772210}e$ |
| 73 | $[73, 73, 3w^{3} - 11w^{2} - 5w + 19]$ | $\phantom{-}\frac{2065265459157261}{384553142260554276320}e^{16} - \frac{44284581858591071}{96138285565138569080}e^{14} + \frac{3072424037632163451}{192276571130277138160}e^{12} - \frac{11072504341946692617}{38455314226055427632}e^{10} + \frac{1107889798027267574591}{384553142260554276320}e^{8} - \frac{3050423227088069642183}{192276571130277138160}e^{6} + \frac{4308739524188569361333}{96138285565138569080}e^{4} - \frac{2652506472620262270337}{48069142782569284540}e^{2} + \frac{36543060815375722569}{2403457139128464227}$ |
| 73 | $[73, 73, -4w^{3} + 13w^{2} + 10w - 17]$ | $-\frac{328680076447127}{96138285565138569080}e^{16} + \frac{15726462982542119}{48069142782569284540}e^{14} - \frac{617556932528158847}{48069142782569284540}e^{12} + \frac{2552689203819501277}{9613828556513856908}e^{10} - \frac{295484409815509024897}{96138285565138569080}e^{8} + \frac{234639528028311354584}{12017285695642321135}e^{6} - \frac{1478395031408156116071}{24034571391284642270}e^{4} + \frac{901064137855851026044}{12017285695642321135}e^{2} - \frac{42238885586665914298}{2403457139128464227}$ |
| 79 | $[79, 79, 3w^{3} - 9w^{2} - 10w + 13]$ | $\phantom{-}\frac{129806292267327}{384553142260554276320}e^{16} - \frac{1673719622135587}{96138285565138569080}e^{14} + \frac{34020877477998617}{192276571130277138160}e^{12} + \frac{193342543570802557}{38455314226055427632}e^{10} - \frac{52780705730875750283}{384553142260554276320}e^{8} + \frac{237501130215287254399}{192276571130277138160}e^{6} - \frac{426935718340652491309}{96138285565138569080}e^{4} + \frac{207224891152525031291}{48069142782569284540}e^{2} - \frac{3556824579252401333}{2403457139128464227}$ |
| 79 | $[79, 79, -2w^{3} + 6w^{2} + 5w - 8]$ | $\phantom{-}\frac{465633800545}{19227657113027713816}e^{16} - \frac{313368740031447}{19227657113027713816}e^{14} + \frac{5098738556268195}{4806914278256928454}e^{12} - \frac{132303589498245499}{4806914278256928454}e^{10} + \frac{6560873114840564197}{19227657113027713816}e^{8} - \frac{37492433950661037083}{19227657113027713816}e^{6} + \frac{15638475099877999863}{4806914278256928454}e^{4} + \frac{17682676485022186652}{2403457139128464227}e^{2} - \frac{16094462557708325920}{2403457139128464227}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31, 31, -2w^{3} + 7w^{2} + 5w - 12]$ | $1$ |