Base field 4.4.11525.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 11x^{2} + 5x + 25\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[31, 31, \frac{1}{5}w^{3} - \frac{1}{5}w^{2} - \frac{1}{5}w + 2]$ |
| Dimension: | $15$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $33$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{15} + 3x^{14} - 46x^{13} - 153x^{12} + 788x^{11} + 2986x^{10} - 6064x^{9} - 28280x^{8} + 18645x^{7} + 136921x^{6} + 1371x^{5} - 330358x^{4} - 102262x^{3} + 354759x^{2} + 112194x - 135162\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, \frac{1}{5}w^{3} + \frac{4}{5}w^{2} - \frac{11}{5}w - 7]$ | $\phantom{-}\frac{648458879986426}{11617322021538489}e^{14} + \frac{34691005289200}{1290813557948721}e^{13} - \frac{61299395746113491}{23234644043076978}e^{12} - \frac{14674041727219783}{7744881347692326}e^{11} + \frac{567959685968975510}{11617322021538489}e^{10} + \frac{1011322333521030779}{23234644043076978}e^{9} - \frac{10465022263720916609}{23234644043076978}e^{8} - \frac{5159217928152594074}{11617322021538489}e^{7} + \frac{16873805916390364139}{7744881347692326}e^{6} + \frac{50066438300553511589}{23234644043076978}e^{5} - \frac{14030814168179654423}{2581627115897442}e^{4} - \frac{55051456248273289696}{11617322021538489}e^{3} + \frac{74426210464172512757}{11617322021538489}e^{2} + \frac{13966524994423229}{3876316990837}e - \frac{3824815366137208253}{1290813557948721}$ |
| 5 | $[5, 5, \frac{1}{5}w^{3} - \frac{6}{5}w^{2} - \frac{1}{5}w + 4]$ | $\phantom{-}e$ |
| 11 | $[11, 11, w + 1]$ | $-\frac{341142110077835}{23234644043076978}e^{14} - \frac{23504759622557}{2581627115897442}e^{13} + \frac{16161226334248259}{23234644043076978}e^{12} + \frac{4596754946262343}{7744881347692326}e^{11} - \frac{150040535659728818}{11617322021538489}e^{10} - \frac{306588215063392301}{23234644043076978}e^{9} + \frac{2770741190414478467}{23234644043076978}e^{8} + \frac{1541697507154334204}{11617322021538489}e^{7} - \frac{2242481201273055760}{3872440673846163}e^{6} - \frac{7453191988529544667}{11617322021538489}e^{5} + \frac{1882260094439940841}{1290813557948721}e^{4} + \frac{32933039536334980781}{23234644043076978}e^{3} - \frac{20442417483081585113}{11617322021538489}e^{2} - \frac{12722311914699650}{11628950972511}e + \frac{1103316410181545720}{1290813557948721}$ |
| 11 | $[11, 11, \frac{1}{5}w^{3} - \frac{1}{5}w^{2} - \frac{11}{5}w + 2]$ | $\phantom{-}\frac{253047512375369}{23234644043076978}e^{14} + \frac{12095898329659}{1290813557948721}e^{13} - \frac{12072622318233065}{23234644043076978}e^{12} - \frac{4279372950843145}{7744881347692326}e^{11} + \frac{112657351128395915}{11617322021538489}e^{10} + \frac{271245590165287493}{23234644043076978}e^{9} - \frac{2087620185913109657}{23234644043076978}e^{8} - \frac{1325016438410893964}{11617322021538489}e^{7} + \frac{1694136489749879626}{3872440673846163}e^{6} + \frac{12595428313782816047}{23234644043076978}e^{5} - \frac{1428715997475653527}{1290813557948721}e^{4} - \frac{13786591149707214823}{11617322021538489}e^{3} + \frac{15735208318877399711}{11617322021538489}e^{2} + \frac{3544341288840522}{3876316990837}e - \frac{882411957903427493}{1290813557948721}$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{1648034331649766}{11617322021538489}e^{14} + \frac{203776239923209}{2581627115897442}e^{13} - \frac{78028290372889295}{11617322021538489}e^{12} - \frac{20495301667598656}{3872440673846163}e^{11} + \frac{1447935536239604725}{11617322021538489}e^{10} + \frac{1382895120127799384}{11617322021538489}e^{9} - \frac{13356875424172597874}{11617322021538489}e^{8} - \frac{13965774436566101935}{11617322021538489}e^{7} + \frac{21575573930334745865}{3872440673846163}e^{6} + \frac{134933994914872717927}{23234644043076978}e^{5} - \frac{18011341068341366456}{1290813557948721}e^{4} - \frac{296672673366550330111}{23234644043076978}e^{3} + \frac{193007483868522967939}{11617322021538489}e^{2} + \frac{113452226231652292}{11628950972511}e - \frac{10153844000473779181}{1290813557948721}$ |
| 19 | $[19, 19, -\frac{1}{5}w^{3} + \frac{1}{5}w^{2} + \frac{1}{5}w + 1]$ | $-\frac{828544013323727}{23234644043076978}e^{14} - \frac{42477437319869}{2581627115897442}e^{13} + \frac{39116389555895261}{23234644043076978}e^{12} + \frac{9186578772257389}{7744881347692326}e^{11} - \frac{362084197947724943}{11617322021538489}e^{10} - \frac{640128659790162821}{23234644043076978}e^{9} + \frac{6666807016222122533}{23234644043076978}e^{8} + \frac{3292102651415065850}{11617322021538489}e^{7} - \frac{5372455359407862460}{3872440673846163}e^{6} - \frac{16102251317189010898}{11617322021538489}e^{5} + \frac{4468520036116516705}{1290813557948721}e^{4} + \frac{71490060965726881355}{23234644043076978}e^{3} - \frac{47569780698384916226}{11617322021538489}e^{2} - \frac{27541332646479503}{11628950972511}e + \frac{2481774007415097008}{1290813557948721}$ |
| 19 | $[19, 19, \frac{1}{5}w^{3} - \frac{1}{5}w^{2} - \frac{11}{5}w]$ | $-\frac{1712954407508431}{7744881347692326}e^{14} - \frac{52965655733071}{430271185982907}e^{13} + \frac{81103566692349349}{7744881347692326}e^{12} + \frac{10650140089298638}{1290813557948721}e^{11} - \frac{1504958763191411915}{7744881347692326}e^{10} - \frac{1436922570066476173}{7744881347692326}e^{9} + \frac{6940665054740277728}{3872440673846163}e^{8} + \frac{14509680051357388577}{7744881347692326}e^{7} - \frac{11208481992692761139}{1290813557948721}e^{6} - \frac{35044944410800460399}{3872440673846163}e^{5} + \frac{6234982101330996009}{286847457321938}e^{4} + \frac{154091269179315178615}{7744881347692326}e^{3} - \frac{100152951581051449399}{3872440673846163}e^{2} - \frac{176724011101316686}{11628950972511}e + \frac{5264786200397323972}{430271185982907}$ |
| 19 | $[19, 19, -\frac{2}{5}w^{3} + \frac{2}{5}w^{2} + \frac{17}{5}w]$ | $\phantom{-}\frac{72592466344199}{2581627115897442}e^{14} + \frac{5400220244053}{286847457321938}e^{13} - \frac{3452519557010195}{2581627115897442}e^{12} - \frac{1022151929061439}{860542371965814}e^{11} + \frac{64339545086795209}{2581627115897442}e^{10} + \frac{33535297821278029}{1290813557948721}e^{9} - \frac{596039643550805111}{2581627115897442}e^{8} - \frac{667857835492363405}{2581627115897442}e^{7} + \frac{967281544629592493}{860542371965814}e^{6} + \frac{1602650718043566298}{1290813557948721}e^{5} - \frac{812185692682174295}{286847457321938}e^{4} - \frac{3517522218896599276}{1290813557948721}e^{3} + \frac{8780677209836579593}{2581627115897442}e^{2} + \frac{8091715720805802}{3876316990837}e - \frac{235279111142555392}{143423728660969}$ |
| 19 | $[19, 19, w - 1]$ | $-\frac{426300903102502}{11617322021538489}e^{14} - \frac{27117851942884}{1290813557948721}e^{13} + \frac{20205344531131480}{11617322021538489}e^{12} + \frac{5405298365154365}{3872440673846163}e^{11} - \frac{375441278047922777}{11617322021538489}e^{10} - \frac{363133387267822981}{11617322021538489}e^{9} + \frac{3468836377112672881}{11617322021538489}e^{8} + \frac{3658050145142539001}{11617322021538489}e^{7} - \frac{5612489477449498087}{3872440673846163}e^{6} - \frac{17639956734930843295}{11617322021538489}e^{5} + \frac{4689921008723021362}{1290813557948721}e^{4} + \frac{38708101626869921899}{11617322021538489}e^{3} - \frac{50183067117033731168}{11617322021538489}e^{2} - \frac{29513791910455265}{11628950972511}e + \frac{2625546797415301868}{1290813557948721}$ |
| 29 | $[29, 29, w^{2} - w - 8]$ | $-\frac{736447563016444}{11617322021538489}e^{14} - \frac{76774345736873}{2581627115897442}e^{13} + \frac{69546483838795961}{23234644043076978}e^{12} + \frac{16403056471370455}{7744881347692326}e^{11} - \frac{643695585898487396}{11617322021538489}e^{10} - \frac{1134422888519817695}{23234644043076978}e^{9} + \frac{11847014489568142091}{23234644043076978}e^{8} + \frac{5792560559197375697}{11617322021538489}e^{7} - \frac{19079195779078322243}{7744881347692326}e^{6} - \frac{28092100520191646893}{11617322021538489}e^{5} + \frac{15848181898187271647}{2581627115897442}e^{4} + \frac{123406669879146122525}{23234644043076978}e^{3} - \frac{84005186014175388308}{11617322021538489}e^{2} - \frac{46900080667553374}{11628950972511}e + \frac{4307873923571437358}{1290813557948721}$ |
| 29 | $[29, 29, \frac{2}{5}w^{3} - \frac{2}{5}w^{2} - \frac{7}{5}w + 2]$ | $-\frac{555487914503357}{11617322021538489}e^{14} - \frac{43634438946842}{1290813557948721}e^{13} + \frac{52878782251143925}{23234644043076978}e^{12} + \frac{8142449746843297}{3872440673846163}e^{11} - \frac{493045443680615149}{11617322021538489}e^{10} - \frac{530870825755012481}{11617322021538489}e^{9} + \frac{4571457252327287288}{11617322021538489}e^{8} + \frac{5271227801431011415}{11617322021538489}e^{7} - \frac{7430977641166892453}{3872440673846163}e^{6} - \frac{25278976505879761046}{11617322021538489}e^{5} + \frac{12527522640501073639}{2581627115897442}e^{4} + \frac{55511031105199459931}{11617322021538489}e^{3} - \frac{136688942284368028607}{23234644043076978}e^{2} - \frac{42669312835575010}{11628950972511}e + \frac{3716954439391796302}{1290813557948721}$ |
| 31 | $[31, 31, \frac{1}{5}w^{3} - \frac{1}{5}w^{2} - \frac{1}{5}w + 2]$ | $\phantom{-}1$ |
| 31 | $[31, 31, \frac{2}{5}w^{3} - \frac{2}{5}w^{2} - \frac{17}{5}w + 3]$ | $-\frac{3067218405274447}{23234644043076978}e^{14} - \frac{93564500088770}{1290813557948721}e^{13} + \frac{145270992200397805}{23234644043076978}e^{12} + \frac{37741838644386881}{7744881347692326}e^{11} - \frac{1348361785754097523}{11617322021538489}e^{10} - \frac{2549460447152407711}{23234644043076978}e^{9} + \frac{24886354298447871271}{23234644043076978}e^{8} + \frac{12878879549205939808}{11617322021538489}e^{7} - \frac{20105345912779954340}{3872440673846163}e^{6} - \frac{124416609964952670319}{23234644043076978}e^{5} + \frac{16778481485744733218}{1290813557948721}e^{4} + \frac{136657633406778566669}{11617322021538489}e^{3} - \frac{179367600034377051529}{11617322021538489}e^{2} - \frac{104282848939258058}{11628950972511}e + \frac{9363162825626653408}{1290813557948721}$ |
| 61 | $[61, 61, -\frac{2}{5}w^{3} - \frac{3}{5}w^{2} + \frac{12}{5}w + 5]$ | $\phantom{-}\frac{841237380542636}{11617322021538489}e^{14} + \frac{104295201198037}{2581627115897442}e^{13} - \frac{39789091615163219}{11617322021538489}e^{12} - \frac{21068161417922843}{7744881347692326}e^{11} + \frac{1475358789135841451}{23234644043076978}e^{10} + \frac{712466739505028249}{11617322021538489}e^{9} - \frac{13598696200478418367}{23234644043076978}e^{8} - \frac{14421844583513905025}{23234644043076978}e^{7} + \frac{10976338079539826168}{3872440673846163}e^{6} + \frac{34933947200355208154}{11617322021538489}e^{5} - \frac{18329456191255797385}{2581627115897442}e^{4} - \frac{77138302503528911300}{11617322021538489}e^{3} + \frac{98494732510735198330}{11617322021538489}e^{2} + \frac{19809424727495381}{3876316990837}e - \frac{5233587807317762350}{1290813557948721}$ |
| 61 | $[61, 61, -\frac{4}{5}w^{3} - \frac{6}{5}w^{2} + \frac{39}{5}w + 15]$ | $-\frac{723011508765199}{23234644043076978}e^{14} - \frac{51821397028993}{2581627115897442}e^{13} + \frac{34419851101220221}{23234644043076978}e^{12} + \frac{9859525795249613}{7744881347692326}e^{11} - \frac{320989526754055207}{11617322021538489}e^{10} - \frac{648551090909368171}{23234644043076978}e^{9} + \frac{5951108754736633891}{23234644043076978}e^{8} + \frac{3233042306515665328}{11617322021538489}e^{7} - \frac{4829496967106321786}{3872440673846163}e^{6} - \frac{15527026581268288529}{11617322021538489}e^{5} + \frac{4051208323673080727}{1290813557948721}e^{4} + \frac{68175907154509735885}{23234644043076978}e^{3} - \frac{43703410822177488934}{11617322021538489}e^{2} - \frac{26107859309711980}{11628950972511}e + \frac{2335968071782417993}{1290813557948721}$ |
| 61 | $[61, 61, \frac{3}{5}w^{3} + \frac{2}{5}w^{2} - \frac{18}{5}w - 3]$ | $\phantom{-}\frac{1144202317611508}{11617322021538489}e^{14} + \frac{138027939082853}{2581627115897442}e^{13} - \frac{108395237149214825}{23234644043076978}e^{12} - \frac{27917539865276209}{7744881347692326}e^{11} + \frac{2012502355175758375}{23234644043076978}e^{10} + \frac{944021694760915681}{11617322021538489}e^{9} - \frac{18576692898513608387}{23234644043076978}e^{8} - \frac{19086994690023496723}{23234644043076978}e^{7} + \frac{15013856646134400976}{3872440673846163}e^{6} + \frac{46123983403563899299}{11617322021538489}e^{5} - \frac{12538096019509716817}{1290813557948721}e^{4} - \frac{101442417052268635420}{11617322021538489}e^{3} + \frac{268403537665310232643}{23234644043076978}e^{2} + \frac{77626817202780478}{11628950972511}e - \frac{7020254848843972259}{1290813557948721}$ |
| 61 | $[61, 61, \frac{7}{5}w^{3} + \frac{3}{5}w^{2} - \frac{62}{5}w - 13]$ | $\phantom{-}\frac{1000056263603696}{11617322021538489}e^{14} + \frac{61256637724106}{1290813557948721}e^{13} - \frac{47361804375814604}{11617322021538489}e^{12} - \frac{12382929353169343}{3872440673846163}e^{11} + \frac{879325230975754270}{11617322021538489}e^{10} + \frac{838211211870505586}{11617322021538489}e^{9} - \frac{8118523453595747759}{11617322021538489}e^{8} - \frac{8491296852440525638}{11617322021538489}e^{7} + \frac{13131273811483171991}{3872440673846163}e^{6} + \frac{41172058210694454239}{11617322021538489}e^{5} - \frac{10983449643342357089}{1290813557948721}e^{4} - \frac{90926855043709196786}{11617322021538489}e^{3} + \frac{118094969502384716521}{11617322021538489}e^{2} + \frac{69884730575463332}{11628950972511}e - \frac{6253762713206412988}{1290813557948721}$ |
| 71 | $[71, 71, \frac{1}{5}w^{3} - \frac{6}{5}w^{2} - \frac{6}{5}w + 4]$ | $\phantom{-}\frac{1072585709977165}{23234644043076978}e^{14} + \frac{33079975972159}{2581627115897442}e^{13} - \frac{50288172033092029}{23234644043076978}e^{12} - \frac{9009024547097099}{7744881347692326}e^{11} + \frac{924932001269448647}{23234644043076978}e^{10} + \frac{339316181485033949}{11617322021538489}e^{9} - \frac{8458746942978524455}{23234644043076978}e^{8} - \frac{7223187812898387503}{23234644043076978}e^{7} + \frac{13528003373464843471}{7744881347692326}e^{6} + \frac{17916797674050270872}{11617322021538489}e^{5} - \frac{11127744879192335407}{2581627115897442}e^{4} - \frac{39881124198291906815}{11617322021538489}e^{3} + \frac{115984582299747123653}{23234644043076978}e^{2} + \frac{30501983589118316}{11628950972511}e - \frac{2884225215891903292}{1290813557948721}$ |
| 71 | $[71, 71, w^{2} - 8]$ | $\phantom{-}\frac{77567321270846}{3872440673846163}e^{14} + \frac{2381729796393}{286847457321938}e^{13} - \frac{7362617799204733}{7744881347692326}e^{12} - \frac{783389220246244}{1290813557948721}e^{11} + \frac{137099526186167381}{7744881347692326}e^{10} + \frac{109418310481852453}{7744881347692326}e^{9} - \frac{635204734120909892}{3872440673846163}e^{8} - \frac{1121106563358154781}{7744881347692326}e^{7} + \frac{2060983995080139727}{2581627115897442}e^{6} + \frac{5430527226709622701}{7744881347692326}e^{5} - \frac{859540465653969523}{430271185982907}e^{4} - \frac{5926179307600406114}{3872440673846163}e^{3} + \frac{9012174988871362903}{3872440673846163}e^{2} + \frac{13289089729906376}{11628950972511}e - \frac{434351498130198451}{430271185982907}$ |
| 79 | $[79, 79, \frac{3}{5}w^{3} + \frac{12}{5}w^{2} - \frac{33}{5}w - 22]$ | $-\frac{4183334997114709}{11617322021538489}e^{14} - \frac{508666263329243}{2581627115897442}e^{13} + \frac{197990807586042547}{11617322021538489}e^{12} + \frac{51476816503354220}{3872440673846163}e^{11} - \frac{3672730732735165343}{11617322021538489}e^{10} - \frac{3483087576841184644}{11617322021538489}e^{9} + \frac{33867495335171649880}{11617322021538489}e^{8} + \frac{35232146680775187887}{11617322021538489}e^{7} - \frac{54681607383427602907}{3872440673846163}e^{6} - \frac{340800601577010874199}{23234644043076978}e^{5} + \frac{45620217599251746406}{1290813557948721}e^{4} + \frac{750137601785931151811}{23234644043076978}e^{3} - \frac{488504224694884222088}{11617322021538489}e^{2} - \frac{95752443050607443}{3876316990837}e + \frac{25680512037483476666}{1290813557948721}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31, 31, \frac{1}{5}w^{3} - \frac{1}{5}w^{2} - \frac{1}{5}w + 2]$ | $-1$ |