Base field 4.4.16357.1
Generator \(w\), with minimal polynomial \(x^{4} - 6x^{2} - x + 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{3} + w^{2} + 6w - 3]$ |
| Dimension: | $13$ |
| 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:
| \(x^{13} + 4x^{12} - 21x^{11} - 97x^{10} + 126x^{9} + 825x^{8} - 63x^{7} - 3017x^{6} - 1411x^{5} + 4507x^{4} + 3170x^{3} - 2012x^{2} - 1110x + 473\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -w^{3} + 5w + 2]$ | $-\frac{487723}{1969125}e^{12} - \frac{73193}{131275}e^{11} + \frac{45639}{7375}e^{10} + \frac{442436}{33375}e^{9} - \frac{107616409}{1969125}e^{8} - \frac{216568049}{1969125}e^{7} + \frac{82611067}{393825}e^{6} + \frac{255682417}{656375}e^{5} - \frac{661787936}{1969125}e^{4} - \frac{1092353282}{1969125}e^{3} + \frac{122034896}{656375}e^{2} + \frac{389867444}{1969125}e - \frac{138187286}{1969125}$ |
| 5 | $[5, 5, w - 1]$ | $\phantom{-}\frac{547997}{1969125}e^{12} + \frac{85987}{131275}e^{11} - \frac{51246}{7375}e^{10} - \frac{521854}{33375}e^{9} + \frac{120883901}{1969125}e^{8} + \frac{256743811}{1969125}e^{7} - \frac{93135998}{393825}e^{6} - \frac{305065938}{656375}e^{5} + \frac{757579129}{1969125}e^{4} + \frac{1314691273}{1969125}e^{3} - \frac{149060569}{656375}e^{2} - \frac{481598191}{1969125}e + \frac{178533829}{1969125}$ |
| 11 | $[11, 11, -w^{3} + 5w]$ | $\phantom{-}\frac{267727}{1969125}e^{12} + \frac{50727}{131275}e^{11} - \frac{23911}{7375}e^{10} - \frac{308114}{33375}e^{9} + \frac{52194466}{1969125}e^{8} + \frac{151186376}{1969125}e^{7} - \frac{34809073}{393825}e^{6} - \frac{177473808}{656375}e^{5} + \frac{197947439}{1969125}e^{4} + \frac{731945693}{1969125}e^{3} - \frac{2383779}{656375}e^{2} - \frac{217688606}{1969125}e + \frac{38780564}{1969125}$ |
| 13 | $[13, 13, -w^{3} + w^{2} + 6w - 3]$ | $\phantom{-}1$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{410096}{1969125}e^{12} + \frac{64991}{131275}e^{11} - \frac{38828}{7375}e^{10} - \frac{395722}{33375}e^{9} + \frac{93602393}{1969125}e^{8} + \frac{195791698}{1969125}e^{7} - \frac{75019679}{393825}e^{6} - \frac{235656059}{656375}e^{5} + \frac{662494222}{1969125}e^{4} + \frac{1054391689}{1969125}e^{3} - \frac{158276867}{656375}e^{2} - \frac{448612888}{1969125}e + \frac{182852572}{1969125}$ |
| 19 | $[19, 19, 2w^{3} - w^{2} - 11w + 2]$ | $-\frac{98821}{656375}e^{12} - \frac{38198}{131275}e^{11} + \frac{28734}{7375}e^{10} + \frac{76597}{11125}e^{9} - \frac{23927418}{656375}e^{8} - \frac{37395273}{656375}e^{7} + \frac{20289369}{131275}e^{6} + \frac{133555227}{656375}e^{5} - \frac{196845147}{656375}e^{4} - \frac{200292389}{656375}e^{3} + \frac{159934051}{656375}e^{2} + \frac{91220563}{656375}e - \frac{50908747}{656375}$ |
| 25 | $[25, 5, -w^{2} + w + 3]$ | $-\frac{75836}{656375}e^{12} - \frac{34723}{131275}e^{11} + \frac{20394}{7375}e^{10} + \frac{68427}{11125}e^{9} - \frac{14716538}{656375}e^{8} - \frac{32202393}{656375}e^{7} + \frac{9266469}{131275}e^{6} + \frac{105754532}{656375}e^{5} - \frac{35817102}{656375}e^{4} - \frac{126070074}{656375}e^{3} - \frac{39721959}{656375}e^{2} + \frac{15579083}{656375}e + \frac{10737373}{656375}$ |
| 27 | $[27, 3, w^{3} - w^{2} - 5w + 4]$ | $\phantom{-}\frac{257974}{1969125}e^{12} + \frac{44009}{131275}e^{11} - \frac{24532}{7375}e^{10} - \frac{267443}{33375}e^{9} + \frac{59515042}{1969125}e^{8} + \frac{131419862}{1969125}e^{7} - \frac{47945821}{393825}e^{6} - \frac{155794221}{656375}e^{5} + \frac{420191318}{1969125}e^{4} + \frac{675986591}{1969125}e^{3} - \frac{94940173}{656375}e^{2} - \frac{263410247}{1969125}e + \frac{95959118}{1969125}$ |
| 31 | $[31, 31, -w - 3]$ | $-\frac{15512}{78765}e^{12} - \frac{14748}{26255}e^{11} + \frac{1441}{295}e^{10} + \frac{17977}{1335}e^{9} - \frac{675424}{15753}e^{8} - \frac{1773233}{15753}e^{7} + \frac{2585543}{15753}e^{6} + \frac{10554326}{26255}e^{5} - \frac{20922154}{78765}e^{4} - \frac{45905308}{78765}e^{3} + \frac{857573}{5251}e^{2} + \frac{17845777}{78765}e - \frac{6349987}{78765}$ |
| 37 | $[37, 37, -w^{3} - w^{2} + 6w + 4]$ | $-\frac{810922}{1969125}e^{12} - \frac{130857}{131275}e^{11} + \frac{75921}{7375}e^{10} + \frac{797654}{33375}e^{9} - \frac{179701051}{1969125}e^{8} - \frac{394992686}{1969125}e^{7} + \frac{139749703}{393825}e^{6} + \frac{474216513}{656375}e^{5} - \frac{1167647804}{1969125}e^{4} - \frac{2085520073}{1969125}e^{3} + \frac{248151019}{656375}e^{2} + \frac{820424291}{1969125}e - \frac{299383979}{1969125}$ |
| 41 | $[41, 41, w^{3} + w^{2} - 6w - 5]$ | $\phantom{-}\frac{7432}{656375}e^{12} - \frac{8749}{131275}e^{11} - \frac{2803}{7375}e^{10} + \frac{17901}{11125}e^{9} + \frac{3058131}{656375}e^{8} - \frac{8620809}{656375}e^{7} - \frac{3460368}{131275}e^{6} + \frac{28202091}{656375}e^{5} + \frac{46480124}{656375}e^{4} - \frac{31080337}{656375}e^{3} - \frac{51736617}{656375}e^{2} - \frac{822821}{656375}e + \frac{9347049}{656375}$ |
| 43 | $[43, 43, w^{3} - 7w - 2]$ | $\phantom{-}\frac{1394857}{1969125}e^{12} + \frac{219037}{131275}e^{11} - \frac{129551}{7375}e^{10} - \frac{1329374}{33375}e^{9} + \frac{302272231}{1969125}e^{8} + \frac{654553241}{1969125}e^{7} - \frac{228900778}{393825}e^{6} - \frac{779306403}{656375}e^{5} + \frac{1806796499}{1969125}e^{4} + \frac{3371527688}{1969125}e^{3} - \frac{333832139}{656375}e^{2} - \frac{1252093796}{1969125}e + \frac{418342649}{1969125}$ |
| 47 | $[47, 47, -2w^{3} + 11w + 2]$ | $\phantom{-}\frac{1295656}{1969125}e^{12} + \frac{212996}{131275}e^{11} - \frac{119808}{7375}e^{10} - \frac{1295717}{33375}e^{9} + \frac{277495123}{1969125}e^{8} + \frac{639470453}{1969125}e^{7} - \frac{207249274}{393825}e^{6} - \frac{762767799}{656375}e^{5} + \frac{1585600592}{1969125}e^{4} + \frac{3300769454}{1969125}e^{3} - \frac{272418212}{656375}e^{2} - \frac{1219579268}{1969125}e + \frac{380843792}{1969125}$ |
| 61 | $[61, 61, w^{2} - 3]$ | $\phantom{-}\frac{154057}{1969125}e^{12} + \frac{20417}{131275}e^{11} - \frac{14651}{7375}e^{10} - \frac{116549}{33375}e^{9} + \frac{35315281}{1969125}e^{8} + \frac{51837791}{1969125}e^{7} - \frac{27641428}{393825}e^{6} - \frac{52384353}{656375}e^{5} + \frac{214573274}{1969125}e^{4} + \frac{172971863}{1969125}e^{3} - \frac{24853214}{656375}e^{2} - \frac{22202321}{1969125}e - \frac{9715426}{1969125}$ |
| 67 | $[67, 67, w^{2} + w - 4]$ | $-\frac{450223}{656375}e^{12} - \frac{227394}{131275}e^{11} + \frac{124167}{7375}e^{10} + \frac{461336}{11125}e^{9} - \frac{94944109}{656375}e^{8} - \frac{227802999}{656375}e^{7} + \frac{69688317}{131275}e^{6} + \frac{815191576}{656375}e^{5} - \frac{512295061}{656375}e^{4} - \frac{1171605532}{656375}e^{3} + \frac{233080588}{656375}e^{2} + \frac{422416244}{656375}e - \frac{122070686}{656375}$ |
| 79 | $[79, 79, -4w^{3} + 2w^{2} + 22w - 9]$ | $\phantom{-}\frac{359272}{1969125}e^{12} + \frac{59207}{131275}e^{11} - \frac{31696}{7375}e^{10} - \frac{356504}{33375}e^{9} + \frac{67016026}{1969125}e^{8} + \frac{173039936}{1969125}e^{7} - \frac{40797898}{393825}e^{6} - \frac{199416513}{656375}e^{5} + \frac{145939304}{1969125}e^{4} + \frac{783159623}{1969125}e^{3} + \frac{61966806}{656375}e^{2} - \frac{174596591}{1969125}e - \frac{41198896}{1969125}$ |
| 97 | $[97, 97, -3w^{3} + 2w^{2} + 19w - 6]$ | $\phantom{-}\frac{676401}{656375}e^{12} + \frac{336188}{131275}e^{11} - \frac{188779}{7375}e^{10} - \frac{678957}{11125}e^{9} + \frac{147146908}{656375}e^{8} + \frac{332731213}{656375}e^{7} - \frac{111374429}{131275}e^{6} - \frac{1178330287}{656375}e^{5} + \frac{865200432}{656375}e^{4} + \frac{1677557659}{656375}e^{3} - \frac{446160881}{656375}e^{2} - \frac{602095178}{656375}e + \frac{187859157}{656375}$ |
| 97 | $[97, 97, -w^{3} + 6w - 3]$ | $\phantom{-}\frac{16841}{78765}e^{12} + \frac{9592}{26255}e^{11} - \frac{1699}{295}e^{10} - \frac{11713}{1335}e^{9} + \frac{4501967}{78765}e^{8} + \frac{5872051}{78765}e^{7} - \frac{20782111}{78765}e^{6} - \frac{7330987}{26255}e^{5} + \frac{9120335}{15753}e^{4} + \frac{36058903}{78765}e^{3} - \frac{14371012}{26255}e^{2} - \frac{19745557}{78765}e + \frac{12611647}{78765}$ |
| 97 | $[97, 97, -3w^{3} + 16w]$ | $\phantom{-}\frac{1252639}{1969125}e^{12} + \frac{214159}{131275}e^{11} - \frac{113702}{7375}e^{10} - \frac{1298573}{33375}e^{9} + \frac{254760487}{1969125}e^{8} + \frac{636994682}{1969125}e^{7} - \frac{178231771}{393825}e^{6} - \frac{750408506}{656375}e^{5} + \frac{1155267698}{1969125}e^{4} + \frac{3146517251}{1969125}e^{3} - \frac{97096928}{656375}e^{2} - \frac{1020430217}{1969125}e + \frac{222139673}{1969125}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{3} + w^{2} + 6w - 3]$ | $-1$ |