Base field 3.3.1937.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 8x - 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[13, 13, -w + 2]$ |
| Dimension: | $18$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $44$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{18} - 24x^{16} + 231x^{14} - 1151x^{12} + 3216x^{10} - 5122x^{8} + 4480x^{6} - 1904x^{4} + 280x^{2} - 8\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w - 2]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w + 1]$ | $\phantom{-}\frac{843}{2668}e^{17} - \frac{18841}{2668}e^{15} + \frac{163539}{2668}e^{13} - \frac{174533}{667}e^{11} + \frac{1539713}{2668}e^{9} - \frac{424861}{667}e^{7} + \frac{411245}{1334}e^{5} - \frac{33431}{667}e^{3} + \frac{6575}{667}e$ |
| 7 | $[7, 7, w - 3]$ | $\phantom{-}\frac{465}{2668}e^{16} - \frac{5283}{1334}e^{14} + \frac{93935}{2668}e^{12} - \frac{415737}{2668}e^{10} + \frac{484913}{1334}e^{8} - \frac{584303}{1334}e^{6} + \frac{161181}{667}e^{4} - \frac{25488}{667}e^{2} - \frac{2324}{667}$ |
| 8 | $[8, 2, 2]$ | $-\frac{1033}{2668}e^{16} + \frac{11719}{1334}e^{14} - \frac{208089}{2668}e^{12} + \frac{920653}{2668}e^{10} - \frac{538972}{667}e^{8} + \frac{661763}{667}e^{6} - \frac{394005}{667}e^{4} + \frac{91934}{667}e^{2} - \frac{4191}{667}$ |
| 9 | $[9, 3, w^{2} - 3w - 2]$ | $\phantom{-}\frac{641}{1334}e^{17} - \frac{30133}{2668}e^{15} + \frac{140749}{1334}e^{13} - \frac{1342629}{2668}e^{11} + \frac{3516327}{2668}e^{9} - \frac{1271944}{667}e^{7} + \frac{1924109}{1334}e^{5} - \frac{323941}{667}e^{3} + \frac{30513}{667}e$ |
| 13 | $[13, 13, w + 3]$ | $-\frac{1389}{2668}e^{16} + \frac{15471}{1334}e^{14} - \frac{267339}{2668}e^{12} + \frac{1133457}{2668}e^{10} - \frac{1235765}{1334}e^{8} + \frac{1330487}{1334}e^{6} - \frac{292642}{667}e^{4} + \frac{18076}{667}e^{2} + \frac{1322}{667}$ |
| 13 | $[13, 13, -w + 2]$ | $-1$ |
| 19 | $[19, 19, -w^{2} + 2w + 4]$ | $-\frac{1729}{2668}e^{17} + \frac{39765}{2668}e^{15} - \frac{360049}{2668}e^{13} + \frac{819941}{1334}e^{11} - \frac{4007393}{2668}e^{9} + \frac{1304983}{667}e^{7} - \frac{1670683}{1334}e^{5} + \frac{210950}{667}e^{3} - \frac{13790}{667}e$ |
| 23 | $[23, 23, w^{2} - 4w + 1]$ | $\phantom{-}\frac{1163}{2668}e^{16} - \frac{13239}{1334}e^{14} + \frac{236043}{2668}e^{12} - \frac{1048155}{2668}e^{10} + \frac{612486}{667}e^{8} - \frac{732194}{667}e^{6} + \frac{384566}{667}e^{4} - \frac{48131}{667}e^{2} - \frac{2390}{667}$ |
| 25 | $[25, 5, w^{2} - 2w - 1]$ | $-\frac{27}{116}e^{17} + \frac{140}{29}e^{15} - \frac{4255}{116}e^{13} + \frac{13673}{116}e^{11} - \frac{5589}{58}e^{9} - \frac{8363}{29}e^{7} + \frac{19283}{29}e^{5} - \frac{12319}{29}e^{3} + \frac{1442}{29}e$ |
| 31 | $[31, 31, w^{2} - 2w - 9]$ | $-\frac{2101}{2668}e^{17} + \frac{49285}{2668}e^{15} - \frac{459209}{2668}e^{13} + \frac{1091355}{1334}e^{11} - \frac{5691441}{2668}e^{9} + \frac{2047892}{667}e^{7} - \frac{3076613}{1334}e^{5} + \frac{509346}{667}e^{3} - \frac{45014}{667}e$ |
| 37 | $[37, 37, -w^{2} + 3w + 3]$ | $\phantom{-}\frac{9}{1334}e^{17} - \frac{467}{1334}e^{15} + \frac{3846}{667}e^{13} - \frac{29621}{667}e^{11} + \frac{119690}{667}e^{9} - \frac{518225}{1334}e^{7} + \frac{288561}{667}e^{5} - \frac{144482}{667}e^{3} + \frac{25071}{667}e$ |
| 41 | $[41, 41, w^{2} - w - 1]$ | $\phantom{-}\frac{1089}{1334}e^{17} - \frac{24491}{1334}e^{15} + \frac{107187}{667}e^{13} - \frac{462581}{667}e^{11} + \frac{1031768}{667}e^{9} - \frac{2269689}{1334}e^{7} + \frac{490677}{667}e^{5} - \frac{2253}{667}e^{3} - \frac{10597}{667}e$ |
| 41 | $[41, 41, w^{2} - w - 5]$ | $-\frac{370}{667}e^{17} + \frac{16535}{1334}e^{15} - \frac{71660}{667}e^{13} + \frac{608953}{1334}e^{11} - \frac{1323477}{1334}e^{9} + \frac{694802}{667}e^{7} - \frac{266625}{667}e^{5} - \frac{25726}{667}e^{3} + \frac{17201}{667}e$ |
| 41 | $[41, 41, w^{2} - w - 10]$ | $-\frac{129}{667}e^{16} + \frac{2914}{667}e^{14} - \frac{25543}{667}e^{12} + \frac{109210}{667}e^{10} - \frac{232626}{667}e^{8} + \frac{211174}{667}e^{6} - \frac{8619}{667}e^{4} - \frac{69176}{667}e^{2} + \frac{6994}{667}$ |
| 43 | $[43, 43, -2w - 3]$ | $\phantom{-}\frac{284}{667}e^{17} - \frac{12205}{1334}e^{15} + \frac{49740}{667}e^{13} - \frac{380187}{1334}e^{11} + \frac{667803}{1334}e^{9} - \frac{184946}{667}e^{7} - \frac{133318}{667}e^{5} + \frac{156586}{667}e^{3} - \frac{26990}{667}e$ |
| 47 | $[47, 47, -w^{2} - w + 4]$ | $\phantom{-}\frac{777}{2668}e^{16} - \frac{8931}{1334}e^{14} + \frac{160491}{2668}e^{12} - \frac{712137}{2668}e^{10} + \frac{808699}{1334}e^{8} - \frac{862075}{1334}e^{6} + \frac{143063}{667}e^{4} + \frac{30548}{667}e^{2} - \frac{2644}{667}$ |
| 49 | $[49, 7, -w^{2} + 5w - 5]$ | $-\frac{429}{2668}e^{16} + \frac{2508}{667}e^{14} - \frac{92515}{2668}e^{12} + \frac{428227}{2668}e^{10} - \frac{262206}{667}e^{8} + \frac{655073}{1334}e^{6} - \frac{177022}{667}e^{4} + \frac{17331}{667}e^{2} + \frac{3108}{667}$ |
| 59 | $[59, 59, w^{2} - w - 4]$ | $-\frac{829}{1334}e^{17} + \frac{18411}{1334}e^{15} - \frac{157799}{1334}e^{13} + \frac{327742}{667}e^{11} - \frac{1353363}{1334}e^{9} + \frac{616669}{667}e^{7} - \frac{108223}{667}e^{5} - \frac{100674}{667}e^{3} + \frac{8952}{667}e$ |
| 59 | $[59, 59, w^{2} - 3]$ | $-\frac{49}{667}e^{17} + \frac{2685}{2668}e^{15} - \frac{80}{667}e^{13} - \frac{162641}{2668}e^{11} + \frac{1016245}{2668}e^{9} - \frac{650436}{667}e^{7} + \frac{1539915}{1334}e^{5} - \frac{378023}{667}e^{3} + \frac{39383}{667}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w + 2]$ | $1$ |