Base field 3.3.1016.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 6x + 2\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[13, 13, 2w^{2} - 2w - 11]$ |
| Dimension: | $14$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $28$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{14} - 8x^{13} + 11x^{12} + 62x^{11} - 171x^{10} - 118x^{9} + 672x^{8} - 90x^{7} - 1090x^{6} + 365x^{5} + 811x^{4} - 223x^{3} - 257x^{2} + 14x + 17\) |
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| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}e$ |
| 2 | $[2, 2, w - 3]$ | $\phantom{-}\frac{1}{2}e^{13} - \frac{7}{2}e^{12} + 2e^{11} + 33e^{10} - \frac{105}{2}e^{9} - \frac{223}{2}e^{8} + \frac{451}{2}e^{7} + \frac{351}{2}e^{6} - \frac{737}{2}e^{5} - 161e^{4} + \frac{443}{2}e^{3} + 82e^{2} - \frac{43}{2}e - \frac{13}{2}$ |
| 3 | $[3, 3, w^{2} - w - 5]$ | $\phantom{-}\frac{1}{2}e^{13} - \frac{7}{2}e^{12} + 2e^{11} + 33e^{10} - \frac{107}{2}e^{9} - \frac{213}{2}e^{8} + \frac{451}{2}e^{7} + \frac{293}{2}e^{6} - \frac{695}{2}e^{5} - 113e^{4} + \frac{379}{2}e^{3} + 52e^{2} - \frac{25}{2}e - \frac{1}{2}$ |
| 9 | $[9, 3, w^{2} + w - 1]$ | $-2e^{13} + 14e^{12} - 8e^{11} - 131e^{10} + 205e^{9} + 444e^{8} - 862e^{7} - 728e^{6} + 1371e^{5} + 737e^{4} - 793e^{3} - 400e^{2} + 61e + 30$ |
| 13 | $[13, 13, 2w^{2} - 2w - 11]$ | $\phantom{-}1$ |
| 29 | $[29, 29, -w^{2} - 3w - 1]$ | $\phantom{-}\frac{1}{2}e^{13} - \frac{9}{2}e^{12} + 8e^{11} + 33e^{10} - \frac{215}{2}e^{9} - \frac{115}{2}e^{8} + \frac{805}{2}e^{7} - \frac{79}{2}e^{6} - \frac{1221}{2}e^{5} + 109e^{4} + \frac{729}{2}e^{3} - 23e^{2} - \frac{103}{2}e - \frac{3}{2}$ |
| 29 | $[29, 29, -w^{2} + w + 3]$ | $\phantom{-}4e^{13} - 27e^{12} + 11e^{11} + 257e^{10} - 359e^{9} - 895e^{8} + 1534e^{7} + 1514e^{6} - 2432e^{5} - 1525e^{4} + 1389e^{3} + 798e^{2} - 103e - 60$ |
| 29 | $[29, 29, 2w - 5]$ | $\phantom{-}\frac{3}{2}e^{13} - \frac{21}{2}e^{12} + 7e^{11} + 94e^{10} - \frac{327}{2}e^{9} - \frac{553}{2}e^{8} + \frac{1315}{2}e^{7} + \frac{631}{2}e^{6} - \frac{1931}{2}e^{5} - 205e^{4} + \frac{1017}{2}e^{3} + 113e^{2} - \frac{69}{2}e - \frac{11}{2}$ |
| 31 | $[31, 31, -w^{2} + w + 1]$ | $-\frac{5}{2}e^{13} + \frac{33}{2}e^{12} - 4e^{11} - 165e^{10} + \frac{417}{2}e^{9} + \frac{1209}{2}e^{8} - \frac{1877}{2}e^{7} - \frac{2143}{2}e^{6} + \frac{3037}{2}e^{5} + 1106e^{4} - \frac{1717}{2}e^{3} - 574e^{2} + \frac{85}{2}e + \frac{83}{2}$ |
| 37 | $[37, 37, -2w - 1]$ | $\phantom{-}\frac{3}{2}e^{13} - \frac{23}{2}e^{12} + 13e^{11} + 93e^{10} - \frac{421}{2}e^{9} - \frac{473}{2}e^{8} + \frac{1605}{2}e^{7} + \frac{401}{2}e^{6} - \frac{2401}{2}e^{5} - 83e^{4} + \frac{1335}{2}e^{3} + 58e^{2} - \frac{107}{2}e + \frac{11}{2}$ |
| 43 | $[43, 43, -2w - 3]$ | $\phantom{-}2e^{13} - 13e^{12} + 2e^{11} + 132e^{10} - 155e^{9} - 499e^{8} + 719e^{7} + 920e^{6} - 1198e^{5} - 951e^{4} + 690e^{3} + 484e^{2} - 32e - 41$ |
| 47 | $[47, 47, 2w - 3]$ | $-3e^{13} + 20e^{12} - 5e^{11} - 202e^{10} + 250e^{9} + 772e^{8} - 1142e^{7} - 1487e^{6} + 1919e^{5} + 1654e^{4} - 1136e^{3} - 880e^{2} + 63e + 65$ |
| 59 | $[59, 59, 4w^{2} + 8w - 1]$ | $\phantom{-}e^{11} - 6e^{10} + 3e^{9} + 39e^{8} - 51e^{7} - 83e^{6} + 131e^{5} + 80e^{4} - 130e^{3} - 29e^{2} + 48e + 1$ |
| 61 | $[61, 61, -w^{2} - w - 1]$ | $-\frac{3}{2}e^{13} + \frac{19}{2}e^{12} + e^{11} - 105e^{10} + \frac{209}{2}e^{9} + \frac{859}{2}e^{8} - \frac{1069}{2}e^{7} - \frac{1711}{2}e^{6} + \frac{1841}{2}e^{5} + 955e^{4} - \frac{1043}{2}e^{3} - 513e^{2} + \frac{9}{2}e + \frac{85}{2}$ |
| 67 | $[67, 67, w^{2} + w - 5]$ | $\phantom{-}\frac{1}{2}e^{13} - \frac{7}{2}e^{12} + 2e^{11} + 33e^{10} - \frac{107}{2}e^{9} - \frac{215}{2}e^{8} + \frac{461}{2}e^{7} + \frac{287}{2}e^{6} - \frac{733}{2}e^{5} - 80e^{4} + \frac{375}{2}e^{3} + 14e^{2} + \frac{9}{2}e + \frac{19}{2}$ |
| 71 | $[71, 71, w^{2} - w - 9]$ | $\phantom{-}\frac{7}{2}e^{13} - \frac{49}{2}e^{12} + 15e^{11} + 223e^{10} - \frac{711}{2}e^{9} - \frac{1467}{2}e^{8} + \frac{2907}{2}e^{7} + \frac{2333}{2}e^{6} - \frac{4513}{2}e^{5} - 1157e^{4} + \frac{2521}{2}e^{3} + 631e^{2} - \frac{151}{2}e - \frac{89}{2}$ |
| 73 | $[73, 73, -2w^{2} + 2w + 9]$ | $\phantom{-}5e^{13} - 35e^{12} + 21e^{11} + 321e^{10} - 509e^{9} - 1059e^{8} + 2081e^{7} + 1687e^{6} - 3186e^{5} - 1711e^{4} + 1720e^{3} + 963e^{2} - 70e - 70$ |
| 73 | $[73, 73, 6w^{2} + 10w - 7]$ | $-e^{13} + 8e^{12} - 12e^{11} - 54e^{10} + 155e^{9} + 87e^{8} - 530e^{7} + 52e^{6} + 734e^{5} - 149e^{4} - 403e^{3} + 52e^{2} + 53e - 10$ |
| 73 | $[73, 73, 3w^{2} - 3w - 17]$ | $-\frac{3}{2}e^{13} + \frac{21}{2}e^{12} - 7e^{11} - 93e^{10} + \frac{315}{2}e^{9} + \frac{559}{2}e^{8} - \frac{1239}{2}e^{7} - \frac{729}{2}e^{6} + \frac{1785}{2}e^{5} + 318e^{4} - \frac{915}{2}e^{3} - 196e^{2} + \frac{51}{2}e + \frac{43}{2}$ |
| 79 | $[79, 79, -2w^{2} + 2w + 15]$ | $\phantom{-}\frac{9}{2}e^{13} - \frac{63}{2}e^{12} + 18e^{11} + 297e^{10} - \frac{947}{2}e^{9} - \frac{1987}{2}e^{8} + \frac{4003}{2}e^{7} + \frac{3131}{2}e^{6} - \frac{6319}{2}e^{5} - 1564e^{4} + \frac{3641}{2}e^{3} + 873e^{2} - \frac{309}{2}e - \frac{131}{2}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, 2w^{2} - 2w - 11]$ | $-1$ |