Base field 5.5.186037.1
Generator \(w\), with minimal polynomial \(x^{5} - x^{4} - 6x^{3} + 2x^{2} + 5x - 2\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{4} + w^{3} + 6w^{2} - w - 3]$ |
| 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:
| \(x^{13} - x^{12} - 16x^{11} + 14x^{10} + 94x^{9} - 70x^{8} - 248x^{7} + 148x^{6} + 289x^{5} - 123x^{4} - 124x^{3} + 34x^{2} + 8x - 2\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w^{4} - w^{3} - 6w^{2} + w + 5]$ | $\phantom{-}\frac{9}{2}e^{12} - \frac{7}{2}e^{11} - \frac{145}{2}e^{10} + \frac{93}{2}e^{9} + \frac{859}{2}e^{8} - \frac{429}{2}e^{7} - \frac{2291}{2}e^{6} + \frac{777}{2}e^{5} + 1353e^{4} - 211e^{3} - 585e^{2} - 2e + 34$ |
| 13 | $[13, 13, -w^{4} + w^{3} + 6w^{2} - w - 3]$ | $-1$ |
| 16 | $[16, 2, w^{4} - w^{3} - 6w^{2} + 2w + 5]$ | $\phantom{-}\frac{1}{2}e^{11} - \frac{3}{2}e^{10} - \frac{11}{2}e^{9} + \frac{35}{2}e^{8} + \frac{41}{2}e^{7} - \frac{141}{2}e^{6} - \frac{63}{2}e^{5} + \frac{233}{2}e^{4} + 20e^{3} - 68e^{2} - 6e + 5$ |
| 19 | $[19, 19, -w^{4} + 7w^{2} + 2w - 3]$ | $\phantom{-}\frac{3}{2}e^{12} - \frac{1}{2}e^{11} - \frac{51}{2}e^{10} + \frac{15}{2}e^{9} + \frac{315}{2}e^{8} - \frac{75}{2}e^{7} - \frac{865}{2}e^{6} + \frac{133}{2}e^{5} + 519e^{4} - 22e^{3} - 225e^{2} - 10e + 15$ |
| 23 | $[23, 23, w^{3} - w^{2} - 4w + 1]$ | $-7e^{12} + 5e^{11} + 114e^{10} - 68e^{9} - 681e^{8} + 322e^{7} + 1826e^{6} - 603e^{5} - 2161e^{4} + 355e^{3} + 932e^{2} - 17e - 55$ |
| 23 | $[23, 23, -w^{4} + 7w^{2} + 4w - 7]$ | $-\frac{1}{2}e^{12} + \frac{1}{2}e^{11} + \frac{17}{2}e^{10} - \frac{15}{2}e^{9} - \frac{107}{2}e^{8} + \frac{79}{2}e^{7} + \frac{309}{2}e^{6} - \frac{171}{2}e^{5} - 207e^{4} + 68e^{3} + 112e^{2} - 13e - 12$ |
| 25 | $[25, 5, -w^{2} + w + 3]$ | $\phantom{-}\frac{3}{2}e^{12} - \frac{3}{2}e^{11} - \frac{47}{2}e^{10} + \frac{39}{2}e^{9} + \frac{273}{2}e^{8} - \frac{179}{2}e^{7} - \frac{723}{2}e^{6} + \frac{341}{2}e^{5} + 432e^{4} - 121e^{3} - 194e^{2} + 24e + 11$ |
| 31 | $[31, 31, w^{2} - w - 1]$ | $\phantom{-}\frac{7}{2}e^{12} - \frac{5}{2}e^{11} - \frac{115}{2}e^{10} + \frac{69}{2}e^{9} + \frac{695}{2}e^{8} - \frac{333}{2}e^{7} - \frac{1895}{2}e^{6} + \frac{643}{2}e^{5} + 1150e^{4} - 204e^{3} - 514e^{2} + 15e + 32$ |
| 31 | $[31, 31, w^{4} - 6w^{2} - 2w + 3]$ | $-\frac{13}{2}e^{12} + \frac{9}{2}e^{11} + \frac{211}{2}e^{10} - \frac{117}{2}e^{9} - \frac{1265}{2}e^{8} + \frac{527}{2}e^{7} + \frac{3437}{2}e^{6} - \frac{927}{2}e^{5} - 2083e^{4} + 238e^{3} + 921e^{2} + 6e - 53$ |
| 31 | $[31, 31, w^{3} - 2w^{2} - 4w + 3]$ | $-\frac{3}{2}e^{12} + \frac{3}{2}e^{11} + \frac{47}{2}e^{10} - \frac{41}{2}e^{9} - \frac{267}{2}e^{8} + \frac{195}{2}e^{7} + \frac{667}{2}e^{6} - \frac{367}{2}e^{5} - 356e^{4} + 109e^{3} + 136e^{2} - 6e - 8$ |
| 67 | $[67, 67, 2w^{3} - 3w^{2} - 8w + 3]$ | $\phantom{-}\frac{5}{2}e^{12} - \frac{3}{2}e^{11} - \frac{83}{2}e^{10} + \frac{43}{2}e^{9} + \frac{501}{2}e^{8} - \frac{211}{2}e^{7} - \frac{1343}{2}e^{6} + \frac{393}{2}e^{5} + 786e^{4} - 98e^{3} - 334e^{2} - 11e + 14$ |
| 79 | $[79, 79, -w^{4} + 6w^{2} + 3w - 1]$ | $\phantom{-}\frac{9}{2}e^{12} - \frac{7}{2}e^{11} - \frac{147}{2}e^{10} + \frac{95}{2}e^{9} + \frac{885}{2}e^{8} - \frac{447}{2}e^{7} - \frac{2411}{2}e^{6} + \frac{823}{2}e^{5} + 1470e^{4} - 228e^{3} - 668e^{2} + 4e + 44$ |
| 83 | $[83, 83, 2w^{4} - 14w^{2} - 7w + 11]$ | $-\frac{39}{2}e^{12} + \frac{29}{2}e^{11} + \frac{633}{2}e^{10} - \frac{387}{2}e^{9} - \frac{3781}{2}e^{8} + \frac{1791}{2}e^{7} + \frac{10181}{2}e^{6} - \frac{3241}{2}e^{5} - 6081e^{4} + 865e^{3} + 2662e^{2} + 17e - 160$ |
| 83 | $[83, 83, w^{4} - 2w^{3} - 4w^{2} + 6w + 1]$ | $-e^{12} + 18e^{10} - 2e^{9} - 115e^{8} + 18e^{7} + 320e^{6} - 46e^{5} - 384e^{4} + 38e^{3} + 170e^{2} - 20e - 16$ |
| 83 | $[83, 83, -w^{4} + 8w^{2} + 3w - 7]$ | $\phantom{-}3e^{12} - 2e^{11} - 49e^{10} + 26e^{9} + 295e^{8} - 116e^{7} - 801e^{6} + 198e^{5} + 959e^{4} - 94e^{3} - 406e^{2} + 19$ |
| 97 | $[97, 97, w^{4} - 2w^{3} - 4w^{2} + 4w + 1]$ | $-\frac{37}{2}e^{12} + \frac{27}{2}e^{11} + \frac{599}{2}e^{10} - \frac{355}{2}e^{9} - \frac{3573}{2}e^{8} + \frac{1615}{2}e^{7} + \frac{9625}{2}e^{6} - \frac{2855}{2}e^{5} - 5766e^{4} + 718e^{3} + 2539e^{2} + 38e - 156$ |
| 101 | $[101, 101, 2w^{4} - 14w^{2} - 6w + 11]$ | $-\frac{27}{2}e^{12} + \frac{21}{2}e^{11} + \frac{437}{2}e^{10} - \frac{283}{2}e^{9} - \frac{2599}{2}e^{8} + \frac{1329}{2}e^{7} + \frac{6953}{2}e^{6} - \frac{2475}{2}e^{5} - 4118e^{4} + 723e^{3} + 1794e^{2} - 21e - 108$ |
| 101 | $[101, 101, -w^{4} + 8w^{2} + 2w - 9]$ | $-\frac{15}{2}e^{12} + \frac{11}{2}e^{11} + \frac{247}{2}e^{10} - \frac{157}{2}e^{9} - \frac{1487}{2}e^{8} + \frac{783}{2}e^{7} + \frac{4007}{2}e^{6} - \frac{1567}{2}e^{5} - 2384e^{4} + 532e^{3} + 1042e^{2} - 67e - 62$ |
| 107 | $[107, 107, -w^{3} + 2w^{2} + 4w - 1]$ | $-\frac{7}{2}e^{12} + \frac{3}{2}e^{11} + \frac{117}{2}e^{10} - \frac{39}{2}e^{9} - \frac{719}{2}e^{8} + \frac{167}{2}e^{7} + \frac{1997}{2}e^{6} - \frac{233}{2}e^{5} - 1241e^{4} - 5e^{3} + 570e^{2} + 40e - 37$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{4} + w^{3} + 6w^{2} - w - 3]$ | $1$ |