Base field 4.4.18097.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} + 6x + 4\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ |
| Dimension: | $10$ |
| 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:
| \(x^{10} - 18x^{8} + 118x^{6} - 333x^{4} + 345x^{2} - 16\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w + 1]$ | $\phantom{-}e$ |
| 4 | $[4, 2, w]$ | $\phantom{-}\frac{1}{4}e^{9} - \frac{7}{2}e^{7} + \frac{33}{2}e^{5} - \frac{117}{4}e^{3} + \frac{57}{4}e$ |
| 4 | $[4, 2, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{7}{2}w - 3]$ | $-\frac{3}{4}e^{9} + \frac{19}{2}e^{7} - \frac{79}{2}e^{5} + \frac{235}{4}e^{3} - \frac{71}{4}e$ |
| 7 | $[7, 7, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + \frac{5}{2}w - 2]$ | $-2e^{8} + 25e^{6} - 100e^{4} + 130e^{2} - 8$ |
| 13 | $[13, 13, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ | $\phantom{-}1$ |
| 17 | $[17, 17, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w]$ | $\phantom{-}\frac{1}{2}e^{9} - 6e^{7} + 23e^{5} - \frac{59}{2}e^{3} + \frac{5}{2}e$ |
| 27 | $[27, 3, -w^{3} + 7w + 1]$ | $-2e^{9} + 25e^{7} - 99e^{5} + 122e^{3} + 7e$ |
| 31 | $[31, 31, w + 3]$ | $-e^{6} + 9e^{4} - 19e^{2}$ |
| 31 | $[31, 31, -w^{2} + 5]$ | $-e^{9} + 11e^{7} - 36e^{5} + 33e^{3} + 4e$ |
| 37 | $[37, 37, w^{2} - 3]$ | $\phantom{-}\frac{5}{2}e^{9} - 30e^{7} + 112e^{5} - \frac{251}{2}e^{3} - \frac{33}{2}e$ |
| 41 | $[41, 41, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{3}{2}w - 2]$ | $-\frac{3}{2}e^{9} + 18e^{7} - 67e^{5} + \frac{149}{2}e^{3} + \frac{17}{2}e$ |
| 47 | $[47, 47, w^{3} - 5w - 3]$ | $\phantom{-}2e^{9} - 24e^{7} + 93e^{5} - 129e^{3} + 40e$ |
| 53 | $[53, 53, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w]$ | $\phantom{-}\frac{1}{2}e^{9} - 6e^{7} + 21e^{5} - \frac{25}{2}e^{3} - \frac{57}{2}e$ |
| 61 | $[61, 61, w^{3} - w^{2} - 5w + 3]$ | $\phantom{-}e^{6} - 8e^{4} + 13e^{2} - 2$ |
| 83 | $[83, 83, w^{3} + w^{2} - 6w - 7]$ | $\phantom{-}2e^{9} - 26e^{7} + 111e^{5} - 169e^{3} + 53e$ |
| 83 | $[83, 83, -w^{3} + 5w + 1]$ | $\phantom{-}2e^{8} - 24e^{6} + 91e^{4} - 112e^{2} + 4$ |
| 83 | $[83, 83, 2w - 1]$ | $\phantom{-}2e^{9} - 26e^{7} + 114e^{5} - 191e^{3} + 85e$ |
| 83 | $[83, 83, -\frac{3}{2}w^{3} - \frac{1}{2}w^{2} + \frac{19}{2}w + 2]$ | $\phantom{-}9e^{8} - 109e^{6} + 416e^{4} - 506e^{2} + 20$ |
| 89 | $[89, 89, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w - 2]$ | $-5e^{8} + 60e^{6} - 225e^{4} + 263e^{2} - 6$ |
| 89 | $[89, 89, w^{3} - 5w + 5]$ | $\phantom{-}e^{8} - 15e^{6} + 72e^{4} - 107e^{2} - 6$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ | $-1$ |