Base field 4.4.4913.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 6x^{2} + x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, -\frac{1}{2}w^{3} + w^{2} + 3w - \frac{1}{2}]$ |
| Dimension: | $2$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $5$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{2} + 3x + 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{1}{2}w^{3} + 3w + \frac{5}{2}]$ | $\phantom{-}e$ |
| 4 | $[4, 2, w^{3} - w^{2} - 6w]$ | $\phantom{-}e - 1$ |
| 13 | $[13, 13, -\frac{1}{2}w^{3} + w^{2} + 3w - \frac{1}{2}]$ | $\phantom{-}1$ |
| 13 | $[13, 13, \frac{1}{2}w^{3} - w^{2} - 3w + \frac{7}{2}]$ | $\phantom{-}e - 3$ |
| 13 | $[13, 13, -\frac{1}{2}w^{3} + 4w - \frac{1}{2}]$ | $-6e - 9$ |
| 13 | $[13, 13, -\frac{1}{2}w^{3} + 4w + \frac{5}{2}]$ | $\phantom{-}4e + 7$ |
| 17 | $[17, 17, \frac{1}{2}w^{3} - w^{2} - w + \frac{3}{2}]$ | $-2e - 3$ |
| 47 | $[47, 47, -w^{2} + 2]$ | $\phantom{-}5e + 8$ |
| 47 | $[47, 47, -w^{3} + 2w^{2} + 5w - 5]$ | $\phantom{-}3e + 11$ |
| 47 | $[47, 47, -\frac{3}{2}w^{3} + w^{2} + 10w + \frac{3}{2}]$ | $\phantom{-}5e + 8$ |
| 47 | $[47, 47, -\frac{1}{2}w^{3} + 5w - \frac{1}{2}]$ | $-e - 14$ |
| 67 | $[67, 67, \frac{3}{2}w^{3} - 2w^{2} - 8w + \frac{3}{2}]$ | $\phantom{-}4e - 3$ |
| 67 | $[67, 67, \frac{1}{2}w^{3} - 2w - \frac{5}{2}]$ | $\phantom{-}2e$ |
| 67 | $[67, 67, -\frac{1}{2}w^{3} + w^{2} + w - \frac{5}{2}]$ | $-6e - 17$ |
| 67 | $[67, 67, -\frac{3}{2}w^{3} + w^{2} + 9w - \frac{1}{2}]$ | $-6e - 17$ |
| 81 | $[81, 3, -3]$ | $-e - 4$ |
| 89 | $[89, 89, w^{2} - 2w - 4]$ | $-e - 3$ |
| 89 | $[89, 89, \frac{3}{2}w^{3} - 2w^{2} - 9w + \frac{3}{2}]$ | $\phantom{-}5e + 16$ |
| 89 | $[89, 89, -\frac{1}{2}w^{3} + w^{2} + 4w - \frac{7}{2}]$ | $\phantom{-}e - 9$ |
| 89 | $[89, 89, -w^{3} + 7w + 1]$ | $-6e - 11$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -\frac{1}{2}w^{3} + w^{2} + 3w - \frac{1}{2}]$ | $-1$ |