Base field 4.4.10309.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 6x^{2} + 8x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, w + 1]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $12$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} - 2x^{3} - 10x^{2} + 16x + 12\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, w^{3} - 5w + 3]$ | $\phantom{-}e$ |
| 9 | $[9, 3, w^{3} - 5w + 2]$ | $-e - 2$ |
| 13 | $[13, 13, w + 1]$ | $\phantom{-}1$ |
| 13 | $[13, 13, w^{3} - 6w + 4]$ | $-e + 2$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{1}{2}e^{3} - 4e - 3$ |
| 17 | $[17, 17, w^{2} + w - 2]$ | $-e^{2} - e + 7$ |
| 17 | $[17, 17, w^{2} + w - 5]$ | $\phantom{-}2e - 2$ |
| 23 | $[23, 23, w^{3} - w^{2} - 6w + 6]$ | $-e^{3} + 8e - 2$ |
| 23 | $[23, 23, w^{3} + w^{2} - 4w - 1]$ | $-e^{2} + e + 4$ |
| 25 | $[25, 5, -w^{2} + 3]$ | $\phantom{-}e^{2} - 8$ |
| 25 | $[25, 5, -w^{3} - w^{2} + 5w + 1]$ | $\phantom{-}e^{3} - 7e - 2$ |
| 29 | $[29, 29, -w^{2} - 2w + 3]$ | $-\frac{1}{2}e^{3} + e^{2} + 3e - 7$ |
| 29 | $[29, 29, -w^{3} + w^{2} + 7w - 7]$ | $-\frac{1}{2}e^{3} + e^{2} + 3e - 7$ |
| 43 | $[43, 43, 2w^{3} + w^{2} - 10w + 3]$ | $-\frac{1}{2}e^{3} - e^{2} + 6e + 1$ |
| 43 | $[43, 43, -w^{3} + w^{2} + 5w - 7]$ | $\phantom{-}e^{3} + 2e^{2} - 10e - 11$ |
| 53 | $[53, 53, w^{3} - w^{2} - 7w + 5]$ | $-2e^{2} + 12$ |
| 53 | $[53, 53, w^{2} + 2w - 5]$ | $\phantom{-}e^{2} + e - 12$ |
| 61 | $[61, 61, 2w^{3} + w^{2} - 10w]$ | $-e^{3} + 8e + 3$ |
| 61 | $[61, 61, w^{3} - 7w + 3]$ | $-\frac{1}{2}e^{3} + e^{2} + 4e - 7$ |
| 61 | $[61, 61, 2w^{3} + w^{2} - 9w + 3]$ | $-e^{2} - 3$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, w + 1]$ | $-1$ |