Base field 4.4.3981.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 4x^{2} + 2x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{3} + w^{2} + 4w]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $4$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} - 2x^{3} - 9x^{2} + 22x - 9\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w^{3} - w^{2} - 3w + 1]$ | $\phantom{-}e^{3} - 9e + 6$ |
| 9 | $[9, 3, -w^{2} + 2]$ | $-e^{2} + 5$ |
| 13 | $[13, 13, -w^{3} + w^{2} + 4w]$ | $-1$ |
| 16 | $[16, 2, 2]$ | $-e^{3} + 10e - 8$ |
| 23 | $[23, 23, w^{2} - 2w - 2]$ | $-e^{2} - 2e + 6$ |
| 37 | $[37, 37, w^{3} - 4w + 1]$ | $\phantom{-}2e^{3} - 18e + 10$ |
| 37 | $[37, 37, w^{3} - w^{2} - 5w + 1]$ | $-2e^{3} + 16e - 10$ |
| 41 | $[41, 41, w^{3} - 5w + 1]$ | $-2e^{3} + 16e - 6$ |
| 43 | $[43, 43, 2w^{3} - w^{2} - 7w]$ | $-2e^{3} + e^{2} + 18e - 14$ |
| 53 | $[53, 53, 2w - 3]$ | $-2e + 6$ |
| 59 | $[59, 59, -2w^{3} + w^{2} + 9w - 1]$ | $\phantom{-}2e^{3} - 2e^{2} - 22e + 24$ |
| 67 | $[67, 67, -w - 3]$ | $\phantom{-}e^{2} + 2e - 5$ |
| 67 | $[67, 67, w^{3} + w^{2} - 5w - 4]$ | $-2e^{3} + 2e^{2} + 22e - 22$ |
| 71 | $[71, 71, 2w^{3} - 3w^{2} - 7w + 5]$ | $\phantom{-}2e^{3} - 2e^{2} - 19e + 24$ |
| 73 | $[73, 73, w^{3} - 6w]$ | $-2e^{3} + 2e^{2} + 18e - 14$ |
| 73 | $[73, 73, -w^{3} - w^{2} + 5w + 3]$ | $-2e^{2} - 2e + 16$ |
| 79 | $[79, 79, w^{3} - 3w - 4]$ | $\phantom{-}2e^{3} - 2e^{2} - 16e + 20$ |
| 83 | $[83, 83, w^{3} - 2w^{2} - 3w + 1]$ | $\phantom{-}2e^{3} - 2e^{2} - 16e + 24$ |
| 83 | $[83, 83, -2w^{3} + 2w^{2} + 6w - 3]$ | $\phantom{-}e^{3} - 7e + 6$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{3} + w^{2} + 4w]$ | $1$ |