Base field 5.5.176281.1
Generator \(w\), with minimal polynomial \(x^{5} - x^{4} - 5x^{3} + 3x^{2} + 4x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 2w - 2]$ |
| Dimension: | $3$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $20$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{3} + 4x^{2} + x - 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, w^{4} - w^{3} - 4w^{2} + 2w + 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w^{2} - w - 2]$ | $-e^{2} - 4e - 1$ |
| 7 | $[7, 7, -w^{2} + 2]$ | $-2e^{2} - 7e$ |
| 11 | $[11, 11, -w^{2} + 3]$ | $\phantom{-}e^{2} + 5e + 1$ |
| 13 | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 2w - 2]$ | $-1$ |
| 23 | $[23, 23, -w^{4} + 5w^{2} + w - 4]$ | $-e^{2} - 2e - 1$ |
| 31 | $[31, 31, -w^{4} + w^{3} + 5w^{2} - 2w - 2]$ | $\phantom{-}e^{2} + 5e - 6$ |
| 32 | $[32, 2, 2]$ | $\phantom{-}4e^{2} + 13e + 1$ |
| 37 | $[37, 37, w^{3} - 3w - 1]$ | $-4e - 8$ |
| 41 | $[41, 41, w^{4} - w^{3} - 3w^{2} + w + 1]$ | $\phantom{-}2e^{2} + 9e + 2$ |
| 47 | $[47, 47, -w^{3} + w^{2} + 4w]$ | $\phantom{-}3e^{2} + 8e - 3$ |
| 61 | $[61, 61, -w^{4} + 6w^{2} + w - 4]$ | $-3e - 11$ |
| 67 | $[67, 67, -w^{4} + w^{3} + 5w^{2} - 4w - 3]$ | $-2e^{2} - 7e + 1$ |
| 71 | $[71, 71, w^{4} - 6w^{2} + 7]$ | $\phantom{-}2e^{2} + 7e + 2$ |
| 73 | $[73, 73, -w^{4} + w^{3} + 4w^{2} - 4w - 2]$ | $\phantom{-}2e^{2} + 5e + 1$ |
| 83 | $[83, 83, -w^{4} + 2w^{3} + 4w^{2} - 6w - 2]$ | $\phantom{-}2e^{2} + 6e + 1$ |
| 83 | $[83, 83, -w^{4} + 5w^{2} + 2w - 4]$ | $\phantom{-}3$ |
| 83 | $[83, 83, w^{4} - 5w^{2} + 2]$ | $-2e^{2} - 8e - 3$ |
| 89 | $[89, 89, -2w^{4} + 2w^{3} + 9w^{2} - 6w - 4]$ | $-5e^{2} - 20e - 5$ |
| 101 | $[101, 101, -2w^{4} + 2w^{3} + 8w^{2} - 3w - 3]$ | $-7e^{2} - 21e + 5$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{4} + w^{3} + 4w^{2} - 2w - 2]$ | $1$ |