Base field 4.4.18432.1
Generator \(w\), with minimal polynomial \(x^{4} - 12x^{2} + 18\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[8, 2, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 2w - 2]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $8$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} - 12x^{2} - 16x - 4\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + 3w + 4]$ | $\phantom{-}0$ |
| 7 | $[7, 7, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 3w - 3]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -\frac{1}{3}w^{2} + w + 1]$ | $-\frac{1}{2}e^{3} + 6e + 6$ |
| 7 | $[7, 7, \frac{1}{3}w^{2} + w - 1]$ | $-\frac{1}{2}e^{3} + e^{2} + 4e$ |
| 7 | $[7, 7, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - 3w + 3]$ | $\phantom{-}e^{3} - e^{2} - 11e - 6$ |
| 9 | $[9, 3, w - 3]$ | $\phantom{-}0$ |
| 41 | $[41, 41, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - 3w + 1]$ | $-e^{3} - e^{2} + 14e + 14$ |
| 41 | $[41, 41, -\frac{1}{3}w^{2} + w + 3]$ | $-e^{3} + e^{2} + 14e + 2$ |
| 41 | $[41, 41, \frac{1}{3}w^{2} + w - 3]$ | $\phantom{-}3e^{3} - 3e^{2} - 34e - 22$ |
| 41 | $[41, 41, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + 3w + 1]$ | $-e^{3} + 3e^{2} + 6e - 10$ |
| 47 | $[47, 47, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 4w - 1]$ | $-e^{3} + 2e^{2} + 6e - 4$ |
| 47 | $[47, 47, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + 2w - 3]$ | $\phantom{-}e^{3} - 10e - 16$ |
| 47 | $[47, 47, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - 2w - 3]$ | $\phantom{-}3e^{3} - 4e^{2} - 30e - 16$ |
| 47 | $[47, 47, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - 4w + 1]$ | $-3e^{3} + 2e^{2} + 34e + 20$ |
| 89 | $[89, 89, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + 3w - 3]$ | $\phantom{-}2e^{3} - 4e^{2} - 16e$ |
| 89 | $[89, 89, \frac{2}{3}w^{2} + w - 5]$ | $-4e$ |
| 89 | $[89, 89, \frac{2}{3}w^{2} - w - 5]$ | $-4e^{3} + 4e^{2} + 44e + 24$ |
| 89 | $[89, 89, \frac{1}{3}w^{3} + \frac{2}{3}w^{2} - 3w - 3]$ | $\phantom{-}2e^{3} - 24e - 24$ |
| 97 | $[97, 97, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - 6w + 5]$ | $-2e^{3} + 3e^{2} + 22e + 2$ |
| 97 | $[97, 97, -w^{3} - \frac{5}{3}w^{2} + 10w + 15]$ | $\phantom{-}e^{3} + e^{2} - 16e - 22$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + 3w + 4]$ | $-1$ |