Base field \(\Q(\sqrt{2}, \sqrt{7})\)
Generator \(w\), with minimal polynomial \(x^{4} - 8x^{2} + 9\); narrow class number \(4\) and class number \(1\).
Form
Weight: | $[2, 2, 2, 2]$ |
Level: | $[8, 2, -\frac{1}{3}w^{3} - w^{2} - \frac{1}{3}w + 1]$ |
Dimension: | $2$ |
CM: | no |
Base change: | yes |
Newspace dimension: | $4$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{2} - 32\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
2 | $[2, 2, \frac{1}{3}w^{3} + w^{2} - \frac{2}{3}w - 2]$ | $\phantom{-}0$ |
7 | $[7, 7, \frac{1}{3}w^{3} - \frac{8}{3}w + 2]$ | $\phantom{-}0$ |
7 | $[7, 7, -w + 2]$ | $\phantom{-}0$ |
9 | $[9, 3, w]$ | $-2$ |
9 | $[9, 3, -\frac{1}{3}w^{3} + \frac{8}{3}w]$ | $-2$ |
25 | $[25, 5, \frac{1}{3}w^{3} - w^{2} - \frac{5}{3}w + 4]$ | $-2$ |
25 | $[25, 5, -\frac{1}{3}w^{3} - w^{2} + \frac{5}{3}w + 4]$ | $-2$ |
31 | $[31, 31, \frac{1}{3}w^{3} + w^{2} - \frac{5}{3}w - 2]$ | $-e$ |
31 | $[31, 31, \frac{1}{3}w^{3} + w^{2} - \frac{5}{3}w - 6]$ | $-e$ |
31 | $[31, 31, \frac{1}{3}w^{3} - w^{2} - \frac{5}{3}w + 6]$ | $\phantom{-}e$ |
31 | $[31, 31, \frac{1}{3}w^{3} - w^{2} - \frac{5}{3}w + 2]$ | $\phantom{-}e$ |
47 | $[47, 47, \frac{1}{3}w^{3} + w^{2} - \frac{8}{3}w - 3]$ | $\phantom{-}e$ |
47 | $[47, 47, -w^{2} - w + 5]$ | $\phantom{-}e$ |
47 | $[47, 47, w^{2} - w - 5]$ | $-e$ |
47 | $[47, 47, -\frac{1}{3}w^{3} + w^{2} + \frac{8}{3}w - 3]$ | $-e$ |
103 | $[103, 103, \frac{2}{3}w^{3} + w^{2} - \frac{10}{3}w - 2]$ | $\phantom{-}3e$ |
103 | $[103, 103, -\frac{2}{3}w^{3} + w^{2} + \frac{10}{3}w - 6]$ | $-3e$ |
103 | $[103, 103, \frac{2}{3}w^{3} + w^{2} - \frac{10}{3}w - 6]$ | $\phantom{-}3e$ |
103 | $[103, 103, \frac{1}{3}w^{3} + 2w^{2} + \frac{7}{3}w - 1]$ | $-3e$ |
113 | $[113, 113, -\frac{2}{3}w^{3} + \frac{16}{3}w + 1]$ | $-10$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$2$ | $[2,2,\frac{1}{3}w^{3}+w^{2}-\frac{2}{3}w-2]$ | $-1$ |