Base field \(\Q(\sqrt{3}, \sqrt{5})\)
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 7x^{2} + 8x + 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[49, 7, \frac{5}{7}w^{3} - \frac{11}{7}w^{2} - \frac{23}{7}w + \frac{25}{7}]$ |
| Dimension: | $2$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $10$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{2} - 12\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{2}{7}w^{3} + \frac{3}{7}w^{2} + \frac{19}{7}w - \frac{17}{7}]$ | $-1$ |
| 9 | $[9, 3, -\frac{2}{7}w^{3} + \frac{3}{7}w^{2} + \frac{19}{7}w - \frac{10}{7}]$ | $-2$ |
| 11 | $[11, 11, -w - 1]$ | $\phantom{-}e$ |
| 11 | $[11, 11, \frac{4}{7}w^{3} - \frac{6}{7}w^{2} - \frac{31}{7}w + \frac{27}{7}]$ | $-e$ |
| 11 | $[11, 11, \frac{4}{7}w^{3} - \frac{6}{7}w^{2} - \frac{31}{7}w + \frac{6}{7}]$ | $-e$ |
| 11 | $[11, 11, -w + 2]$ | $\phantom{-}e$ |
| 25 | $[25, 5, \frac{4}{7}w^{3} - \frac{6}{7}w^{2} - \frac{24}{7}w + \frac{13}{7}]$ | $-10$ |
| 49 | $[49, 7, \frac{5}{7}w^{3} - \frac{11}{7}w^{2} - \frac{23}{7}w + \frac{25}{7}]$ | $\phantom{-}1$ |
| 49 | $[49, 7, w^{3} - w^{2} - 7w + 1]$ | $\phantom{-}2$ |
| 59 | $[59, 59, \frac{3}{7}w^{3} - \frac{1}{7}w^{2} - \frac{32}{7}w + \frac{15}{7}]$ | $\phantom{-}3e$ |
| 59 | $[59, 59, -\frac{3}{7}w^{3} + \frac{8}{7}w^{2} + \frac{25}{7}w - \frac{43}{7}]$ | $\phantom{-}e$ |
| 59 | $[59, 59, -\frac{3}{7}w^{3} + \frac{1}{7}w^{2} + \frac{32}{7}w + \frac{13}{7}]$ | $\phantom{-}e$ |
| 59 | $[59, 59, \frac{3}{7}w^{3} - \frac{8}{7}w^{2} - \frac{25}{7}w + \frac{15}{7}]$ | $\phantom{-}3e$ |
| 61 | $[61, 61, -\frac{8}{7}w^{3} + \frac{12}{7}w^{2} + \frac{55}{7}w - \frac{26}{7}]$ | $\phantom{-}2$ |
| 61 | $[61, 61, -\frac{4}{7}w^{3} + \frac{6}{7}w^{2} + \frac{17}{7}w - \frac{6}{7}]$ | $-10$ |
| 61 | $[61, 61, \frac{6}{7}w^{3} - \frac{9}{7}w^{2} - \frac{50}{7}w + \frac{16}{7}]$ | $-10$ |
| 61 | $[61, 61, \frac{8}{7}w^{3} - \frac{12}{7}w^{2} - \frac{55}{7}w + \frac{33}{7}]$ | $\phantom{-}2$ |
| 71 | $[71, 71, -\frac{3}{7}w^{3} + \frac{1}{7}w^{2} + \frac{25}{7}w - \frac{15}{7}]$ | $-2e$ |
| 71 | $[71, 71, -w^{3} + w^{2} + 8w - 2]$ | $-4e$ |
| 71 | $[71, 71, w^{3} - 2w^{2} - 7w + 6]$ | $-4e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $49$ | $[49, 7, \frac{5}{7}w^{3} - \frac{11}{7}w^{2} - \frac{23}{7}w + \frac{25}{7}]$ | $-1$ |