Base field \(\Q(\sqrt{5}, \sqrt{21})\)
Generator \(w\), with minimal polynomial \(x^{4} - 13x^{2} + 16\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[9, 3, -\frac{1}{8}w^{3} + \frac{17}{8}w + \frac{3}{2}]$ |
| 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:
| \(x^{2} - 2x - 7\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, \frac{1}{4}w^{3} - \frac{1}{2}w^{2} - \frac{11}{4}w + 5]$ | $-e + 2$ |
| 4 | $[4, 2, \frac{1}{8}w^{3} - \frac{1}{2}w^{2} - \frac{5}{8}w + \frac{3}{2}]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -\frac{3}{8}w^{3} + \frac{35}{8}w - \frac{1}{2}]$ | $\phantom{-}e - 1$ |
| 5 | $[5, 5, \frac{3}{8}w^{3} - \frac{35}{8}w - \frac{1}{2}]$ | $\phantom{-}e - 1$ |
| 9 | $[9, 3, -\frac{1}{8}w^{3} + \frac{17}{8}w + \frac{3}{2}]$ | $-1$ |
| 41 | $[41, 41, \frac{1}{8}w^{3} - \frac{1}{8}w + \frac{3}{2}]$ | $-e + 9$ |
| 41 | $[41, 41, \frac{3}{8}w^{3} - \frac{35}{8}w + \frac{3}{2}]$ | $-e - 7$ |
| 41 | $[41, 41, \frac{3}{8}w^{3} - \frac{35}{8}w - \frac{3}{2}]$ | $-e - 7$ |
| 41 | $[41, 41, -\frac{1}{8}w^{3} + \frac{1}{8}w + \frac{3}{2}]$ | $-e + 9$ |
| 49 | $[49, 7, \frac{1}{8}w^{3} - \frac{17}{8}w + \frac{7}{2}]$ | $-2$ |
| 59 | $[59, 59, \frac{5}{8}w^{3} - \frac{53}{8}w - \frac{5}{2}]$ | $\phantom{-}0$ |
| 59 | $[59, 59, -\frac{7}{8}w^{3} + \frac{3}{2}w^{2} + \frac{83}{8}w - \frac{35}{2}]$ | $\phantom{-}0$ |
| 59 | $[59, 59, -\frac{3}{4}w^{3} + \frac{3}{2}w^{2} + \frac{33}{4}w - 16]$ | $\phantom{-}0$ |
| 59 | $[59, 59, \frac{5}{8}w^{3} - \frac{53}{8}w + \frac{5}{2}]$ | $\phantom{-}0$ |
| 79 | $[79, 79, \frac{3}{8}w^{3} - w^{2} - \frac{35}{8}w + \frac{21}{2}]$ | $-4e + 4$ |
| 79 | $[79, 79, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + 3w - 5]$ | $\phantom{-}4e - 4$ |
| 79 | $[79, 79, \frac{7}{8}w^{3} - \frac{79}{8}w - \frac{1}{2}]$ | $\phantom{-}4e - 4$ |
| 79 | $[79, 79, -\frac{3}{8}w^{3} - w^{2} + \frac{35}{8}w + \frac{21}{2}]$ | $-4e + 4$ |
| 89 | $[89, 89, \frac{1}{8}w^{3} - \frac{1}{8}w - \frac{5}{2}]$ | $\phantom{-}e + 7$ |
| 89 | $[89, 89, \frac{3}{8}w^{3} - \frac{35}{8}w + \frac{5}{2}]$ | $\phantom{-}e - 9$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $9$ | $[9,3,-\frac{1}{8}w^{3}+\frac{17}{8}w+\frac{3}{2}]$ | $1$ |