Base field 4.4.4525.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} + 3x + 9\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[31, 31, \frac{2}{3}w^{3} + \frac{1}{3}w^{2} - \frac{11}{3}w - 2]$ |
| Dimension: | $3$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $7$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{3} + 2x^{2} - 5x - 5\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, -\frac{1}{3}w^{3} - \frac{2}{3}w^{2} + \frac{7}{3}w + 4]$ | $\phantom{-}e$ |
| 5 | $[5, 5, \frac{1}{3}w^{3} - \frac{4}{3}w^{2} - \frac{1}{3}w + 3]$ | $-e - 1$ |
| 9 | $[9, 3, -w]$ | $\phantom{-}e^{2} + 2e - 4$ |
| 9 | $[9, 3, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{7}{3}w + 1]$ | $-e^{2} + 4$ |
| 16 | $[16, 2, 2]$ | $-e - 5$ |
| 19 | $[19, 19, \frac{2}{3}w^{3} - \frac{2}{3}w^{2} - \frac{11}{3}w]$ | $-1$ |
| 19 | $[19, 19, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + \frac{1}{3}w + 1]$ | $-e^{2} - 3e + 2$ |
| 31 | $[31, 31, \frac{2}{3}w^{3} + \frac{1}{3}w^{2} - \frac{11}{3}w - 2]$ | $\phantom{-}1$ |
| 31 | $[31, 31, \frac{2}{3}w^{3} - \frac{5}{3}w^{2} - \frac{5}{3}w + 5]$ | $\phantom{-}e^{2} - e - 7$ |
| 41 | $[41, 41, -\frac{1}{3}w^{3} + \frac{4}{3}w^{2} + \frac{7}{3}w - 6]$ | $\phantom{-}e^{2} - 7$ |
| 41 | $[41, 41, -\frac{1}{3}w^{3} + \frac{4}{3}w^{2} + \frac{7}{3}w - 3]$ | $-e^{2} - 4e$ |
| 61 | $[61, 61, \frac{1}{3}w^{3} - \frac{4}{3}w^{2} + \frac{2}{3}w + 4]$ | $\phantom{-}e - 8$ |
| 61 | $[61, 61, \frac{2}{3}w^{3} + \frac{1}{3}w^{2} - \frac{14}{3}w - 2]$ | $-e - 9$ |
| 71 | $[71, 71, -\frac{1}{3}w^{3} + \frac{4}{3}w^{2} + \frac{1}{3}w - 7]$ | $-e^{2} + e + 1$ |
| 71 | $[71, 71, \frac{1}{3}w^{3} + \frac{2}{3}w^{2} - \frac{7}{3}w]$ | $-3e^{2} - 5e + 7$ |
| 89 | $[89, 89, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + \frac{10}{3}w - 3]$ | $-e^{2} + e + 8$ |
| 89 | $[89, 89, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{10}{3}w - 1]$ | $\phantom{-}2e^{2} + 4e - 4$ |
| 101 | $[101, 101, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{7}{3}w - 3]$ | $\phantom{-}4e + 4$ |
| 101 | $[101, 101, -\frac{2}{3}w^{3} + \frac{5}{3}w^{2} + \frac{11}{3}w - 4]$ | $-e^{2} - 5e - 1$ |
| 101 | $[101, 101, \frac{2}{3}w^{3} - \frac{5}{3}w^{2} - \frac{8}{3}w + 3]$ | $-2e^{2} - e + 14$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31, 31, \frac{2}{3}w^{3} + \frac{1}{3}w^{2} - \frac{11}{3}w - 2]$ | $-1$ |