Base field 4.4.2777.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 4x^{2} + x + 2\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[31, 31, w^{3} - 2w^{2} - w + 3]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $5$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} - x^{3} - 6x^{2} + 4x + 4\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}e$ |
| 8 | $[8, 2, -w^{3} + w^{2} + 4w - 1]$ | $\phantom{-}\frac{1}{2}e^{3} - \frac{1}{2}e^{2} - 3e + 1$ |
| 11 | $[11, 11, w^{3} - 2w^{2} - 2w + 1]$ | $\phantom{-}e^{3} - 5e$ |
| 23 | $[23, 23, -w^{3} + 4w + 1]$ | $-\frac{3}{2}e^{3} + \frac{1}{2}e^{2} + 7e - 1$ |
| 23 | $[23, 23, -w^{2} + 2w + 3]$ | $-e^{3} + 5e + 4$ |
| 31 | $[31, 31, w^{3} - 2w^{2} - w + 3]$ | $-1$ |
| 37 | $[37, 37, -w^{3} + 3w + 3]$ | $-e^{3} - 2e^{2} + 3e + 10$ |
| 37 | $[37, 37, -2w^{3} + 3w^{2} + 6w - 3]$ | $\phantom{-}e^{3} - 2e^{2} - 3e + 6$ |
| 41 | $[41, 41, w^{3} - 2w^{2} - 3w + 1]$ | $-\frac{1}{2}e^{3} - \frac{5}{2}e^{2} + 5e + 7$ |
| 41 | $[41, 41, -2w^{3} + 2w^{2} + 6w - 1]$ | $\phantom{-}2e^{3} + e^{2} - 11e - 4$ |
| 43 | $[43, 43, -w^{2} + w + 5]$ | $-e^{3} - e^{2} + 5e + 1$ |
| 47 | $[47, 47, 2w^{2} - 3w - 5]$ | $-\frac{3}{2}e^{3} + \frac{3}{2}e^{2} + 6e + 1$ |
| 53 | $[53, 53, -w^{3} + 3w^{2} + w - 7]$ | $-e^{3} - e^{2} + 7e + 5$ |
| 53 | $[53, 53, -2w^{2} + 2w + 5]$ | $\phantom{-}e^{2} - e$ |
| 59 | $[59, 59, 2w^{2} - w - 7]$ | $\phantom{-}3e^{2} - e - 12$ |
| 61 | $[61, 61, 2w^{2} - w - 3]$ | $-\frac{1}{2}e^{3} - \frac{3}{2}e^{2} + 2e + 11$ |
| 61 | $[61, 61, 2w^{2} - w - 5]$ | $-\frac{3}{2}e^{3} + \frac{5}{2}e^{2} + 5e - 7$ |
| 67 | $[67, 67, 2w^{3} - 2w^{2} - 7w + 3]$ | $-e^{3} + 9e + 4$ |
| 67 | $[67, 67, 2w^{3} - 2w^{2} - 7w - 1]$ | $\phantom{-}e^{3} - 9e + 4$ |
| 71 | $[71, 71, 2w^{3} - 4w^{2} - 4w + 7]$ | $-e^{3} + e^{2} - e - 3$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31, 31, w^{3} - 2w^{2} - w + 3]$ | $1$ |