Base field 4.4.8069.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 5x^{2} + 5x + 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{2} + 3]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $8$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} + 7x^{3} + 13x^{2} - x - 13\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, w^{3} - 4w]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w + 1]$ | $-2e^{3} - 9e^{2} - 5e + 8$ |
| 7 | $[7, 7, -w^{3} + 4w - 1]$ | $\phantom{-}e^{2} + 4e + 1$ |
| 13 | $[13, 13, -w^{2} + 3]$ | $\phantom{-}1$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}e^{3} + 5e^{2} + e - 11$ |
| 17 | $[17, 17, w^{3} + w^{2} - 4w - 2]$ | $\phantom{-}e^{3} + 3e^{2} - 2e - 6$ |
| 17 | $[17, 17, -w^{3} + 5w - 2]$ | $\phantom{-}e + 3$ |
| 19 | $[19, 19, -w^{2} - w + 4]$ | $-3e^{3} - 15e^{2} - 11e + 14$ |
| 19 | $[19, 19, -w^{2} - w + 1]$ | $\phantom{-}e^{3} + 4e^{2} + e - 5$ |
| 29 | $[29, 29, 2w^{3} - w^{2} - 9w + 5]$ | $\phantom{-}3e^{3} + 13e^{2} + 4e - 22$ |
| 41 | $[41, 41, -w^{3} + w^{2} + 5w - 3]$ | $-e^{3} - 4e^{2} - 5e - 4$ |
| 43 | $[43, 43, w^{3} - w^{2} - 4w + 2]$ | $-3e^{3} - 13e^{2} - 5e + 12$ |
| 43 | $[43, 43, w^{3} - 6w]$ | $-e^{3} - 6e^{2} - 8e + 3$ |
| 47 | $[47, 47, -w^{3} - w^{2} + 5w]$ | $\phantom{-}e^{3} + 8e^{2} + 10e - 13$ |
| 49 | $[49, 7, w^{2} + 2w - 2]$ | $\phantom{-}4e^{3} + 16e^{2} + 3e - 22$ |
| 59 | $[59, 59, 2w^{3} - 8w + 3]$ | $\phantom{-}2e^{3} + 9e^{2} + e - 16$ |
| 67 | $[67, 67, w^{2} - w - 4]$ | $\phantom{-}7e^{3} + 30e^{2} + 10e - 38$ |
| 79 | $[79, 79, w^{3} - w^{2} - 4w + 1]$ | $\phantom{-}5e^{3} + 23e^{2} + 11e - 28$ |
| 81 | $[81, 3, -3]$ | $-5e^{3} - 25e^{2} - 14e + 35$ |
| 97 | $[97, 97, w^{3} + w^{2} - 5w + 1]$ | $-e^{3} - 3e^{2} + 5e + 6$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{2} + 3]$ | $-1$ |