Base field 4.4.5725.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 8x^{2} + 6x + 11\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[59,59,\frac{1}{3}w^{3} - \frac{2}{3}w + \frac{7}{3}]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $18$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} + 7x^{3} + 4x^{2} - 30x + 9\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, \frac{1}{3}w^{3} - \frac{8}{3}w - \frac{2}{3}]$ | $\phantom{-}\frac{1}{3}e^{3} + 2e^{2} + \frac{2}{3}e - 6$ |
| 9 | $[9, 3, -w + 1]$ | $\phantom{-}e$ |
| 11 | $[11, 11, w]$ | $\phantom{-}\frac{1}{3}e^{3} + \frac{1}{3}e^{2} - \frac{11}{3}e + 5$ |
| 11 | $[11, 11, -\frac{1}{3}w^{3} + \frac{2}{3}w + \frac{5}{3}]$ | $-\frac{1}{3}e^{3} - \frac{4}{3}e^{2} + \frac{5}{3}e + 2$ |
| 11 | $[11, 11, \frac{1}{3}w^{3} - \frac{8}{3}w + \frac{1}{3}]$ | $-\frac{1}{3}e^{2} - \frac{2}{3}e + 1$ |
| 11 | $[11, 11, -\frac{2}{3}w^{3} + \frac{13}{3}w + \frac{4}{3}]$ | $-\frac{1}{3}e^{3} - \frac{4}{3}e^{2} + \frac{2}{3}e$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{1}{3}e^{3} + \frac{7}{3}e^{2} + \frac{4}{3}e - 8$ |
| 25 | $[25, 5, \frac{2}{3}w^{3} - \frac{10}{3}w - \frac{1}{3}]$ | $-\frac{5}{3}e^{2} - \frac{19}{3}e + 6$ |
| 29 | $[29, 29, -w - 3]$ | $\phantom{-}\frac{1}{3}e^{3} + \frac{2}{3}e^{2} - 3e + 3$ |
| 29 | $[29, 29, -\frac{1}{3}w^{3} + \frac{8}{3}w - \frac{10}{3}]$ | $-\frac{2}{3}e^{3} - \frac{8}{3}e^{2} + \frac{1}{3}e$ |
| 31 | $[31, 31, w^{3} - 6w + 1]$ | $-\frac{1}{3}e^{3} + \frac{2}{3}e^{2} + \frac{23}{3}e - 8$ |
| 31 | $[31, 31, w^{3} - 6w - 2]$ | $\phantom{-}\frac{5}{3}e^{2} + \frac{16}{3}e - 9$ |
| 41 | $[41, 41, \frac{2}{3}w^{3} + w^{2} - \frac{13}{3}w - \frac{10}{3}]$ | $-\frac{1}{3}e^{3} - \frac{2}{3}e^{2} + 3e - 9$ |
| 41 | $[41, 41, \frac{2}{3}w^{3} - \frac{13}{3}w + \frac{5}{3}]$ | $-\frac{7}{3}e^{2} - \frac{23}{3}e + 9$ |
| 59 | $[59, 59, \frac{2}{3}w^{3} - \frac{13}{3}w + \frac{8}{3}]$ | $\phantom{-}\frac{1}{3}e^{3} + \frac{1}{3}e^{2} - \frac{20}{3}e - 2$ |
| 59 | $[59, 59, w^{3} + w^{2} - 6w - 4]$ | $\phantom{-}1$ |
| 79 | $[79, 79, \frac{2}{3}w^{3} + w^{2} - \frac{10}{3}w - \frac{19}{3}]$ | $-\frac{2}{3}e^{3} - 2e^{2} + \frac{11}{3}e + 1$ |
| 79 | $[79, 79, \frac{1}{3}w^{3} + w^{2} - \frac{2}{3}w - \frac{17}{3}]$ | $\phantom{-}e^{2} + 4e - 10$ |
| 89 | $[89, 89, \frac{4}{3}w^{3} - \frac{23}{3}w - \frac{5}{3}]$ | $\phantom{-}e^{3} + \frac{19}{3}e^{2} + \frac{17}{3}e - 17$ |
| 89 | $[89, 89, \frac{1}{3}w^{3} + 2w^{2} - \frac{5}{3}w - \frac{32}{3}]$ | $\phantom{-}2e^{3} + \frac{25}{3}e^{2} - \frac{16}{3}e - 14$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $59$ | $[59,59,\frac{1}{3}w^{3} - \frac{2}{3}w + \frac{7}{3}]$ | $-1$ |