Base field 4.4.14725.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 13x^{2} + 11x + 31\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[29,29,-\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + \frac{5}{3}w + \frac{14}{3}]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $40$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 24x^{6} + 184x^{4} - 448x^{2} + 64\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + \frac{11}{3}w + \frac{11}{3}]$ | $\phantom{-}e$ |
| 9 | $[9, 3, w - 2]$ | $\phantom{-}\frac{1}{16}e^{7} - \frac{9}{8}e^{5} + \frac{23}{4}e^{3} - 8e$ |
| 11 | $[11, 11, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + \frac{5}{3}w + \frac{8}{3}]$ | $\phantom{-}\frac{1}{4}e^{5} - \frac{13}{4}e^{3} + 8e$ |
| 11 | $[11, 11, \frac{2}{3}w^{3} + \frac{2}{3}w^{2} - \frac{19}{3}w - \frac{13}{3}]$ | $-\frac{1}{8}e^{5} + \frac{7}{4}e^{3} - 6e$ |
| 16 | $[16, 2, 2]$ | $\phantom{-}\frac{1}{2}e^{2} - 3$ |
| 19 | $[19, 19, w + 2]$ | $-\frac{1}{16}e^{7} + e^{5} - \frac{19}{4}e^{3} + 9e$ |
| 25 | $[25, 5, -\frac{2}{3}w^{3} - \frac{2}{3}w^{2} + \frac{16}{3}w + \frac{13}{3}]$ | $-e^{2} + 2$ |
| 29 | $[29, 29, -\frac{2}{3}w^{3} - \frac{2}{3}w^{2} + \frac{19}{3}w + \frac{19}{3}]$ | $\phantom{-}\frac{1}{4}e^{6} - \frac{15}{4}e^{4} + 14e^{2} - 10$ |
| 29 | $[29, 29, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + \frac{5}{3}w + \frac{14}{3}]$ | $\phantom{-}1$ |
| 29 | $[29, 29, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - \frac{11}{3}w - \frac{8}{3}]$ | $\phantom{-}\frac{1}{4}e^{5} - 3e^{3} + 5e$ |
| 29 | $[29, 29, w - 1]$ | $\phantom{-}\frac{1}{16}e^{7} - e^{5} + \frac{19}{4}e^{3} - 8e$ |
| 31 | $[31, 31, w]$ | $-\frac{1}{16}e^{7} + \frac{5}{8}e^{5} + \frac{5}{4}e^{3} - 13e$ |
| 31 | $[31, 31, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{13}{3}]$ | $-\frac{1}{8}e^{6} + \frac{9}{4}e^{4} - 11e^{2} + 8$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + \frac{11}{3}w + \frac{5}{3}]$ | $\phantom{-}\frac{1}{2}e^{3} - 3e$ |
| 41 | $[41, 41, \frac{1}{3}w^{3} + \frac{4}{3}w^{2} - \frac{8}{3}w - \frac{20}{3}]$ | $\phantom{-}\frac{1}{8}e^{6} - \frac{3}{2}e^{4} + 3e^{2}$ |
| 41 | $[41, 41, \frac{2}{3}w^{3} + \frac{5}{3}w^{2} - \frac{16}{3}w - \frac{40}{3}]$ | $-e^{2} + 2$ |
| 49 | $[49, 7, -\frac{1}{3}w^{3} - \frac{4}{3}w^{2} + \frac{8}{3}w + \frac{38}{3}]$ | $\phantom{-}\frac{1}{8}e^{6} - \frac{9}{4}e^{4} + 10e^{2} - 10$ |
| 49 | $[49, 7, \frac{2}{3}w^{3} + \frac{5}{3}w^{2} - \frac{16}{3}w - \frac{43}{3}]$ | $-\frac{3}{4}e^{4} + 7e^{2} - 6$ |
| 59 | $[59, 59, \frac{5}{3}w^{3} + \frac{8}{3}w^{2} - \frac{40}{3}w - \frac{49}{3}]$ | $-\frac{3}{16}e^{6} + \frac{5}{2}e^{4} - 6e^{2} - 6$ |
| 59 | $[59, 59, \frac{2}{3}w^{3} - \frac{1}{3}w^{2} - \frac{16}{3}w + \frac{11}{3}]$ | $-\frac{3}{8}e^{6} + \frac{21}{4}e^{4} - 16e^{2} + 4$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $29$ | $[29,29,-\frac{1}{3}w^{3} - \frac{1}{3}w^{2} + \frac{5}{3}w + \frac{14}{3}]$ | $-1$ |