Base field 4.4.19525.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 14x^{2} + 15x + 45\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[25, 5, -\frac{2}{3}w^{2} + \frac{5}{3}w + 5]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $44$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 28x^{6} + 168x^{4} - 344x^{2} + 196\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 5 | $[5, 5, \frac{2}{3}w^{2} + \frac{1}{3}w - 5]$ | $\phantom{-}0$ |
| 5 | $[5, 5, \frac{2}{3}w^{2} - \frac{5}{3}w - 4]$ | $-\frac{5}{56}e^{7} + \frac{9}{4}e^{5} - \frac{35}{4}e^{3} + \frac{47}{7}e$ |
| 9 | $[9, 3, -w + 3]$ | $-\frac{5}{56}e^{7} + \frac{9}{4}e^{5} - \frac{35}{4}e^{3} + \frac{54}{7}e$ |
| 9 | $[9, 3, w + 2]$ | $-2$ |
| 11 | $[11, 11, \frac{2}{3}w^{2} + \frac{1}{3}w - 4]$ | $-\frac{1}{8}e^{6} + 3e^{4} - \frac{35}{4}e^{2} + \frac{5}{2}$ |
| 16 | $[16, 2, 2]$ | $-\frac{3}{56}e^{7} + \frac{11}{8}e^{5} - 6e^{3} + \frac{243}{28}e$ |
| 19 | $[19, 19, \frac{1}{3}w^{3} - \frac{10}{3}w - 3]$ | $\phantom{-}\frac{3}{14}e^{7} - \frac{21}{4}e^{5} + \frac{35}{2}e^{3} - \frac{115}{14}e$ |
| 19 | $[19, 19, \frac{1}{3}w^{3} - w^{2} - \frac{7}{3}w + 6]$ | $-\frac{1}{8}e^{6} + 3e^{4} - \frac{35}{4}e^{2} + \frac{5}{2}$ |
| 29 | $[29, 29, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + \frac{8}{3}w - 2]$ | $-\frac{1}{8}e^{7} + 3e^{5} - \frac{35}{4}e^{3} + \frac{3}{2}e$ |
| 29 | $[29, 29, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - 3w + 1]$ | $\phantom{-}\frac{3}{8}e^{6} - 9e^{4} + \frac{105}{4}e^{2} - \frac{15}{2}$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + 3w + 1]$ | $\phantom{-}\frac{1}{8}e^{6} - \frac{13}{4}e^{4} + \frac{57}{4}e^{2} - 13$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + \frac{8}{3}w - 4]$ | $\phantom{-}\frac{1}{4}e^{4} - \frac{11}{2}e^{2} + \frac{21}{2}$ |
| 49 | $[49, 7, \frac{1}{3}w^{3} - \frac{10}{3}w - 1]$ | $\phantom{-}\frac{5}{28}e^{7} - \frac{9}{2}e^{5} + \frac{35}{2}e^{3} - \frac{94}{7}e$ |
| 49 | $[49, 7, -\frac{1}{3}w^{3} + w^{2} + \frac{7}{3}w - 4]$ | $-\frac{3}{8}e^{6} + \frac{19}{2}e^{4} - \frac{149}{4}e^{2} + \frac{57}{2}$ |
| 59 | $[59, 59, -\frac{1}{3}w^{3} + \frac{5}{3}w^{2} + \frac{5}{3}w - 11]$ | $\phantom{-}\frac{11}{56}e^{7} - 5e^{5} + \frac{83}{4}e^{3} - \frac{351}{14}e$ |
| 59 | $[59, 59, -\frac{2}{3}w^{3} + \frac{5}{3}w^{2} + 5w - 13]$ | $-e^{6} + 25e^{4} - 94e^{2} + 76$ |
| 61 | $[61, 61, -\frac{1}{3}w^{3} + \frac{5}{3}w^{2} + \frac{2}{3}w - 9]$ | $-\frac{5}{56}e^{7} + \frac{9}{4}e^{5} - \frac{35}{4}e^{3} + \frac{54}{7}e$ |
| 61 | $[61, 61, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + 4w + 2]$ | $\phantom{-}\frac{3}{7}e^{7} - \frac{21}{2}e^{5} + 35e^{3} - \frac{115}{7}e$ |
| 61 | $[61, 61, \frac{2}{3}w^{2} - \frac{5}{3}w - 1]$ | $\phantom{-}\frac{3}{56}e^{7} - \frac{3}{2}e^{5} + \frac{35}{4}e^{3} - \frac{167}{14}e$ |
| 61 | $[61, 61, -\frac{1}{3}w^{2} + \frac{4}{3}w + 6]$ | $-\frac{29}{56}e^{7} + 13e^{5} - \frac{197}{4}e^{3} + \frac{513}{14}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $5$ | $[5, 5, \frac{2}{3}w^{2} + \frac{1}{3}w - 5]$ | $-1$ |