Base field 4.4.7625.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 9x^{2} + 4x + 16\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[31,31,-\frac{3}{4}w^{3} + \frac{7}{4}w^{2} + \frac{11}{4}w - 5]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $20$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 30x^{6} + 301x^{4} - 1040x^{2} + 304\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{3}{2}w + 4]$ | $\phantom{-}\frac{1}{2}e^{3} - \frac{11}{2}e$ |
| 4 | $[4, 2, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 5]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -\frac{1}{4}w^{3} - \frac{3}{4}w^{2} + \frac{5}{4}w + 3]$ | $-\frac{1}{4}e^{6} + 5e^{4} - \frac{103}{4}e^{2} + 11$ |
| 11 | $[11, 11, -\frac{1}{4}w^{3} + \frac{1}{4}w^{2} + \frac{9}{4}w]$ | $-\frac{1}{4}e^{6} + 5e^{4} - \frac{103}{4}e^{2} + 11$ |
| 11 | $[11, 11, w - 1]$ | $-\frac{1}{8}e^{6} + \frac{9}{4}e^{4} - \frac{77}{8}e^{2} + \frac{3}{2}$ |
| 29 | $[29, 29, \frac{1}{4}w^{3} - \frac{1}{4}w^{2} - \frac{1}{4}w + 2]$ | $\phantom{-}\frac{3}{16}e^{7} - \frac{27}{8}e^{5} + \frac{215}{16}e^{3} + \frac{51}{4}e$ |
| 29 | $[29, 29, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w + 3]$ | $\phantom{-}\frac{3}{16}e^{7} - \frac{31}{8}e^{5} + \frac{343}{16}e^{3} - \frac{59}{4}e$ |
| 31 | $[31, 31, -\frac{3}{4}w^{3} - \frac{1}{4}w^{2} + \frac{19}{4}w + 4]$ | $-\frac{1}{2}e^{4} + \frac{9}{2}e^{2} + 6$ |
| 31 | $[31, 31, -\frac{3}{4}w^{3} + \frac{7}{4}w^{2} + \frac{11}{4}w - 5]$ | $-1$ |
| 49 | $[49, 7, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{5}{2}w - 5]$ | $-e^{3} + 9e$ |
| 49 | $[49, 7, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ | $\phantom{-}\frac{1}{8}e^{7} - \frac{11}{4}e^{5} + \frac{129}{8}e^{3} - \frac{27}{2}e$ |
| 59 | $[59, 59, -\frac{1}{4}w^{3} + \frac{5}{4}w^{2} + \frac{5}{4}w - 3]$ | $\phantom{-}\frac{3}{16}e^{7} - \frac{27}{8}e^{5} + \frac{215}{16}e^{3} + \frac{51}{4}e$ |
| 59 | $[59, 59, w^{2} - 7]$ | $-\frac{1}{16}e^{7} + \frac{9}{8}e^{5} - \frac{53}{16}e^{3} - \frac{67}{4}e$ |
| 61 | $[61, 61, -\frac{5}{4}w^{3} - \frac{7}{4}w^{2} + \frac{37}{4}w + 15]$ | $\phantom{-}\frac{5}{8}e^{6} - \frac{49}{4}e^{4} + \frac{497}{8}e^{2} - \frac{49}{2}$ |
| 71 | $[71, 71, \frac{3}{4}w^{3} + \frac{9}{4}w^{2} - \frac{11}{4}w - 7]$ | $-\frac{1}{8}e^{6} + \frac{9}{4}e^{4} - \frac{77}{8}e^{2} + \frac{11}{2}$ |
| 71 | $[71, 71, \frac{1}{4}w^{3} - \frac{1}{4}w^{2} - \frac{13}{4}w - 2]$ | $\phantom{-}\frac{1}{2}e^{6} - 10e^{4} + \frac{99}{2}e^{2} - 4$ |
| 79 | $[79, 79, -\frac{1}{4}w^{3} + \frac{5}{4}w^{2} + \frac{1}{4}w - 7]$ | $-\frac{7}{16}e^{7} + \frac{71}{8}e^{5} - \frac{739}{16}e^{3} + \frac{67}{4}e$ |
| 79 | $[79, 79, \frac{1}{4}w^{3} + \frac{3}{4}w^{2} - \frac{9}{4}w - 2]$ | $-\frac{3}{16}e^{7} + \frac{31}{8}e^{5} - \frac{327}{16}e^{3} + \frac{39}{4}e$ |
| 81 | $[81, 3, -3]$ | $\phantom{-}\frac{9}{8}e^{6} - \frac{89}{4}e^{4} + \frac{893}{8}e^{2} - \frac{45}{2}$ |
| 89 | $[89, 89, -w^{2} + 2w + 1]$ | $-\frac{1}{16}e^{7} + \frac{9}{8}e^{5} - \frac{85}{16}e^{3} + \frac{13}{4}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31,31,-\frac{3}{4}w^{3} + \frac{7}{4}w^{2} + \frac{11}{4}w - 5]$ | $1$ |