Base field \(\Q(\sqrt{2}, \sqrt{7})\)
Generator \(w\), with minimal polynomial \(x^{4} - 8x^{2} + 9\); narrow class number \(4\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[18,6,\frac{1}{3}w^{3} - \frac{8}{3}w - 3]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $16$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 35x^{6} + 396x^{4} - 1504x^{2} + 1024\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, \frac{1}{3}w^{3} + w^{2} - \frac{2}{3}w - 2]$ | $-1$ |
| 7 | $[7, 7, \frac{1}{3}w^{3} - \frac{8}{3}w + 2]$ | $\phantom{-}e$ |
| 7 | $[7, 7, -w + 2]$ | $-\frac{1}{64}e^{7} + \frac{27}{64}e^{5} - \frac{53}{16}e^{3} + \frac{13}{2}e$ |
| 9 | $[9, 3, w]$ | $\phantom{-}\frac{1}{16}e^{6} - \frac{23}{16}e^{4} + \frac{17}{2}e^{2} - 10$ |
| 9 | $[9, 3, -\frac{1}{3}w^{3} + \frac{8}{3}w]$ | $-1$ |
| 25 | $[25, 5, \frac{1}{3}w^{3} - w^{2} - \frac{5}{3}w + 4]$ | $\phantom{-}\frac{1}{16}e^{6} - \frac{23}{16}e^{4} + \frac{15}{2}e^{2} - 2$ |
| 25 | $[25, 5, -\frac{1}{3}w^{3} - w^{2} + \frac{5}{3}w + 4]$ | $\phantom{-}e^{2} - 10$ |
| 31 | $[31, 31, \frac{1}{3}w^{3} + w^{2} - \frac{5}{3}w - 2]$ | $\phantom{-}\frac{1}{64}e^{7} - \frac{19}{64}e^{5} + \frac{7}{16}e^{3} + \frac{17}{2}e$ |
| 31 | $[31, 31, \frac{1}{3}w^{3} + w^{2} - \frac{5}{3}w - 6]$ | $-\frac{1}{64}e^{7} + \frac{19}{64}e^{5} - \frac{23}{16}e^{3} + \frac{11}{2}e$ |
| 31 | $[31, 31, \frac{1}{3}w^{3} - w^{2} - \frac{5}{3}w + 6]$ | $-\frac{1}{32}e^{7} + \frac{23}{32}e^{5} - \frac{19}{4}e^{3} + 11e$ |
| 31 | $[31, 31, \frac{1}{3}w^{3} - w^{2} - \frac{5}{3}w + 2]$ | $-\frac{1}{32}e^{7} + \frac{19}{32}e^{5} - \frac{15}{8}e^{3} - 4e$ |
| 47 | $[47, 47, \frac{1}{3}w^{3} + w^{2} - \frac{8}{3}w - 3]$ | $\phantom{-}\frac{1}{8}e^{5} - \frac{15}{8}e^{3} + 4e$ |
| 47 | $[47, 47, -w^{2} - w + 5]$ | $\phantom{-}\frac{1}{64}e^{7} - \frac{19}{64}e^{5} + \frac{7}{16}e^{3} + \frac{17}{2}e$ |
| 47 | $[47, 47, w^{2} - w - 5]$ | $-\frac{1}{32}e^{7} + \frac{19}{32}e^{5} - \frac{15}{8}e^{3} - 4e$ |
| 47 | $[47, 47, -\frac{1}{3}w^{3} + w^{2} + \frac{8}{3}w - 3]$ | $-\frac{1}{64}e^{7} + \frac{35}{64}e^{5} - \frac{83}{16}e^{3} + \frac{19}{2}e$ |
| 103 | $[103, 103, \frac{2}{3}w^{3} + w^{2} - \frac{10}{3}w - 2]$ | $-\frac{1}{32}e^{7} + \frac{31}{32}e^{5} - \frac{17}{2}e^{3} + 16e$ |
| 103 | $[103, 103, -\frac{2}{3}w^{3} + w^{2} + \frac{10}{3}w - 6]$ | $\phantom{-}\frac{1}{4}e^{5} - \frac{19}{4}e^{3} + 18e$ |
| 103 | $[103, 103, \frac{2}{3}w^{3} + w^{2} - \frac{10}{3}w - 6]$ | $-\frac{1}{32}e^{7} + \frac{23}{32}e^{5} - \frac{15}{4}e^{3}$ |
| 103 | $[103, 103, \frac{1}{3}w^{3} + 2w^{2} + \frac{7}{3}w - 1]$ | $\phantom{-}\frac{1}{8}e^{5} - \frac{15}{8}e^{3} + 5e$ |
| 113 | $[113, 113, -\frac{2}{3}w^{3} + \frac{16}{3}w + 1]$ | $-\frac{3}{16}e^{6} + \frac{53}{16}e^{4} - \frac{19}{2}e^{2} - 2$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2,2,-\frac{2}{3}w^{3} - w^{2} + \frac{13}{3}w + 6]$ | $1$ |
| $9$ | $[9,3,-\frac{1}{3}w^{3} + \frac{8}{3}w]$ | $1$ |