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: | $[25,5,\frac{1}{3}w^{3} - w^{2} - \frac{5}{3}w + 4]$ |
Dimension: | $8$ |
CM: | no |
Base change: | no |
Newspace dimension: | $36$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{8} - 31x^{6} + 236x^{4} - 131x^{2} + 1\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
2 | $[2, 2, \frac{1}{3}w^{3} + w^{2} - \frac{2}{3}w - 2]$ | $-\frac{1}{60}e^{6} + \frac{8}{15}e^{4} - \frac{62}{15}e^{2} - \frac{7}{20}$ |
7 | $[7, 7, \frac{1}{3}w^{3} - \frac{8}{3}w + 2]$ | $\phantom{-}e$ |
7 | $[7, 7, -w + 2]$ | $-\frac{7}{20}e^{7} + \frac{163}{15}e^{5} - \frac{1247}{15}e^{3} + \frac{2999}{60}e$ |
9 | $[9, 3, w]$ | $\phantom{-}\frac{1}{20}e^{6} - \frac{22}{15}e^{4} + \frac{31}{3}e^{2} - \frac{329}{60}$ |
9 | $[9, 3, -\frac{1}{3}w^{3} + \frac{8}{3}w]$ | $-\frac{1}{15}e^{6} + 2e^{4} - \frac{217}{15}e^{2} + \frac{47}{15}$ |
25 | $[25, 5, \frac{1}{3}w^{3} - w^{2} - \frac{5}{3}w + 4]$ | $\phantom{-}1$ |
25 | $[25, 5, -\frac{1}{3}w^{3} - w^{2} + \frac{5}{3}w + 4]$ | $\phantom{-}\frac{1}{60}e^{6} - \frac{8}{15}e^{4} + \frac{62}{15}e^{2} - \frac{93}{20}$ |
31 | $[31, 31, \frac{1}{3}w^{3} + w^{2} - \frac{5}{3}w - 2]$ | $\phantom{-}\frac{14}{15}e^{7} - \frac{433}{15}e^{5} + \frac{3277}{15}e^{3} - \frac{557}{5}e$ |
31 | $[31, 31, \frac{1}{3}w^{3} + w^{2} - \frac{5}{3}w - 6]$ | $-\frac{4}{15}e^{7} + \frac{41}{5}e^{5} - \frac{922}{15}e^{3} + \frac{431}{15}e$ |
31 | $[31, 31, \frac{1}{3}w^{3} - w^{2} - \frac{5}{3}w + 6]$ | $\phantom{-}\frac{17}{30}e^{7} - \frac{88}{5}e^{5} + \frac{2014}{15}e^{3} - \frac{2227}{30}e$ |
31 | $[31, 31, \frac{1}{3}w^{3} - w^{2} - \frac{5}{3}w + 2]$ | $\phantom{-}\frac{23}{30}e^{7} - \frac{356}{15}e^{5} + \frac{2696}{15}e^{3} - \frac{911}{10}e$ |
47 | $[47, 47, \frac{1}{3}w^{3} + w^{2} - \frac{8}{3}w - 3]$ | $\phantom{-}\frac{1}{30}e^{7} - \frac{16}{15}e^{5} + \frac{139}{15}e^{3} - \frac{183}{10}e$ |
47 | $[47, 47, -w^{2} - w + 5]$ | $-\frac{19}{60}e^{7} + \frac{49}{5}e^{5} - \frac{1108}{15}e^{3} + \frac{1841}{60}e$ |
47 | $[47, 47, w^{2} - w - 5]$ | $-\frac{37}{60}e^{7} + \frac{96}{5}e^{5} - \frac{443}{3}e^{3} + \frac{5471}{60}e$ |
47 | $[47, 47, -\frac{1}{3}w^{3} + w^{2} + \frac{8}{3}w - 3]$ | $-\frac{1}{20}e^{7} + \frac{22}{15}e^{5} - \frac{28}{3}e^{3} - \frac{511}{60}e$ |
103 | $[103, 103, \frac{2}{3}w^{3} + w^{2} - \frac{10}{3}w - 2]$ | $-\frac{5}{12}e^{7} + \frac{193}{15}e^{5} - \frac{483}{5}e^{3} + \frac{2347}{60}e$ |
103 | $[103, 103, -\frac{2}{3}w^{3} + w^{2} + \frac{10}{3}w - 6]$ | $-\frac{9}{20}e^{7} + 14e^{5} - \frac{537}{5}e^{3} + \frac{1283}{20}e$ |
103 | $[103, 103, \frac{2}{3}w^{3} + w^{2} - \frac{10}{3}w - 6]$ | $\phantom{-}\frac{2}{15}e^{7} - \frac{62}{15}e^{5} + 32e^{3} - \frac{401}{15}e$ |
103 | $[103, 103, \frac{1}{3}w^{3} + 2w^{2} + \frac{7}{3}w - 1]$ | $-\frac{13}{60}e^{7} + \frac{20}{3}e^{5} - \frac{248}{5}e^{3} + \frac{1111}{60}e$ |
113 | $[113, 113, -\frac{2}{3}w^{3} + \frac{16}{3}w + 1]$ | $-\frac{2}{15}e^{4} + \frac{46}{15}e^{2} - \frac{67}{15}$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$25$ | $[25,5,\frac{1}{3}w^{3} - w^{2} - \frac{5}{3}w + 4]$ | $-1$ |