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: | $[31, 31, -\frac{3}{4}w^{3} - \frac{1}{4}w^{2} + \frac{19}{4}w + 4]$ |
Dimension: | $6$ |
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:
\(x^{6} - 10x^{4} + 26x^{2} - 9\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
4 | $[4, 2, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{3}{2}w + 4]$ | $\phantom{-}e$ |
4 | $[4, 2, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - \frac{7}{2}w - 5]$ | $\phantom{-}e$ |
5 | $[5, 5, -\frac{1}{4}w^{3} - \frac{3}{4}w^{2} + \frac{5}{4}w + 3]$ | $\phantom{-}e^{2} - 3$ |
11 | $[11, 11, -\frac{1}{4}w^{3} + \frac{1}{4}w^{2} + \frac{9}{4}w]$ | $\phantom{-}2e^{2} - 6$ |
11 | $[11, 11, w - 1]$ | $-e^{4} + 5e^{2}$ |
29 | $[29, 29, \frac{1}{4}w^{3} - \frac{1}{4}w^{2} - \frac{1}{4}w + 2]$ | $\phantom{-}e^{5} - 8e^{3} + 13e$ |
29 | $[29, 29, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w + 3]$ | $\phantom{-}e^{3} - 3e$ |
31 | $[31, 31, -\frac{3}{4}w^{3} - \frac{1}{4}w^{2} + \frac{19}{4}w + 4]$ | $-1$ |
31 | $[31, 31, -\frac{3}{4}w^{3} + \frac{7}{4}w^{2} + \frac{11}{4}w - 5]$ | $-e^{4} + 6e^{2} - 5$ |
49 | $[49, 7, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{5}{2}w - 5]$ | $\phantom{-}e^{5} - 8e^{3} + 17e$ |
49 | $[49, 7, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{7}{2}w - 1]$ | $-5e^{3} + 23e$ |
59 | $[59, 59, -\frac{1}{4}w^{3} + \frac{5}{4}w^{2} + \frac{5}{4}w - 3]$ | $\phantom{-}e^{5} - 7e^{3} + 10e$ |
59 | $[59, 59, w^{2} - 7]$ | $-e^{5} + 5e^{3}$ |
61 | $[61, 61, -\frac{5}{4}w^{3} - \frac{7}{4}w^{2} + \frac{37}{4}w + 15]$ | $-3e^{4} + 17e^{2} - 8$ |
71 | $[71, 71, \frac{3}{4}w^{3} + \frac{9}{4}w^{2} - \frac{11}{4}w - 7]$ | $\phantom{-}2e^{4} - 12e^{2} + 18$ |
71 | $[71, 71, \frac{1}{4}w^{3} - \frac{1}{4}w^{2} - \frac{13}{4}w - 2]$ | $-2e^{4} + 11e^{2} - 9$ |
79 | $[79, 79, -\frac{1}{4}w^{3} + \frac{5}{4}w^{2} + \frac{1}{4}w - 7]$ | $-4e^{3} + 20e$ |
79 | $[79, 79, \frac{1}{4}w^{3} + \frac{3}{4}w^{2} - \frac{9}{4}w - 2]$ | $-e^{5} + 10e^{3} - 25e$ |
81 | $[81, 3, -3]$ | $\phantom{-}e^{4} - 5e^{2} - 2$ |
89 | $[89, 89, -w^{2} + 2w + 1]$ | $\phantom{-}2e^{5} - 16e^{3} + 32e$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$31$ | $[31, 31, -\frac{3}{4}w^{3} - \frac{1}{4}w^{2} + \frac{19}{4}w + 4]$ | $1$ |