Base field \(\Q(\zeta_{16})^+\)
Generator \(w\), with minimal polynomial \(x^{4} - 4x^{2} + 2\); narrow class number \(1\) and class number \(1\).
Form
Weight: | $[2, 2, 2, 2]$ |
Level: | $[79, 79, -w^{3} - w^{2} + 4w - 1]$ |
Dimension: | $5$ |
CM: | no |
Base change: | no |
Newspace dimension: | $6$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 5x - 4\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
2 | $[2, 2, w]$ | $\phantom{-}e$ |
17 | $[17, 17, -w^{2} - w + 3]$ | $-\frac{1}{2}e^{4} + e^{3} + \frac{3}{2}e^{2} - 3e + 2$ |
17 | $[17, 17, -w^{3} - w^{2} + 3w + 1]$ | $-e^{4} + \frac{1}{2}e^{3} + 6e^{2} - \frac{3}{2}e - 4$ |
17 | $[17, 17, w^{3} - w^{2} - 3w + 1]$ | $\phantom{-}\frac{3}{2}e^{4} - 2e^{3} - \frac{17}{2}e^{2} + 7e + 6$ |
17 | $[17, 17, w^{2} - w - 3]$ | $-e^{4} + e^{3} + 7e^{2} - 5e - 6$ |
31 | $[31, 31, w^{3} + w^{2} - 2w - 3]$ | $-\frac{1}{2}e^{3} - \frac{1}{2}e + 2$ |
31 | $[31, 31, -w^{3} + w^{2} + 4w - 1]$ | $-e^{4} + \frac{3}{2}e^{3} + 5e^{2} - \frac{13}{2}e + 2$ |
31 | $[31, 31, w^{3} + w^{2} - 4w - 1]$ | $\phantom{-}\frac{1}{2}e^{4} - e^{3} - \frac{7}{2}e^{2} + 4e + 8$ |
31 | $[31, 31, -w^{3} + w^{2} + 2w - 3]$ | $-e^{3} + 2e^{2} + 5e - 4$ |
47 | $[47, 47, -2w^{3} + w^{2} + 5w - 1]$ | $-\frac{1}{2}e^{4} + 2e^{3} + \frac{3}{2}e^{2} - 8e$ |
47 | $[47, 47, 2w^{3} + w^{2} - 6w - 1]$ | $-e^{4} + 2e^{3} + 7e^{2} - 10e - 8$ |
47 | $[47, 47, -2w^{3} + w^{2} + 6w - 1]$ | $-2e + 4$ |
47 | $[47, 47, 2w^{3} + w^{2} - 5w - 1]$ | $\phantom{-}e^{4} - \frac{3}{2}e^{3} - 6e^{2} + \frac{17}{2}e + 6$ |
49 | $[49, 7, w^{2} + 1]$ | $\phantom{-}e^{4} - 2e^{3} - 5e^{2} + 10e - 2$ |
49 | $[49, 7, -2w^{2} + 3]$ | $\phantom{-}e^{4} - 9e^{2} + 10$ |
79 | $[79, 79, -w^{3} - w^{2} + 4w - 1]$ | $-1$ |
79 | $[79, 79, -w^{3} + w^{2} + 2w - 5]$ | $\phantom{-}\frac{1}{2}e^{4} - 4e^{3} - \frac{3}{2}e^{2} + 19e + 4$ |
79 | $[79, 79, w^{3} + w^{2} - 2w - 5]$ | $\phantom{-}\frac{1}{2}e^{4} - e^{3} - \frac{7}{2}e^{2} + 5e + 8$ |
79 | $[79, 79, w^{3} - w^{2} - 4w - 1]$ | $-\frac{5}{2}e^{4} + 4e^{3} + \frac{27}{2}e^{2} - 16e - 4$ |
81 | $[81, 3, -3]$ | $\phantom{-}\frac{1}{2}e^{4} - 2e^{3} - \frac{7}{2}e^{2} + 13e - 2$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$79$ | $[79, 79, -w^{3} - w^{2} + 4w - 1]$ | $1$ |