Base field \(\Q(\zeta_{28})^+\)
Generator \(w\), with minimal polynomial \(x^{6} - 7x^{4} + 14x^{2} - 7\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[8, 2, w^{3} + w^{2} - 2w - 1]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $4$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} + 6x^{3} - 95x^{2} - 312x - 236\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 7 | $[7, 7, w]$ | $-\frac{1}{7}e^{3} - \frac{9}{14}e^{2} + \frac{207}{14}e + \frac{162}{7}$ |
| 8 | $[8, 2, w^{3} + w^{2} - 2w - 1]$ | $-1$ |
| 27 | $[27, 3, -w^{5} - w^{4} + 5w^{3} + 5w^{2} - 5w - 6]$ | $-\frac{2}{7}e^{3} - \frac{9}{7}e^{2} + \frac{207}{7}e + \frac{324}{7}$ |
| 27 | $[27, 3, w^{4} - 4w^{2} - w + 1]$ | $-\frac{2}{7}e^{3} - \frac{9}{7}e^{2} + \frac{207}{7}e + \frac{324}{7}$ |
| 29 | $[29, 29, w^{2} + w - 3]$ | $\phantom{-}\frac{2}{7}e^{3} + \frac{9}{7}e^{2} - \frac{200}{7}e - \frac{303}{7}$ |
| 29 | $[29, 29, -w^{4} - w^{3} + 5w^{2} + 3w - 4]$ | $\phantom{-}\frac{2}{7}e^{3} + \frac{9}{7}e^{2} - \frac{200}{7}e - \frac{303}{7}$ |
| 29 | $[29, 29, w^{5} + w^{4} - 5w^{3} - 4w^{2} + 5w + 3]$ | $\phantom{-}\frac{2}{7}e^{3} + \frac{9}{7}e^{2} - \frac{200}{7}e - \frac{303}{7}$ |
| 29 | $[29, 29, -w^{5} + w^{4} + 5w^{3} - 4w^{2} - 5w + 3]$ | $-\frac{2}{7}e^{3} - \frac{9}{7}e^{2} + \frac{200}{7}e + \frac{324}{7}$ |
| 29 | $[29, 29, w^{4} - w^{3} - 5w^{2} + 3w + 4]$ | $-\frac{2}{7}e^{3} - \frac{9}{7}e^{2} + \frac{200}{7}e + \frac{324}{7}$ |
| 29 | $[29, 29, -w^{2} + w + 3]$ | $-\frac{2}{7}e^{3} - \frac{9}{7}e^{2} + \frac{200}{7}e + \frac{324}{7}$ |
| 83 | $[83, 83, w^{4} - w^{3} - 4w^{2} + 2w + 1]$ | $-\frac{3}{14}e^{3} - \frac{5}{7}e^{2} + \frac{321}{14}e + \frac{152}{7}$ |
| 83 | $[83, 83, w^{5} - w^{4} - 4w^{3} + 6w^{2} + 3w - 5]$ | $-\frac{3}{14}e^{3} - \frac{5}{7}e^{2} + \frac{321}{14}e + \frac{152}{7}$ |
| 83 | $[83, 83, -w^{5} - w^{4} + 5w^{3} + 5w^{2} - 4w - 6]$ | $-\frac{3}{14}e^{3} - \frac{17}{14}e^{2} + \frac{150}{7}e + \frac{334}{7}$ |
| 83 | $[83, 83, w^{5} - w^{4} - 5w^{3} + 5w^{2} + 4w - 6]$ | $-\frac{3}{14}e^{3} - \frac{5}{7}e^{2} + \frac{321}{14}e + \frac{152}{7}$ |
| 83 | $[83, 83, -w^{5} + w^{4} + 6w^{3} - 3w^{2} - 9w + 2]$ | $-\frac{3}{14}e^{3} - \frac{17}{14}e^{2} + \frac{150}{7}e + \frac{334}{7}$ |
| 83 | $[83, 83, -w^{4} - w^{3} + 4w^{2} + 2w - 1]$ | $-\frac{3}{14}e^{3} - \frac{17}{14}e^{2} + \frac{150}{7}e + \frac{334}{7}$ |
| 113 | $[113, 113, -w^{5} - w^{4} + 5w^{3} + 3w^{2} - 5w - 1]$ | $\phantom{-}\frac{4}{7}e^{3} + \frac{18}{7}e^{2} - \frac{400}{7}e - \frac{564}{7}$ |
| 113 | $[113, 113, 2w^{4} - w^{3} - 9w^{2} + 3w + 6]$ | $-\frac{4}{7}e^{3} - \frac{18}{7}e^{2} + \frac{400}{7}e + \frac{690}{7}$ |
| 113 | $[113, 113, w^{4} - 6w^{2} - w + 8]$ | $\phantom{-}\frac{4}{7}e^{3} + \frac{18}{7}e^{2} - \frac{400}{7}e - \frac{564}{7}$ |
| 113 | $[113, 113, w^{4} - 6w^{2} + w + 8]$ | $-\frac{4}{7}e^{3} - \frac{18}{7}e^{2} + \frac{400}{7}e + \frac{690}{7}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $8$ | $[8, 2, w^{3} + w^{2} - 2w - 1]$ | $1$ |