Base field \(\Q(\zeta_{13})^+\)
Generator \(w\), with minimal polynomial \(x^{6} - x^{5} - 5x^{4} + 4x^{3} + 6x^{2} - 3x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[53, 53, -w^{4} + w^{3} + 3w^{2} - 2w + 1]$ |
| Dimension: | $3$ |
| 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:
| \(x^{3} - 5x^{2} - 11x + 42\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 13 | $[13, 13, w^{5} - 5w^{3} + 4w]$ | $\phantom{-}e$ |
| 25 | $[25, 5, w^{5} - 5w^{3} + 6w - 1]$ | $\phantom{-}e - 5$ |
| 25 | $[25, 5, -w^{3} + w^{2} + 3w - 1]$ | $\phantom{-}\frac{1}{3}e^{2} - \frac{1}{3}e - 3$ |
| 25 | $[25, 5, w^{5} - 4w^{3} - w^{2} + 3w + 2]$ | $\phantom{-}\frac{1}{3}e^{2} - \frac{4}{3}e - 8$ |
| 27 | $[27, 3, w^{4} - w^{3} - 4w^{2} + 2w + 2]$ | $-e + 2$ |
| 27 | $[27, 3, w^{4} - w^{3} - 4w^{2} + 2w + 3]$ | $\phantom{-}\frac{2}{3}e^{2} - \frac{8}{3}e - 6$ |
| 53 | $[53, 53, -w^{4} + w^{3} + 3w^{2} - 2w + 1]$ | $-1$ |
| 53 | $[53, 53, -w^{4} + w^{3} + 4w^{2} - 3w - 4]$ | $-\frac{2}{3}e^{2} - \frac{1}{3}e + 12$ |
| 53 | $[53, 53, -w^{5} + w^{4} + 4w^{3} - 4w^{2} - 3w + 1]$ | $-2e + 4$ |
| 53 | $[53, 53, w^{3} - 2w - 2]$ | $-\frac{2}{3}e^{2} + \frac{8}{3}e + 10$ |
| 53 | $[53, 53, w^{5} - 5w^{3} - w^{2} + 5w]$ | $-\frac{1}{3}e^{2} + \frac{1}{3}e + 7$ |
| 53 | $[53, 53, -w^{4} + 4w^{2} + w - 4]$ | $\phantom{-}e^{2} - 3e - 9$ |
| 64 | $[64, 2, -2]$ | $\phantom{-}2e - 1$ |
| 79 | $[79, 79, -2w^{5} + w^{4} + 9w^{3} - 3w^{2} - 9w + 2]$ | $\phantom{-}4$ |
| 79 | $[79, 79, -w^{5} - w^{4} + 5w^{3} + 4w^{2} - 6w - 1]$ | $\phantom{-}\frac{2}{3}e^{2} + \frac{1}{3}e - 11$ |
| 79 | $[79, 79, w^{3} - w^{2} - 4w + 1]$ | $-e^{2} + e + 17$ |
| 79 | $[79, 79, -2w^{5} + 2w^{4} + 9w^{3} - 7w^{2} - 9w + 3]$ | $-\frac{1}{3}e^{2} + \frac{7}{3}e + 7$ |
| 79 | $[79, 79, -2w^{4} + w^{3} + 7w^{2} - 3w - 3]$ | $-\frac{4}{3}e^{2} + \frac{7}{3}e + 15$ |
| 79 | $[79, 79, -w^{5} + 6w^{3} - w^{2} - 8w + 1]$ | $\phantom{-}\frac{4}{3}e^{2} - \frac{10}{3}e - 12$ |
| 103 | $[103, 103, 2w^{4} - 7w^{2} - w + 3]$ | $-\frac{1}{3}e^{2} + \frac{10}{3}e - 2$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $53$ | $[53, 53, -w^{4} + w^{3} + 3w^{2} - 2w + 1]$ | $1$ |