Base field 5.5.170701.1
Generator \(w\), with minimal polynomial \(x^{5} - x^{4} - 6x^{3} + 4x + 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2]$ |
| Level: | $[13, 13, -w^{4} + 2w^{3} + 5w^{2} - 5w - 3]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $19$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} + x^{3} - 11x^{2} - 14x - 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -w^{4} + w^{3} + 6w^{2} - w - 3]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w^{4} - w^{3} - 6w^{2} + w + 2]$ | $-\frac{1}{7}e^{3} - \frac{3}{7}e^{2} + \frac{12}{7}e + \frac{24}{7}$ |
| 8 | $[8, 2, -2w^{4} + 3w^{3} + 10w^{2} - 5w - 3]$ | $-\frac{2}{7}e^{3} + \frac{1}{7}e^{2} + \frac{17}{7}e - \frac{1}{7}$ |
| 11 | $[11, 11, w^{4} - w^{3} - 6w^{2} + 2]$ | $\phantom{-}\frac{2}{7}e^{3} - \frac{1}{7}e^{2} - \frac{24}{7}e - \frac{13}{7}$ |
| 13 | $[13, 13, -w^{4} + 2w^{3} + 5w^{2} - 5w - 3]$ | $-1$ |
| 23 | $[23, 23, -w^{2} + 2]$ | $-\frac{1}{7}e^{3} - \frac{3}{7}e^{2} + \frac{19}{7}e + \frac{17}{7}$ |
| 23 | $[23, 23, -w + 2]$ | $-\frac{9}{7}e^{3} + \frac{1}{7}e^{2} + \frac{94}{7}e + \frac{34}{7}$ |
| 31 | $[31, 31, -w^{4} + 2w^{3} + 5w^{2} - 6w - 4]$ | $-\frac{4}{7}e^{3} + \frac{2}{7}e^{2} + \frac{27}{7}e - \frac{9}{7}$ |
| 43 | $[43, 43, -w^{3} + 2w^{2} + 3w - 2]$ | $\phantom{-}\frac{12}{7}e^{3} + \frac{1}{7}e^{2} - \frac{130}{7}e - \frac{71}{7}$ |
| 47 | $[47, 47, -w^{4} + 2w^{3} + 5w^{2} - 4w - 4]$ | $-\frac{4}{7}e^{3} + \frac{9}{7}e^{2} + \frac{27}{7}e - \frac{72}{7}$ |
| 53 | $[53, 53, w^{2} - w - 4]$ | $\phantom{-}\frac{5}{7}e^{3} + \frac{1}{7}e^{2} - \frac{32}{7}e - \frac{8}{7}$ |
| 53 | $[53, 53, -w^{4} + 2w^{3} + 4w^{2} - 4w - 3]$ | $-\frac{1}{7}e^{3} + \frac{4}{7}e^{2} + \frac{5}{7}e - \frac{18}{7}$ |
| 53 | $[53, 53, -w^{3} + w^{2} + 4w - 1]$ | $\phantom{-}\frac{3}{7}e^{3} + \frac{2}{7}e^{2} - \frac{50}{7}e - \frac{37}{7}$ |
| 71 | $[71, 71, w^{4} - w^{3} - 7w^{2} + 3w + 6]$ | $\phantom{-}2e^{3} - 21e - 13$ |
| 71 | $[71, 71, -2w^{4} + 2w^{3} + 11w^{2} - 2]$ | $\phantom{-}\frac{12}{7}e^{3} + \frac{8}{7}e^{2} - \frac{137}{7}e - \frac{99}{7}$ |
| 71 | $[71, 71, -w^{4} + 3w^{3} + 3w^{2} - 9w - 1]$ | $\phantom{-}\frac{5}{7}e^{3} - \frac{6}{7}e^{2} - \frac{39}{7}e - \frac{8}{7}$ |
| 73 | $[73, 73, 2w^{4} - 4w^{3} - 9w^{2} + 10w + 5]$ | $-\frac{6}{7}e^{3} - \frac{11}{7}e^{2} + \frac{72}{7}e + \frac{95}{7}$ |
| 79 | $[79, 79, 2w^{4} - 3w^{3} - 11w^{2} + 4w + 8]$ | $\phantom{-}\frac{4}{7}e^{3} - \frac{9}{7}e^{2} - \frac{41}{7}e - \frac{26}{7}$ |
| 83 | $[83, 83, -3w^{4} + 5w^{3} + 15w^{2} - 9w - 10]$ | $\phantom{-}\frac{15}{7}e^{3} + \frac{3}{7}e^{2} - \frac{166}{7}e - \frac{80}{7}$ |
| 89 | $[89, 89, w^{3} - w^{2} - 5w + 1]$ | $\phantom{-}\frac{1}{7}e^{3} + \frac{3}{7}e^{2} - \frac{26}{7}e - \frac{24}{7}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, -w^{4} + 2w^{3} + 5w^{2} - 5w - 3]$ | $1$ |