Base field 6.6.1312625.1
Generator \(w\), with minimal polynomial \(x^{6} - x^{5} - 7x^{4} + 7x^{3} + 12x^{2} - 12x - 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2, 2, 2]$ |
| Level: | $[41, 41, 2w^{5} + w^{4} - 13w^{3} - 5w^{2} + 19w + 4]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $30$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} - x^{3} - 12x^{2} + 8x - 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -w^{5} + 6w^{3} + w^{2} - 8w - 2]$ | $\phantom{-}e$ |
| 11 | $[11, 11, w + 1]$ | $-e$ |
| 16 | $[16, 2, -w^{5} + 6w^{3} + w^{2} - 7w - 2]$ | $\phantom{-}\frac{4}{5}e^{3} - \frac{1}{5}e^{2} - 9e + \frac{12}{5}$ |
| 19 | $[19, 19, -w^{3} + w^{2} + 4w - 3]$ | $-\frac{7}{5}e^{3} + \frac{3}{5}e^{2} + 17e - \frac{36}{5}$ |
| 29 | $[29, 29, 2w^{5} - 13w^{3} + 19w - 2]$ | $\phantom{-}\frac{6}{5}e^{3} - \frac{4}{5}e^{2} - 15e + \frac{13}{5}$ |
| 31 | $[31, 31, -2w^{5} + 12w^{3} + w^{2} - 16w - 2]$ | $-\frac{8}{5}e^{3} + \frac{7}{5}e^{2} + 18e - \frac{39}{5}$ |
| 41 | $[41, 41, -w^{5} + 7w^{3} + w^{2} - 12w - 2]$ | $-\frac{4}{5}e^{3} + \frac{1}{5}e^{2} + 11e - \frac{7}{5}$ |
| 41 | $[41, 41, 2w^{5} + w^{4} - 13w^{3} - 5w^{2} + 19w + 4]$ | $-1$ |
| 59 | $[59, 59, -w^{5} + 7w^{3} + w^{2} - 11w]$ | $-\frac{3}{5}e^{3} + \frac{2}{5}e^{2} + 7e - \frac{34}{5}$ |
| 61 | $[61, 61, w^{5} - 7w^{3} + 10w]$ | $\phantom{-}\frac{13}{5}e^{3} - \frac{12}{5}e^{2} - 29e + \frac{59}{5}$ |
| 61 | $[61, 61, -2w^{5} + 12w^{3} + w^{2} - 16w - 3]$ | $-5e^{3} + 4e^{2} + 59e - 21$ |
| 71 | $[71, 71, -2w^{5} - w^{4} + 12w^{3} + 5w^{2} - 16w - 4]$ | $\phantom{-}\frac{11}{5}e^{3} - \frac{9}{5}e^{2} - 26e + \frac{93}{5}$ |
| 71 | $[71, 71, -3w^{5} - w^{4} + 20w^{3} + 6w^{2} - 31w - 5]$ | $\phantom{-}\frac{7}{5}e^{3} - \frac{3}{5}e^{2} - 19e + \frac{31}{5}$ |
| 71 | $[71, 71, -w^{5} + 7w^{3} - w^{2} - 12w + 2]$ | $-\frac{1}{5}e^{3} + \frac{4}{5}e^{2} + e - \frac{3}{5}$ |
| 79 | $[79, 79, w^{5} + w^{4} - 7w^{3} - 5w^{2} + 10w + 4]$ | $\phantom{-}\frac{4}{5}e^{3} - \frac{1}{5}e^{2} - 11e + \frac{7}{5}$ |
| 79 | $[79, 79, -w^{5} - w^{4} + 8w^{3} + 4w^{2} - 15w + 1]$ | $-\frac{11}{5}e^{3} + \frac{9}{5}e^{2} + 27e - \frac{3}{5}$ |
| 79 | $[79, 79, 2w^{5} + w^{4} - 13w^{3} - 5w^{2} + 20w + 3]$ | $-\frac{19}{5}e^{3} + \frac{11}{5}e^{2} + 45e - \frac{37}{5}$ |
| 89 | $[89, 89, 2w^{5} + w^{4} - 14w^{3} - 5w^{2} + 23w + 1]$ | $-\frac{18}{5}e^{3} + \frac{22}{5}e^{2} + 44e - \frac{129}{5}$ |
| 89 | $[89, 89, -2w^{5} - w^{4} + 12w^{3} + 5w^{2} - 15w - 4]$ | $-e^{3} + 14e + 6$ |
| 89 | $[89, 89, 2w^{5} + w^{4} - 11w^{3} - 5w^{2} + 13w + 3]$ | $\phantom{-}\frac{8}{5}e^{3} - \frac{2}{5}e^{2} - 19e + \frac{29}{5}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $41$ | $[41, 41, 2w^{5} + w^{4} - 13w^{3} - 5w^{2} + 19w + 4]$ | $1$ |