Base field 3.3.993.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 6x + 3\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[13, 13, w^{2} - w - 4]$ |
| Dimension: | $7$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $16$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{7} + x^{6} - 12x^{5} - 7x^{4} + 40x^{3} + 17x^{2} - 36x - 9\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, -w]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w - 1]$ | $-\frac{2}{3}e^{6} - \frac{5}{3}e^{5} + 4e^{4} + \frac{26}{3}e^{3} - \frac{5}{3}e^{2} - \frac{22}{3}e - 5$ |
| 5 | $[5, 5, -w + 2]$ | $\phantom{-}2e^{6} + 6e^{5} - 11e^{4} - 34e^{3} + 4e^{2} + 33e + 7$ |
| 7 | $[7, 7, w + 1]$ | $-e^{6} - 3e^{5} + 5e^{4} + 16e^{3} + e^{2} - 13e - 6$ |
| 8 | $[8, 2, 2]$ | $\phantom{-}\frac{1}{3}e^{6} - \frac{2}{3}e^{5} - 5e^{4} + \frac{20}{3}e^{3} + \frac{46}{3}e^{2} - \frac{31}{3}e - 8$ |
| 13 | $[13, 13, w^{2} - w - 4]$ | $-1$ |
| 17 | $[17, 17, -w^{2} + 2w + 1]$ | $-2e^{6} - 5e^{5} + 13e^{4} + 27e^{3} - 14e^{2} - 26e - 1$ |
| 25 | $[25, 5, -w^{2} - w + 4]$ | $\phantom{-}\frac{2}{3}e^{6} + \frac{8}{3}e^{5} - 3e^{4} - \frac{53}{3}e^{3} - \frac{7}{3}e^{2} + \frac{67}{3}e + 7$ |
| 31 | $[31, 31, w^{2} - w - 1]$ | $-\frac{20}{3}e^{6} - \frac{56}{3}e^{5} + 40e^{4} + \frac{317}{3}e^{3} - \frac{89}{3}e^{2} - \frac{322}{3}e - 16$ |
| 31 | $[31, 31, 2w^{2} - w - 14]$ | $\phantom{-}\frac{13}{3}e^{6} + \frac{46}{3}e^{5} - 20e^{4} - \frac{274}{3}e^{3} - \frac{17}{3}e^{2} + \frac{290}{3}e + 17$ |
| 31 | $[31, 31, w^{2} - 2]$ | $\phantom{-}e^{6} + 2e^{5} - 8e^{4} - 11e^{3} + 14e^{2} + 10e - 5$ |
| 37 | $[37, 37, -3w^{2} + w + 19]$ | $\phantom{-}\frac{17}{3}e^{6} + \frac{59}{3}e^{5} - 26e^{4} - \frac{347}{3}e^{3} - \frac{31}{3}e^{2} + \frac{358}{3}e + 29$ |
| 49 | $[49, 7, w^{2} - 2w - 4]$ | $-e^{5} - e^{4} + 8e^{3} + e^{2} - 13e + 7$ |
| 53 | $[53, 53, -w - 4]$ | $-\frac{10}{3}e^{6} - \frac{34}{3}e^{5} + 16e^{4} + \frac{199}{3}e^{3} + \frac{5}{3}e^{2} - \frac{200}{3}e - 18$ |
| 61 | $[61, 61, 2w^{2} + w - 7]$ | $-\frac{4}{3}e^{6} - \frac{13}{3}e^{5} + 6e^{4} + \frac{70}{3}e^{3} + \frac{11}{3}e^{2} - \frac{62}{3}e - 5$ |
| 71 | $[71, 71, 2w^{2} - 2w - 13]$ | $\phantom{-}9e^{6} + 26e^{5} - 50e^{4} - 144e^{3} + 17e^{2} + 135e + 40$ |
| 73 | $[73, 73, w - 5]$ | $\phantom{-}\frac{7}{3}e^{6} + \frac{28}{3}e^{5} - 7e^{4} - \frac{163}{3}e^{3} - \frac{74}{3}e^{2} + \frac{164}{3}e + 29$ |
| 83 | $[83, 83, w^{2} - 3w - 2]$ | $\phantom{-}\frac{11}{3}e^{6} + \frac{26}{3}e^{5} - 23e^{4} - \frac{131}{3}e^{3} + \frac{47}{3}e^{2} + \frac{106}{3}e + 12$ |
| 89 | $[89, 89, w^{2} + 2w - 4]$ | $\phantom{-}6e^{6} + 19e^{5} - 30e^{4} - 108e^{3} - 3e^{2} + 108e + 30$ |
| 97 | $[97, 97, -w^{2} + 2w + 7]$ | $\phantom{-}e^{6} + 8e^{5} + 4e^{4} - 53e^{3} - 37e^{2} + 64e + 20$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, w^{2} - w - 4]$ | $1$ |