Base field 3.3.1300.1
Generator \(w\), with minimal polynomial \(x^{3} - 10x - 10\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[13, 13, w - 3]$ |
| Dimension: | $9$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $26$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{9} + 3x^{8} - 6x^{7} - 21x^{6} + 7x^{5} + 40x^{4} + 5x^{3} - 18x^{2} - 3x + 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w - 2]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -w^{2} + 2w + 5]$ | $-e^{8} - 3e^{7} + 6e^{6} + 21e^{5} - 7e^{4} - 39e^{3} - 4e^{2} + 14e + 1$ |
| 7 | $[7, 7, w + 3]$ | $-\frac{1}{3}e^{8} - \frac{1}{3}e^{7} + \frac{11}{3}e^{6} + \frac{8}{3}e^{5} - \frac{41}{3}e^{4} - 6e^{3} + \frac{55}{3}e^{2} + \frac{7}{3}e - \frac{14}{3}$ |
| 11 | $[11, 11, -2w^{2} + 5w + 9]$ | $\phantom{-}\frac{4}{3}e^{8} + \frac{10}{3}e^{7} - \frac{29}{3}e^{6} - \frac{71}{3}e^{5} + \frac{59}{3}e^{4} + 46e^{3} - \frac{25}{3}e^{2} - \frac{67}{3}e - \frac{4}{3}$ |
| 13 | $[13, 13, -w^{2} + w + 7]$ | $\phantom{-}\frac{1}{3}e^{8} + \frac{4}{3}e^{7} - \frac{5}{3}e^{6} - \frac{29}{3}e^{5} + \frac{2}{3}e^{4} + 19e^{3} + \frac{5}{3}e^{2} - \frac{25}{3}e + \frac{5}{3}$ |
| 13 | $[13, 13, w - 3]$ | $-1$ |
| 17 | $[17, 17, -w^{2} + 2w + 7]$ | $-\frac{4}{3}e^{8} - \frac{7}{3}e^{7} + \frac{35}{3}e^{6} + \frac{47}{3}e^{5} - \frac{98}{3}e^{4} - 26e^{3} + \frac{88}{3}e^{2} + \frac{22}{3}e - \frac{17}{3}$ |
| 17 | $[17, 17, -2w^{2} + 5w + 7]$ | $-\frac{5}{3}e^{8} - \frac{14}{3}e^{7} + \frac{31}{3}e^{6} + \frac{94}{3}e^{5} - \frac{43}{3}e^{4} - 54e^{3} - \frac{7}{3}e^{2} + \frac{53}{3}e + \frac{2}{3}$ |
| 17 | $[17, 17, -w^{2} + 3w + 3]$ | $\phantom{-}\frac{2}{3}e^{8} + \frac{5}{3}e^{7} - \frac{16}{3}e^{6} - \frac{37}{3}e^{5} + \frac{40}{3}e^{4} + 24e^{3} - \frac{35}{3}e^{2} - \frac{23}{3}e + \frac{7}{3}$ |
| 19 | $[19, 19, -w + 1]$ | $-\frac{1}{3}e^{8} - \frac{4}{3}e^{7} + \frac{2}{3}e^{6} + \frac{26}{3}e^{5} + \frac{19}{3}e^{4} - 16e^{3} - \frac{47}{3}e^{2} + \frac{28}{3}e + \frac{10}{3}$ |
| 27 | $[27, 3, -3]$ | $\phantom{-}\frac{2}{3}e^{8} + \frac{8}{3}e^{7} - \frac{7}{3}e^{6} - \frac{55}{3}e^{5} - \frac{20}{3}e^{4} + 33e^{3} + \frac{70}{3}e^{2} - \frac{29}{3}e - \frac{26}{3}$ |
| 31 | $[31, 31, w^{2} - 2w - 9]$ | $\phantom{-}2e^{8} + 4e^{7} - 16e^{6} - 27e^{5} + 40e^{4} + 47e^{3} - 32e^{2} - 14e + 6$ |
| 37 | $[37, 37, w^{2} - 7]$ | $\phantom{-}e^{8} + 2e^{7} - 7e^{6} - 12e^{5} + 14e^{4} + 16e^{3} - 8e^{2} - e - 1$ |
| 41 | $[41, 41, w^{2} - w - 11]$ | $-e^{7} - 2e^{6} + 8e^{5} + 12e^{4} - 22e^{3} - 16e^{2} + 20e$ |
| 47 | $[47, 47, w^{2} - 3]$ | $\phantom{-}\frac{5}{3}e^{8} + \frac{14}{3}e^{7} - \frac{28}{3}e^{6} - \frac{91}{3}e^{5} + \frac{25}{3}e^{4} + 49e^{3} + \frac{22}{3}e^{2} - \frac{35}{3}e + \frac{4}{3}$ |
| 49 | $[49, 7, w^{2} - 3w - 1]$ | $\phantom{-}\frac{2}{3}e^{8} + \frac{5}{3}e^{7} - \frac{13}{3}e^{6} - \frac{28}{3}e^{5} + \frac{22}{3}e^{4} + 6e^{3} - \frac{5}{3}e^{2} + \frac{43}{3}e + \frac{4}{3}$ |
| 59 | $[59, 59, w^{2} - 4w + 1]$ | $-\frac{1}{3}e^{8} - \frac{4}{3}e^{7} + \frac{2}{3}e^{6} + \frac{23}{3}e^{5} + \frac{22}{3}e^{4} - 6e^{3} - \frac{65}{3}e^{2} - \frac{29}{3}e + \frac{34}{3}$ |
| 67 | $[67, 67, w^{2} - 3w - 7]$ | $\phantom{-}e^{8} + 3e^{7} - 5e^{6} - 19e^{5} + 27e^{3} + 20e^{2} + 3e - 12$ |
| 71 | $[71, 71, 4w + 9]$ | $\phantom{-}\frac{10}{3}e^{8} + \frac{25}{3}e^{7} - \frac{68}{3}e^{6} - \frac{170}{3}e^{5} + \frac{119}{3}e^{4} + 100e^{3} - \frac{34}{3}e^{2} - \frac{106}{3}e + \frac{17}{3}$ |
| 73 | $[73, 73, -4w^{2} + 8w + 23]$ | $-e^{7} - 4e^{6} + 5e^{5} + 29e^{4} - e^{3} - 56e^{2} - 14e + 16$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $13$ | $[13, 13, w - 3]$ | $1$ |