Base field 3.3.761.1
Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 6x - 1\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2]$ |
| Level: | $[19, 19, -w^{2} + 2w + 5]$ |
| Dimension: | $8$ |
| 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:
| \(x^{8} + 2x^{7} - 16x^{6} - 32x^{5} + 72x^{4} + 156x^{3} - 64x^{2} - 240x - 104\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w + 1]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w - 1]$ | $\phantom{-}e^{7} + \frac{3}{4}e^{6} - 17e^{5} - 11e^{4} + 86e^{3} + 51e^{2} - 126e - 88$ |
| 8 | $[8, 2, 2]$ | $\phantom{-}\frac{3}{2}e^{7} + e^{6} - \frac{51}{2}e^{5} - \frac{29}{2}e^{4} + \frac{259}{2}e^{3} + 68e^{2} - 191e - 125$ |
| 9 | $[9, 3, -w^{2} + 2w + 4]$ | $\phantom{-}\frac{3}{4}e^{7} + \frac{1}{2}e^{6} - \frac{25}{2}e^{5} - 7e^{4} + \frac{123}{2}e^{3} + 31e^{2} - 87e - 56$ |
| 11 | $[11, 11, -w^{2} + 2w + 2]$ | $\phantom{-}\frac{1}{4}e^{7} + \frac{1}{4}e^{6} - 4e^{5} - 3e^{4} + \frac{37}{2}e^{3} + 9e^{2} - 23e - 10$ |
| 13 | $[13, 13, -w^{2} + w + 4]$ | $-\frac{1}{2}e^{7} - \frac{1}{2}e^{6} + \frac{17}{2}e^{5} + \frac{15}{2}e^{4} - 44e^{3} - 35e^{2} + 69e + 56$ |
| 19 | $[19, 19, w + 3]$ | $-\frac{7}{2}e^{7} - \frac{5}{2}e^{6} + 59e^{5} + \frac{71}{2}e^{4} - 295e^{3} - 159e^{2} + 422e + 276$ |
| 19 | $[19, 19, -w^{2} + 2w + 5]$ | $-1$ |
| 19 | $[19, 19, -w^{2} + 3w + 2]$ | $\phantom{-}\frac{1}{2}e^{7} + \frac{1}{2}e^{6} - 8e^{5} - 7e^{4} + \frac{73}{2}e^{3} + 28e^{2} - 45e - 38$ |
| 23 | $[23, 23, w^{2} - w - 3]$ | $-\frac{11}{4}e^{7} - 2e^{6} + \frac{93}{2}e^{5} + \frac{57}{2}e^{4} - \frac{467}{2}e^{3} - 128e^{2} + 336e + 220$ |
| 23 | $[23, 23, -w^{2} + 2]$ | $-\frac{1}{2}e^{7} - \frac{1}{4}e^{6} + 8e^{5} + 3e^{4} - \frac{73}{2}e^{3} - 11e^{2} + 44e + 18$ |
| 23 | $[23, 23, -w + 4]$ | $-\frac{7}{4}e^{7} - \frac{5}{4}e^{6} + \frac{59}{2}e^{5} + \frac{35}{2}e^{4} - 147e^{3} - 77e^{2} + 207e + 132$ |
| 31 | $[31, 31, w^{2} - 5]$ | $-\frac{9}{4}e^{7} - \frac{7}{4}e^{6} + \frac{75}{2}e^{5} + 25e^{4} - 185e^{3} - 110e^{2} + 263e + 178$ |
| 43 | $[43, 43, w^{2} - 3w - 3]$ | $\phantom{-}e^{7} + e^{6} - \frac{33}{2}e^{5} - 14e^{4} + \frac{161}{2}e^{3} + 58e^{2} - 113e - 86$ |
| 49 | $[49, 7, w^{2} - 6]$ | $-\frac{1}{4}e^{7} + 5e^{5} + \frac{1}{2}e^{4} - 31e^{3} - 7e^{2} + 58e + 26$ |
| 53 | $[53, 53, 2w - 5]$ | $-\frac{7}{4}e^{7} - \frac{5}{4}e^{6} + \frac{59}{2}e^{5} + 18e^{4} - 148e^{3} - 84e^{2} + 217e + 152$ |
| 61 | $[61, 61, 2w^{2} - 2w - 9]$ | $-e^{7} - \frac{3}{4}e^{6} + \frac{33}{2}e^{5} + 10e^{4} - \frac{161}{2}e^{3} - 41e^{2} + 113e + 68$ |
| 71 | $[71, 71, 2w - 3]$ | $-\frac{15}{4}e^{7} - 3e^{6} + 63e^{5} + \frac{87}{2}e^{4} - 314e^{3} - 195e^{2} + 452e + 318$ |
| 73 | $[73, 73, 2w^{2} - 5w - 5]$ | $\phantom{-}\frac{1}{2}e^{7} - 9e^{5} + \frac{1}{2}e^{4} + \frac{97}{2}e^{3} - e^{2} - 74e - 22$ |
| 83 | $[83, 83, w^{2} - w - 9]$ | $-\frac{5}{4}e^{7} - \frac{3}{4}e^{6} + 21e^{5} + 11e^{4} - 104e^{3} - 53e^{2} + 145e + 96$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19, 19, -w^{2} + 2w + 5]$ | $1$ |