Base field 4.4.7600.1
Generator \(w\), with minimal polynomial \(x^{4} - 9x^{2} + 19\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[19, 19, -w]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $8$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} - 8x^{3} + 16x^{2} - 4x - 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -w^{3} + w^{2} + 5w - 6]$ | $\phantom{-}\frac{1}{2}e^{3} - \frac{7}{2}e^{2} + \frac{11}{2}e + \frac{1}{2}$ |
| 9 | $[9, 3, -w^{2} - w + 4]$ | $\phantom{-}e$ |
| 9 | $[9, 3, w^{2} - w - 4]$ | $\phantom{-}\frac{1}{2}e^{3} - \frac{7}{2}e^{2} + \frac{9}{2}e + \frac{7}{2}$ |
| 11 | $[11, 11, w + 1]$ | $\phantom{-}\frac{1}{2}e^{3} - \frac{9}{2}e^{2} + \frac{19}{2}e + \frac{3}{2}$ |
| 11 | $[11, 11, w - 1]$ | $-\frac{1}{2}e^{3} + \frac{9}{2}e^{2} - \frac{19}{2}e + \frac{5}{2}$ |
| 19 | $[19, 19, -w]$ | $-1$ |
| 19 | $[19, 19, -w^{2} - w + 6]$ | $-e^{3} + 9e^{2} - 20e + 3$ |
| 19 | $[19, 19, -w^{2} + w + 6]$ | $\phantom{-}\frac{1}{2}e^{3} - \frac{11}{2}e^{2} + \frac{29}{2}e - \frac{5}{2}$ |
| 25 | $[25, 5, 2w^{2} - 9]$ | $\phantom{-}\frac{1}{2}e^{3} - \frac{7}{2}e^{2} + \frac{11}{2}e + \frac{1}{2}$ |
| 29 | $[29, 29, -w^{3} + 4w + 2]$ | $\phantom{-}\frac{1}{2}e^{3} - \frac{7}{2}e^{2} + \frac{5}{2}e + \frac{13}{2}$ |
| 29 | $[29, 29, -w^{3} + 4w - 2]$ | $-e^{3} + 7e^{2} - 8e - 4$ |
| 41 | $[41, 41, 2w^{2} - w - 7]$ | $-2e^{3} + 15e^{2} - 28e + 5$ |
| 41 | $[41, 41, w^{3} - w^{2} - 6w + 4]$ | $-2e^{3} + 13e^{2} - 16e - 3$ |
| 61 | $[61, 61, -w^{3} + 3w^{2} + 6w - 14]$ | $-\frac{1}{2}e^{3} + \frac{3}{2}e^{2} + \frac{3}{2}e + \frac{7}{2}$ |
| 61 | $[61, 61, w^{3} + 2w^{2} - 5w - 8]$ | $\phantom{-}e^{3} - 5e^{2} - e + 7$ |
| 61 | $[61, 61, -w^{3} + 2w^{2} + 5w - 8]$ | $\phantom{-}e^{3} - 9e^{2} + 23e - 9$ |
| 61 | $[61, 61, w^{3} + 3w^{2} - 6w - 14]$ | $-3e^{3} + 23e^{2} - 40e + 2$ |
| 89 | $[89, 89, -w^{3} + w^{2} + 6w - 9]$ | $\phantom{-}2e^{3} - 15e^{2} + 26e - 4$ |
| 89 | $[89, 89, 2w^{3} - w^{2} - 10w + 10]$ | $\phantom{-}e^{3} - 6e^{2} + 7e - 3$ |
| 109 | $[109, 109, -w^{3} + 5w^{2} + 7w - 23]$ | $-\frac{3}{2}e^{3} + \frac{23}{2}e^{2} - \frac{49}{2}e + \frac{13}{2}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19, 19, -w]$ | $1$ |