Base field 4.4.8725.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 10x^{2} + 2x + 19\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[19,19,\frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{13}{3}w + \frac{17}{3}]$ |
| Dimension: | $2$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $10$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{2} - 13\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 9 | $[9, 3, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{10}{3}]$ | $-\frac{1}{2}e - \frac{3}{2}$ |
| 9 | $[9, 3, w + 1]$ | $\phantom{-}e$ |
| 11 | $[11, 11, w^{2} - w - 6]$ | $-\frac{1}{2}e + \frac{1}{2}$ |
| 11 | $[11, 11, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{7}{3}w + \frac{23}{3}]$ | $\phantom{-}e + 2$ |
| 16 | $[16, 2, 2]$ | $-\frac{1}{2}e - \frac{7}{2}$ |
| 19 | $[19, 19, w]$ | $-7$ |
| 19 | $[19, 19, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{8}{3}w + \frac{7}{3}]$ | $-\frac{3}{2}e + \frac{1}{2}$ |
| 19 | $[19, 19, \frac{1}{3}w^{3} - \frac{2}{3}w^{2} - \frac{2}{3}w + \frac{10}{3}]$ | $\phantom{-}2$ |
| 19 | $[19, 19, -\frac{2}{3}w^{3} + \frac{4}{3}w^{2} + \frac{13}{3}w - \frac{17}{3}]$ | $\phantom{-}1$ |
| 25 | $[25, 5, \frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{10}{3}w + \frac{11}{3}]$ | $-1$ |
| 31 | $[31, 31, w + 3]$ | $-1$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} + \frac{2}{3}w^{2} + \frac{8}{3}w - \frac{16}{3}]$ | $-\frac{3}{2}e - \frac{5}{2}$ |
| 31 | $[31, 31, \frac{2}{3}w^{3} - \frac{7}{3}w^{2} - \frac{10}{3}w + \frac{23}{3}]$ | $\phantom{-}e - 5$ |
| 31 | $[31, 31, -\frac{1}{3}w^{3} + \frac{5}{3}w^{2} + \frac{5}{3}w - \frac{25}{3}]$ | $\phantom{-}e - 5$ |
| 59 | $[59, 59, -\frac{2}{3}w^{3} + \frac{1}{3}w^{2} + \frac{10}{3}w + \frac{1}{3}]$ | $-\frac{3}{2}e - \frac{9}{2}$ |
| 59 | $[59, 59, \frac{5}{3}w^{3} - \frac{13}{3}w^{2} - \frac{25}{3}w + \frac{47}{3}]$ | $-3$ |
| 61 | $[61, 61, \frac{1}{3}w^{3} + \frac{1}{3}w^{2} - \frac{11}{3}w - \frac{8}{3}]$ | $-\frac{1}{2}e - \frac{19}{2}$ |
| 61 | $[61, 61, -\frac{2}{3}w^{3} + \frac{7}{3}w^{2} + \frac{4}{3}w - \frac{26}{3}]$ | $-\frac{1}{2}e - \frac{1}{2}$ |
| 71 | $[71, 71, \frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{13}{3}w + \frac{5}{3}]$ | $-2e + 5$ |
| 71 | $[71, 71, -w^{3} + 3w^{2} + 5w - 10]$ | $\phantom{-}\frac{5}{2}e + \frac{1}{2}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19,19,\frac{2}{3}w^{3} - \frac{4}{3}w^{2} - \frac{13}{3}w + \frac{17}{3}]$ | $-1$ |