Base field 4.4.7600.1
Generator \(w\), with minimal polynomial \(x^4 - 9 x^2 + 19\); 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:
| \(x^4 + 8 x^3 + 16 x^2 + 4 x - 1\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -w^3 + w^2 + 5 w - 6]$ | $\phantom{-}\frac{1}{2} e^3 + \frac{7}{2} e^2 + \frac{11}{2} e - \frac{1}{2}$ |
| 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 | $[19, 19, -w]$ | $\phantom{-}1$ |
| 19 | $[19, 19, -w^2 - w + 6]$ | $\phantom{-}e^3 + 9 e^2 + 20 e + 3$ |
| 19 | $[19, 19, -w^2 + w + 6]$ | $-\frac{1}{2} e^3 - \frac{11}{2} e^2 - \frac{29}{2} e - \frac{5}{2}$ |
| 25 | $[25, 5, 2 w^2 - 9]$ | $-\frac{1}{2} e^3 - \frac{7}{2} e^2 - \frac{11}{2} e + \frac{1}{2}$ |
| 29 | $[29, 29, -w^3 + 4 w + 2]$ | $-\frac{1}{2} e^3 - \frac{7}{2} e^2 - \frac{5}{2} e + \frac{13}{2}$ |
| 29 | $[29, 29, -w^3 + 4 w - 2]$ | $\phantom{-}e^3 + 7 e^2 + 8 e - 4$ |
| 41 | $[41, 41, 2 w^2 - w - 7]$ | $-2 e^3 - 15 e^2 - 28 e - 5$ |
| 41 | $[41, 41, w^3 - w^2 - 6 w + 4]$ | $-2 e^3 - 13 e^2 - 16 e + 3$ |
| 61 | $[61, 61, -w^3 + 3 w^2 + 6 w - 14]$ | $-\frac{1}{2} e^3 - \frac{3}{2} e^2 + \frac{3}{2} e - \frac{7}{2}$ |
| 61 | $[61, 61, w^3 + 2 w^2 - 5 w - 8]$ | $\phantom{-}e^3 + 5 e^2 - e - 7$ |
| 61 | $[61, 61, -w^3 + 2 w^2 + 5 w - 8]$ | $\phantom{-}e^3 + 9 e^2 + 23 e + 9$ |
| 61 | $[61, 61, w^3 + 3 w^2 - 6 w - 14]$ | $-3 e^3 - 23 e^2 - 40 e - 2$ |
| 89 | $[89, 89, -w^3 + w^2 + 6 w - 9]$ | $-2 e^3 - 15 e^2 - 26 e - 4$ |
| 89 | $[89, 89, 2 w^3 - w^2 - 10 w + 10]$ | $-e^3 - 6 e^2 - 7 e - 3$ |
| 109 | $[109, 109, -w^3 + 5 w^2 + 7 w - 23]$ | $-\frac{3}{2} e^3 - \frac{23}{2} e^2 - \frac{49}{2} e - \frac{13}{2}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19, 19, -w]$ | $-1$ |