Base field \(\Q(\sqrt{5}, \sqrt{6})\)
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 13x^{2} + 14x + 19\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[19, 19, -w]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $24$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 14x^{6} + 58x^{4} - 63x^{2} + 13\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{2}{19}w^{3} + \frac{3}{19}w^{2} + \frac{37}{19}w - 3]$ | $\phantom{-}\frac{2}{13}e^{6} - \frac{21}{13}e^{4} + \frac{62}{13}e^{2} - 4$ |
| 5 | $[5, 5, \frac{3}{19}w^{3} + \frac{5}{19}w^{2} - \frac{27}{19}w - 1]$ | $\phantom{-}e$ |
| 5 | $[5, 5, -\frac{3}{19}w^{3} + \frac{14}{19}w^{2} + \frac{8}{19}w - 2]$ | $-\frac{3}{13}e^{7} + \frac{38}{13}e^{5} - \frac{132}{13}e^{3} + 6e$ |
| 9 | $[9, 3, -\frac{2}{19}w^{3} + \frac{3}{19}w^{2} + \frac{37}{19}w - 4]$ | $\phantom{-}\frac{1}{13}e^{6} - \frac{4}{13}e^{4} - \frac{21}{13}e^{2} + 2$ |
| 19 | $[19, 19, -w]$ | $\phantom{-}1$ |
| 19 | $[19, 19, \frac{4}{19}w^{3} - \frac{6}{19}w^{2} - \frac{55}{19}w + 1]$ | $-\frac{5}{13}e^{6} + \frac{46}{13}e^{4} - \frac{90}{13}e^{2} + 1$ |
| 19 | $[19, 19, \frac{4}{19}w^{3} - \frac{6}{19}w^{2} - \frac{55}{19}w + 2]$ | $\phantom{-}e^{4} - 8e^{2} + 7$ |
| 19 | $[19, 19, w - 1]$ | $\phantom{-}\frac{5}{13}e^{6} - \frac{72}{13}e^{4} + \frac{272}{13}e^{2} - 13$ |
| 29 | $[29, 29, -\frac{5}{19}w^{3} - \frac{2}{19}w^{2} + \frac{64}{19}w + 2]$ | $\phantom{-}\frac{5}{13}e^{7} - \frac{72}{13}e^{5} + \frac{272}{13}e^{3} - 10e$ |
| 29 | $[29, 29, \frac{10}{19}w^{3} - \frac{34}{19}w^{2} - \frac{71}{19}w + 9]$ | $-\frac{6}{13}e^{7} + \frac{63}{13}e^{5} - \frac{173}{13}e^{3} + 7e$ |
| 29 | $[29, 29, \frac{4}{19}w^{3} - \frac{6}{19}w^{2} - \frac{55}{19}w + 4]$ | $\phantom{-}\frac{11}{13}e^{7} - \frac{135}{13}e^{5} + \frac{471}{13}e^{3} - 28e$ |
| 29 | $[29, 29, -w + 3]$ | $\phantom{-}e^{3} - 8e$ |
| 49 | $[49, 7, \frac{5}{19}w^{3} - \frac{17}{19}w^{2} - \frac{64}{19}w + 11]$ | $-\frac{8}{13}e^{6} + \frac{84}{13}e^{4} - \frac{209}{13}e^{2} - 1$ |
| 49 | $[49, 7, \frac{6}{19}w^{3} - \frac{9}{19}w^{2} - \frac{92}{19}w - 1]$ | $-\frac{8}{13}e^{6} + \frac{84}{13}e^{4} - \frac{235}{13}e^{2} + 12$ |
| 71 | $[71, 71, \frac{3}{19}w^{3} + \frac{5}{19}w^{2} - \frac{46}{19}w - 1]$ | $-\frac{7}{13}e^{7} + \frac{93}{13}e^{5} - \frac{334}{13}e^{3} + 16e$ |
| 71 | $[71, 71, \frac{6}{19}w^{3} - \frac{9}{19}w^{2} - \frac{73}{19}w]$ | $\phantom{-}\frac{1}{13}e^{7} - \frac{17}{13}e^{5} + \frac{57}{13}e^{3} + 2e$ |
| 71 | $[71, 71, \frac{6}{19}w^{3} - \frac{9}{19}w^{2} - \frac{73}{19}w + 4]$ | $-\frac{2}{13}e^{7} + \frac{21}{13}e^{5} - \frac{75}{13}e^{3} + 14e$ |
| 71 | $[71, 71, -\frac{3}{19}w^{3} + \frac{14}{19}w^{2} + \frac{27}{19}w - 3]$ | $\phantom{-}\frac{4}{13}e^{7} - \frac{29}{13}e^{5} + \frac{7}{13}e^{3} + 7e$ |
| 101 | $[101, 101, \frac{7}{19}w^{3} - \frac{1}{19}w^{2} - \frac{63}{19}w - 3]$ | $\phantom{-}\frac{2}{13}e^{7} - \frac{8}{13}e^{5} - \frac{81}{13}e^{3} + 25e$ |
| 101 | $[101, 101, -\frac{9}{19}w^{3} + \frac{23}{19}w^{2} + \frac{81}{19}w - 5]$ | $-\frac{2}{13}e^{7} + \frac{47}{13}e^{5} - \frac{296}{13}e^{3} + 33e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19, 19, -w]$ | $-1$ |