Base field \(\Q(\sqrt{5}, \sqrt{7})\)
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 15x^{2} + 16x + 29\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[16, 2, 2]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $30$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} + 8x^{3} + 6x^{2} - 40x - 20\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} - \frac{3}{23}w + \frac{1}{23}]$ | $\phantom{-}0$ |
| 9 | $[9, 3, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w + \frac{24}{23}]$ | $-\frac{1}{6}e^{3} - e^{2} + \frac{5}{3}$ |
| 9 | $[9, 3, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w - \frac{68}{23}]$ | $\phantom{-}\frac{1}{6}e^{3} + e^{2} - \frac{11}{3}$ |
| 19 | $[19, 19, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} - \frac{3}{23}w + \frac{24}{23}]$ | $\phantom{-}e$ |
| 19 | $[19, 19, -\frac{6}{23}w^{3} + \frac{9}{23}w^{2} + \frac{83}{23}w - \frac{20}{23}]$ | $\phantom{-}e + 4$ |
| 19 | $[19, 19, \frac{6}{23}w^{3} - \frac{9}{23}w^{2} - \frac{83}{23}w + \frac{66}{23}]$ | $-\frac{1}{6}e^{3} - e^{2} + e + \frac{20}{3}$ |
| 19 | $[19, 19, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} - \frac{3}{23}w - \frac{22}{23}]$ | $-\frac{1}{6}e^{3} - e^{2} + e + \frac{8}{3}$ |
| 25 | $[25, 5, -\frac{4}{23}w^{3} + \frac{6}{23}w^{2} + \frac{40}{23}w - \frac{21}{23}]$ | $-4$ |
| 29 | $[29, 29, w]$ | $-\frac{1}{3}e^{3} - \frac{7}{3}e^{2} + \frac{2}{3}e + 9$ |
| 29 | $[29, 29, -\frac{4}{23}w^{3} + \frac{6}{23}w^{2} + \frac{63}{23}w - \frac{21}{23}]$ | $\phantom{-}\frac{1}{3}e^{2} - \frac{2}{3}e - \frac{23}{3}$ |
| 29 | $[29, 29, \frac{4}{23}w^{3} - \frac{6}{23}w^{2} - \frac{63}{23}w + \frac{44}{23}]$ | $\phantom{-}\frac{1}{3}e^{3} + \frac{5}{3}e^{2} - \frac{10}{3}e - \frac{29}{3}$ |
| 29 | $[29, 29, -w + 1]$ | $\phantom{-}\frac{1}{3}e^{2} + \frac{10}{3}e + \frac{1}{3}$ |
| 31 | $[31, 31, w + 2]$ | $\phantom{-}\frac{1}{6}e^{3} + \frac{2}{3}e^{2} - \frac{7}{3}e - 8$ |
| 31 | $[31, 31, -\frac{4}{23}w^{3} + \frac{6}{23}w^{2} + \frac{63}{23}w + \frac{25}{23}]$ | $-\frac{1}{3}e^{2} - \frac{7}{3}e + \frac{14}{3}$ |
| 31 | $[31, 31, \frac{4}{23}w^{3} - \frac{6}{23}w^{2} - \frac{63}{23}w + \frac{90}{23}]$ | $\phantom{-}\frac{1}{6}e^{3} + \frac{4}{3}e^{2} + \frac{1}{3}e - \frac{4}{3}$ |
| 31 | $[31, 31, -w + 3]$ | $\phantom{-}\frac{1}{3}e^{2} + \frac{1}{3}e - \frac{26}{3}$ |
| 49 | $[49, 7, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w - \frac{22}{23}]$ | $\phantom{-}6$ |
| 59 | $[59, 59, -\frac{20}{23}w^{3} + \frac{30}{23}w^{2} + \frac{269}{23}w - \frac{128}{23}]$ | $-\frac{2}{3}e^{2} - \frac{14}{3}e + \frac{4}{3}$ |
| 59 | $[59, 59, -\frac{8}{23}w^{3} + \frac{12}{23}w^{2} + \frac{103}{23}w - \frac{88}{23}]$ | $\phantom{-}\frac{1}{3}e^{3} + \frac{4}{3}e^{2} - \frac{14}{3}e - 8$ |
| 59 | $[59, 59, -\frac{8}{23}w^{3} + \frac{12}{23}w^{2} + \frac{103}{23}w - \frac{19}{23}]$ | $\phantom{-}\frac{2}{3}e^{2} + \frac{2}{3}e - \frac{28}{3}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4,2,-\frac{2}{23}w^{3}+\frac{3}{23}w^{2}-\frac{3}{23}w+\frac{1}{23}]$ | $-1$ |