Base field \(\Q(\sqrt{2}, \sqrt{17})\)
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 11x^{2} + 12x + 2\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[4, 2, \frac{2}{9}w^{3} - \frac{1}{3}w^{2} - \frac{19}{9}w + \frac{10}{9}]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $6$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} - 96x^{2} + 1936\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, \frac{4}{9}w^{3} - \frac{2}{3}w^{2} - \frac{47}{9}w + \frac{20}{9}]$ | $\phantom{-}1$ |
| 2 | $[2, 2, \frac{5}{9}w^{3} - \frac{4}{3}w^{2} - \frac{52}{9}w + \frac{79}{9}]$ | $\phantom{-}1$ |
| 9 | $[9, 3, \frac{1}{3}w^{3} - \frac{11}{3}w - \frac{1}{3}]$ | $-\frac{1}{44}e^{3} + \frac{13}{11}e$ |
| 9 | $[9, 3, w^{3} - w^{2} - 12w - 1]$ | $-\frac{1}{44}e^{3} + \frac{13}{11}e$ |
| 17 | $[17, 17, -\frac{1}{9}w^{3} - \frac{1}{3}w^{2} + \frac{14}{9}w + \frac{49}{9}]$ | $\phantom{-}e$ |
| 17 | $[17, 17, -\frac{1}{9}w^{3} + \frac{2}{3}w^{2} + \frac{5}{9}w - \frac{59}{9}]$ | $\phantom{-}\frac{1}{44}e^{3} - \frac{24}{11}e$ |
| 25 | $[25, 5, -\frac{7}{9}w^{3} + \frac{2}{3}w^{2} + \frac{89}{9}w + \frac{19}{9}]$ | $\phantom{-}\frac{1}{22}e^{3} - \frac{26}{11}e$ |
| 25 | $[25, 5, -\frac{5}{9}w^{3} + \frac{1}{3}w^{2} + \frac{52}{9}w + \frac{11}{9}]$ | $\phantom{-}\frac{1}{22}e^{3} - \frac{26}{11}e$ |
| 47 | $[47, 47, -\frac{1}{9}w^{3} + \frac{2}{3}w^{2} + \frac{5}{9}w - \frac{41}{9}]$ | $-\frac{1}{2}e^{2} + 26$ |
| 47 | $[47, 47, \frac{1}{9}w^{3} + \frac{1}{3}w^{2} - \frac{14}{9}w - \frac{13}{9}]$ | $-\frac{1}{2}e^{2} + 26$ |
| 47 | $[47, 47, -\frac{1}{9}w^{3} + \frac{2}{3}w^{2} + \frac{5}{9}w - \frac{23}{9}]$ | $\phantom{-}\frac{1}{2}e^{2} - 22$ |
| 47 | $[47, 47, \frac{1}{9}w^{3} + \frac{1}{3}w^{2} - \frac{14}{9}w - \frac{31}{9}]$ | $\phantom{-}\frac{1}{2}e^{2} - 22$ |
| 49 | $[49, 7, -\frac{4}{9}w^{3} + \frac{2}{3}w^{2} + \frac{38}{9}w - \frac{29}{9}]$ | $\phantom{-}\frac{1}{2}e^{2} - 20$ |
| 49 | $[49, 7, \frac{4}{9}w^{3} - \frac{2}{3}w^{2} - \frac{38}{9}w + \frac{11}{9}]$ | $-\frac{1}{2}e^{2} + 28$ |
| 89 | $[89, 89, \frac{20}{9}w^{3} - \frac{16}{3}w^{2} - \frac{208}{9}w + \frac{325}{9}]$ | $-2$ |
| 89 | $[89, 89, \frac{2}{3}w^{3} - w^{2} - \frac{25}{3}w + \frac{7}{3}]$ | $-2$ |
| 89 | $[89, 89, -\frac{2}{3}w^{3} + w^{2} + \frac{25}{3}w - \frac{19}{3}]$ | $-2$ |
| 89 | $[89, 89, -2w^{3} + 3w^{2} + 23w - 11]$ | $-2$ |
| 103 | $[103, 103, \frac{8}{9}w^{3} - \frac{4}{3}w^{2} - \frac{94}{9}w + \frac{67}{9}]$ | $-\frac{1}{11}e^{3} + \frac{63}{11}e$ |
| 103 | $[103, 103, -\frac{38}{9}w^{3} + \frac{28}{3}w^{2} + \frac{406}{9}w - \frac{541}{9}]$ | $-\frac{3}{44}e^{3} + \frac{28}{11}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, \frac{4}{9}w^{3} - \frac{2}{3}w^{2} - \frac{47}{9}w + \frac{20}{9}]$ | $-1$ |
| $2$ | $[2, 2, \frac{5}{9}w^{3} - \frac{4}{3}w^{2} - \frac{52}{9}w + \frac{79}{9}]$ | $-1$ |