Base field \(\Q(\sqrt{209}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 52\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[8,4,22w - 170]$ |
| Dimension: | $5$ |
| 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:
| \(x^{5} - 3x^{4} - 15x^{3} + 54x^{2} - 32x + 4\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -11w - 74]$ | $\phantom{-}1$ |
| 2 | $[2, 2, -11w + 85]$ | $\phantom{-}0$ |
| 5 | $[5, 5, 4w - 31]$ | $-\frac{7}{18}e^{4} + \frac{17}{18}e^{3} + \frac{107}{18}e^{2} - \frac{161}{9}e + \frac{65}{9}$ |
| 5 | $[5, 5, -4w - 27]$ | $\phantom{-}e$ |
| 9 | $[9, 3, 3]$ | $\phantom{-}e - 1$ |
| 11 | $[11, 11, -70w - 471]$ | $\phantom{-}\frac{1}{18}e^{4} - \frac{5}{18}e^{3} - \frac{23}{18}e^{2} + \frac{41}{9}e + \frac{1}{9}$ |
| 13 | $[13, 13, -2w - 13]$ | $\phantom{-}\frac{1}{6}e^{4} + \frac{1}{6}e^{3} - \frac{17}{6}e^{2} - \frac{1}{3}e + \frac{10}{3}$ |
| 13 | $[13, 13, -2w + 15]$ | $-\frac{2}{3}e^{4} + \frac{4}{3}e^{3} + \frac{31}{3}e^{2} - \frac{80}{3}e + \frac{32}{3}$ |
| 19 | $[19, 19, 92w - 711]$ | $\phantom{-}\frac{1}{6}e^{4} - \frac{5}{6}e^{3} - \frac{17}{6}e^{2} + \frac{44}{3}e - \frac{14}{3}$ |
| 23 | $[23, 23, -26w + 201]$ | $\phantom{-}\frac{5}{9}e^{4} - \frac{16}{9}e^{3} - \frac{79}{9}e^{2} + \frac{284}{9}e - \frac{125}{9}$ |
| 23 | $[23, 23, -26w - 175]$ | $\phantom{-}\frac{11}{18}e^{4} - \frac{19}{18}e^{3} - \frac{181}{18}e^{2} + \frac{181}{9}e - \frac{7}{9}$ |
| 29 | $[29, 29, 18w - 139]$ | $\phantom{-}\frac{1}{6}e^{4} - \frac{5}{6}e^{3} - \frac{17}{6}e^{2} + \frac{41}{3}e - \frac{2}{3}$ |
| 29 | $[29, 29, 18w + 121]$ | $-\frac{7}{6}e^{4} + \frac{17}{6}e^{3} + \frac{113}{6}e^{2} - \frac{152}{3}e + \frac{44}{3}$ |
| 41 | $[41, 41, -10w - 67]$ | $\phantom{-}e^{4} - 3e^{3} - 16e^{2} + 56e - 20$ |
| 41 | $[41, 41, 10w - 77]$ | $\phantom{-}\frac{1}{2}e^{4} - \frac{1}{2}e^{3} - \frac{15}{2}e^{2} + 12e$ |
| 47 | $[47, 47, 2w - 17]$ | $\phantom{-}\frac{2}{9}e^{4} - \frac{1}{9}e^{3} - \frac{28}{9}e^{2} + \frac{56}{9}e - \frac{59}{9}$ |
| 47 | $[47, 47, 2w + 15]$ | $\phantom{-}\frac{13}{18}e^{4} - \frac{47}{18}e^{3} - \frac{209}{18}e^{2} + \frac{398}{9}e - \frac{131}{9}$ |
| 49 | $[49, 7, -7]$ | $\phantom{-}\frac{1}{2}e^{4} - \frac{1}{2}e^{3} - \frac{15}{2}e^{2} + 12e - 5$ |
| 79 | $[79, 79, 40w - 309]$ | $\phantom{-}\frac{1}{2}e^{4} - \frac{1}{2}e^{3} - \frac{17}{2}e^{2} + 14e + 8$ |
| 79 | $[79, 79, 40w + 269]$ | $\phantom{-}\frac{11}{6}e^{4} - \frac{25}{6}e^{3} - \frac{181}{6}e^{2} + \frac{235}{3}e - \frac{58}{3}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2,2,11w - 85]$ | $-1$ |
| $2$ | $[2,2,11w + 74]$ | $-1$ |