Base field \(\Q(\sqrt{185}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 46\); narrow class number \(2\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[8,8,-w + 3]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $42$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} - 3x^{7} - 10x^{6} + 33x^{5} + 21x^{4} - 92x^{3} + 4x^{2} + 44x + 4\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}e$ |
| 2 | $[2, 2, w + 1]$ | $\phantom{-}0$ |
| 5 | $[5, 5, w + 2]$ | $\phantom{-}e^{7} - \frac{3}{4}e^{6} - \frac{47}{4}e^{5} + 7e^{4} + \frac{147}{4}e^{3} - \frac{51}{4}e^{2} - \frac{43}{2}e - \frac{1}{2}$ |
| 9 | $[9, 3, 3]$ | $-\frac{9}{4}e^{7} + \frac{7}{4}e^{6} + 26e^{5} - \frac{63}{4}e^{4} - \frac{313}{4}e^{3} + \frac{51}{2}e^{2} + \frac{81}{2}e + 5$ |
| 11 | $[11, 11, 2w + 13]$ | $\phantom{-}\frac{1}{4}e^{7} + \frac{1}{4}e^{6} - 3e^{5} - \frac{13}{4}e^{4} + \frac{41}{4}e^{3} + \frac{21}{2}e^{2} - \frac{17}{2}e - 3$ |
| 11 | $[11, 11, -2w + 15]$ | $\phantom{-}\frac{3}{4}e^{7} - \frac{1}{4}e^{6} - 9e^{5} + \frac{5}{4}e^{4} + \frac{115}{4}e^{3} + \frac{7}{2}e^{2} - \frac{31}{2}e - 5$ |
| 13 | $[13, 13, w + 4]$ | $-2e^{7} + \frac{3}{2}e^{6} + \frac{47}{2}e^{5} - 14e^{4} - \frac{147}{2}e^{3} + \frac{53}{2}e^{2} + 44e - 1$ |
| 13 | $[13, 13, w + 8]$ | $\phantom{-}\frac{1}{2}e^{6} - \frac{1}{2}e^{5} - 6e^{4} + \frac{9}{2}e^{3} + \frac{35}{2}e^{2} - 8e - 3$ |
| 17 | $[17, 17, w + 3]$ | $-\frac{1}{2}e^{7} + \frac{3}{4}e^{6} + \frac{21}{4}e^{5} - \frac{15}{2}e^{4} - \frac{51}{4}e^{3} + \frac{65}{4}e^{2} + \frac{3}{2}e - \frac{1}{2}$ |
| 17 | $[17, 17, w + 13]$ | $\phantom{-}\frac{3}{2}e^{7} - \frac{5}{4}e^{6} - \frac{67}{4}e^{5} + \frac{23}{2}e^{4} + \frac{189}{4}e^{3} - \frac{83}{4}e^{2} - \frac{37}{2}e + \frac{3}{2}$ |
| 23 | $[23, 23, w]$ | $\phantom{-}\frac{3}{2}e^{7} - \frac{3}{4}e^{6} - \frac{73}{4}e^{5} + \frac{11}{2}e^{4} + \frac{243}{4}e^{3} - \frac{5}{4}e^{2} - \frac{89}{2}e - \frac{19}{2}$ |
| 23 | $[23, 23, w + 22]$ | $-e^{7} + \frac{1}{4}e^{6} + \frac{49}{4}e^{5} - e^{4} - \frac{169}{4}e^{3} - \frac{23}{4}e^{2} + \frac{71}{2}e + \frac{19}{2}$ |
| 37 | $[37, 37, w + 18]$ | $-e^{7} + \frac{1}{2}e^{6} + \frac{23}{2}e^{5} - 4e^{4} - \frac{69}{2}e^{3} + \frac{5}{2}e^{2} + 17e + 11$ |
| 41 | $[41, 41, -2w + 13]$ | $\phantom{-}\frac{11}{4}e^{7} - \frac{9}{4}e^{6} - 32e^{5} + \frac{81}{4}e^{4} + \frac{391}{4}e^{3} - \frac{67}{2}e^{2} - \frac{107}{2}e - 7$ |
| 41 | $[41, 41, -2w - 11]$ | $-\frac{13}{4}e^{7} + \frac{7}{4}e^{6} + 39e^{5} - \frac{59}{4}e^{4} - \frac{509}{4}e^{3} + \frac{35}{2}e^{2} + \frac{177}{2}e + 15$ |
| 43 | $[43, 43, w + 11]$ | $\phantom{-}\frac{1}{2}e^{7} + \frac{3}{4}e^{6} - \frac{27}{4}e^{5} - \frac{19}{2}e^{4} + \frac{109}{4}e^{3} + \frac{125}{4}e^{2} - \frac{61}{2}e - \frac{29}{2}$ |
| 43 | $[43, 43, w + 31]$ | $-4e^{7} + \frac{11}{4}e^{6} + \frac{187}{4}e^{5} - 24e^{4} - \frac{571}{4}e^{3} + \frac{143}{4}e^{2} + \frac{147}{2}e + \frac{21}{2}$ |
| 49 | $[49, 7, -7]$ | $\phantom{-}\frac{19}{4}e^{7} - \frac{13}{4}e^{6} - 56e^{5} + \frac{113}{4}e^{4} + \frac{707}{4}e^{3} - \frac{83}{2}e^{2} - \frac{219}{2}e - 13$ |
| 71 | $[71, 71, -2w + 17]$ | $-\frac{3}{2}e^{7} + \frac{1}{2}e^{6} + 18e^{5} - \frac{7}{2}e^{4} - \frac{115}{2}e^{3} - 3e^{2} + 33e + 17$ |
| 71 | $[71, 71, -2w - 15]$ | $\phantom{-}\frac{1}{2}e^{7} - \frac{3}{2}e^{6} - 6e^{5} + \frac{31}{2}e^{4} + \frac{37}{2}e^{3} - 38e^{2} - 9e + 9$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2,2,-w + 1]$ | $1$ |