Base field \(\Q(\sqrt{481}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 120\); narrow class number \(2\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[4, 2, 2]$ |
| Dimension: | $10$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $80$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{10} - 2x^{9} - 18x^{8} + 33x^{7} + 109x^{6} - 175x^{5} - 258x^{4} + 321x^{3} + 217x^{2} - 160x - 16\) |
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| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}1$ |
| 2 | $[2, 2, w + 1]$ | $\phantom{-}1$ |
| 3 | $[3, 3, -2w - 21]$ | $\phantom{-}e$ |
| 3 | $[3, 3, 2w - 23]$ | $\phantom{-}e$ |
| 5 | $[5, 5, w]$ | $-\frac{1}{12}e^{9} + \frac{3}{2}e^{7} + \frac{1}{12}e^{6} - \frac{35}{4}e^{5} - \frac{1}{4}e^{4} + \frac{55}{3}e^{3} - \frac{19}{12}e^{2} - \frac{41}{4}e + 3$ |
| 5 | $[5, 5, w + 4]$ | $-\frac{1}{12}e^{9} + \frac{3}{2}e^{7} + \frac{1}{12}e^{6} - \frac{35}{4}e^{5} - \frac{1}{4}e^{4} + \frac{55}{3}e^{3} - \frac{19}{12}e^{2} - \frac{41}{4}e + 3$ |
| 13 | $[13, 13, w + 6]$ | $\phantom{-}\frac{1}{6}e^{8} - \frac{5}{2}e^{6} - \frac{2}{3}e^{5} + \frac{21}{2}e^{4} + \frac{11}{2}e^{3} - \frac{61}{6}e^{2} - \frac{53}{6}e$ |
| 19 | $[19, 19, w + 2]$ | $\phantom{-}\frac{1}{6}e^{8} - \frac{1}{6}e^{7} - \frac{8}{3}e^{6} + \frac{8}{3}e^{5} + \frac{25}{2}e^{4} - 13e^{3} - \frac{97}{6}e^{2} + \frac{46}{3}e + 4$ |
| 19 | $[19, 19, w + 16]$ | $\phantom{-}\frac{1}{6}e^{8} - \frac{1}{6}e^{7} - \frac{8}{3}e^{6} + \frac{8}{3}e^{5} + \frac{25}{2}e^{4} - 13e^{3} - \frac{97}{6}e^{2} + \frac{46}{3}e + 4$ |
| 31 | $[31, 31, w + 13]$ | $\phantom{-}\frac{1}{6}e^{9} - \frac{1}{6}e^{8} - 3e^{7} + \frac{8}{3}e^{6} + \frac{35}{2}e^{5} - \frac{41}{3}e^{4} - \frac{223}{6}e^{3} + \frac{67}{3}e^{2} + \frac{76}{3}e - \frac{16}{3}$ |
| 31 | $[31, 31, w + 17]$ | $\phantom{-}\frac{1}{6}e^{9} - \frac{1}{6}e^{8} - 3e^{7} + \frac{8}{3}e^{6} + \frac{35}{2}e^{5} - \frac{41}{3}e^{4} - \frac{223}{6}e^{3} + \frac{67}{3}e^{2} + \frac{76}{3}e - \frac{16}{3}$ |
| 37 | $[37, 37, w + 18]$ | $-\frac{1}{6}e^{9} + \frac{1}{3}e^{8} + \frac{8}{3}e^{7} - \frac{29}{6}e^{6} - \frac{77}{6}e^{5} + \frac{119}{6}e^{4} + \frac{50}{3}e^{3} - \frac{33}{2}e^{2} + \frac{7}{6}e - \frac{16}{3}$ |
| 49 | $[49, 7, -7]$ | $-\frac{1}{12}e^{9} + \frac{3}{2}e^{7} + \frac{1}{12}e^{6} - \frac{35}{4}e^{5} - \frac{1}{4}e^{4} + \frac{58}{3}e^{3} - \frac{19}{12}e^{2} - \frac{69}{4}e + 11$ |
| 53 | $[53, 53, -234w + 2683]$ | $\phantom{-}\frac{1}{12}e^{9} - \frac{5}{3}e^{7} - \frac{1}{4}e^{6} + \frac{133}{12}e^{5} + \frac{13}{4}e^{4} - \frac{167}{6}e^{3} - \frac{137}{12}e^{2} + \frac{257}{12}e + 7$ |
| 53 | $[53, 53, -234w - 2449]$ | $\phantom{-}\frac{1}{12}e^{9} - \frac{5}{3}e^{7} - \frac{1}{4}e^{6} + \frac{133}{12}e^{5} + \frac{13}{4}e^{4} - \frac{167}{6}e^{3} - \frac{137}{12}e^{2} + \frac{257}{12}e + 7$ |
| 59 | $[59, 59, w + 1]$ | $\phantom{-}\frac{1}{6}e^{9} - \frac{19}{6}e^{7} - \frac{1}{3}e^{6} + \frac{119}{6}e^{5} + \frac{7}{2}e^{4} - \frac{277}{6}e^{3} - \frac{53}{6}e^{2} + \frac{95}{3}e$ |
| 59 | $[59, 59, w + 57]$ | $\phantom{-}\frac{1}{6}e^{9} - \frac{19}{6}e^{7} - \frac{1}{3}e^{6} + \frac{119}{6}e^{5} + \frac{7}{2}e^{4} - \frac{277}{6}e^{3} - \frac{53}{6}e^{2} + \frac{95}{3}e$ |
| 89 | $[89, 89, w + 41]$ | $-\frac{1}{6}e^{9} + \frac{1}{3}e^{8} + \frac{8}{3}e^{7} - \frac{31}{6}e^{6} - \frac{79}{6}e^{5} + \frac{49}{2}e^{4} + \frac{68}{3}e^{3} - \frac{69}{2}e^{2} - \frac{119}{6}e + 8$ |
| 89 | $[89, 89, w + 47]$ | $-\frac{1}{6}e^{9} + \frac{1}{3}e^{8} + \frac{8}{3}e^{7} - \frac{31}{6}e^{6} - \frac{79}{6}e^{5} + \frac{49}{2}e^{4} + \frac{68}{3}e^{3} - \frac{69}{2}e^{2} - \frac{119}{6}e + 8$ |
| 97 | $[97, 97, w + 26]$ | $-\frac{1}{4}e^{9} + 4e^{7} + \frac{3}{4}e^{6} - \frac{73}{4}e^{5} - \frac{31}{4}e^{4} + \frac{35}{2}e^{3} + \frac{93}{4}e^{2} + \frac{59}{4}e - 13$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, w]$ | $-1$ |
| $2$ | $[2, 2, w + 1]$ | $-1$ |