Base field \(\Q(\sqrt{193}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 48\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[9, 3, 3]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $35$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + 3x^{7} - 10x^{6} - 35x^{5} + 16x^{4} + 108x^{3} + 45x^{2} - 52x - 24\) |
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| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -9w - 58]$ | $\phantom{-}e$ |
| 2 | $[2, 2, -9w + 67]$ | $\phantom{-}e$ |
| 3 | $[3, 3, -2w + 15]$ | $\phantom{-}1$ |
| 3 | $[3, 3, 2w + 13]$ | $\phantom{-}1$ |
| 7 | $[7, 7, 186w - 1385]$ | $-\frac{1}{2}e^{7} - \frac{1}{2}e^{6} + 6e^{5} + \frac{11}{2}e^{4} - 20e^{3} - 16e^{2} + \frac{29}{2}e + 7$ |
| 7 | $[7, 7, -186w - 1199]$ | $-\frac{1}{2}e^{7} - \frac{1}{2}e^{6} + 6e^{5} + \frac{11}{2}e^{4} - 20e^{3} - 16e^{2} + \frac{29}{2}e + 7$ |
| 23 | $[23, 23, -38w - 245]$ | $\phantom{-}e^{4} - 7e^{2} + 6$ |
| 23 | $[23, 23, 38w - 283]$ | $\phantom{-}e^{4} - 7e^{2} + 6$ |
| 25 | $[25, 5, 5]$ | $-e^{7} - 2e^{6} + 12e^{5} + 22e^{4} - 39e^{3} - 62e^{2} + 22e + 28$ |
| 31 | $[31, 31, 16w - 119]$ | $-e^{7} - 2e^{6} + 12e^{5} + 22e^{4} - 38e^{3} - 62e^{2} + 17e + 22$ |
| 31 | $[31, 31, 16w + 103]$ | $-e^{7} - 2e^{6} + 12e^{5} + 22e^{4} - 38e^{3} - 62e^{2} + 17e + 22$ |
| 43 | $[43, 43, 4w + 25]$ | $-\frac{1}{2}e^{7} - \frac{1}{2}e^{6} + 6e^{5} + \frac{13}{2}e^{4} - 19e^{3} - 23e^{2} + \frac{19}{2}e + 13$ |
| 43 | $[43, 43, -4w + 29]$ | $-\frac{1}{2}e^{7} - \frac{1}{2}e^{6} + 6e^{5} + \frac{13}{2}e^{4} - 19e^{3} - 23e^{2} + \frac{19}{2}e + 13$ |
| 59 | $[59, 59, 12w - 89]$ | $\phantom{-}e^{6} - 12e^{4} - e^{3} + 37e^{2} + 5e - 18$ |
| 59 | $[59, 59, -12w - 77]$ | $\phantom{-}e^{6} - 12e^{4} - e^{3} + 37e^{2} + 5e - 18$ |
| 67 | $[67, 67, 92w + 593]$ | $-e^{7} - 2e^{6} + 11e^{5} + 22e^{4} - 30e^{3} - 61e^{2} + 4e + 17$ |
| 67 | $[67, 67, 92w - 685]$ | $-e^{7} - 2e^{6} + 11e^{5} + 22e^{4} - 30e^{3} - 61e^{2} + 4e + 17$ |
| 83 | $[83, 83, 204w - 1519]$ | $\phantom{-}e^{5} - 7e^{3} + 2e^{2} + 6e - 6$ |
| 83 | $[83, 83, 204w + 1315]$ | $\phantom{-}e^{5} - 7e^{3} + 2e^{2} + 6e - 6$ |
| 97 | $[97, 97, -24w + 179]$ | $-\frac{1}{2}e^{7} - \frac{1}{2}e^{6} + 7e^{5} + \frac{13}{2}e^{4} - 28e^{3} - 23e^{2} + \frac{55}{2}e + 13$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3, 3, -2w + 15]$ | $-1$ |
| $3$ | $[3, 3, 2w + 13]$ | $-1$ |