Base field \(\Q(\sqrt{113}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 28\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[9, 3, 3]$ |
| Dimension: | $10$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $24$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{10} + x^{9} - 15x^{8} - 11x^{7} + 76x^{6} + 36x^{5} - 150x^{4} - 44x^{3} + 98x^{2} + 20x - 16\) |
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| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -w + 6]$ | $\phantom{-}e$ |
| 2 | $[2, 2, w + 5]$ | $\phantom{-}e$ |
| 7 | $[7, 7, 6w - 35]$ | $\phantom{-}\frac{3}{4}e^{9} + \frac{1}{4}e^{8} - \frac{45}{4}e^{7} - \frac{3}{4}e^{6} + 55e^{5} - 10e^{4} - \frac{189}{2}e^{3} + 33e^{2} + \frac{73}{2}e - 14$ |
| 7 | $[7, 7, -6w - 29]$ | $\phantom{-}\frac{3}{4}e^{9} + \frac{1}{4}e^{8} - \frac{45}{4}e^{7} - \frac{3}{4}e^{6} + 55e^{5} - 10e^{4} - \frac{189}{2}e^{3} + 33e^{2} + \frac{73}{2}e - 14$ |
| 9 | $[9, 3, 3]$ | $-1$ |
| 11 | $[11, 11, 4w + 19]$ | $-e^{9} - e^{8} + 14e^{7} + 9e^{6} - 64e^{5} - 16e^{4} + 106e^{3} - 6e^{2} - 43e + 6$ |
| 11 | $[11, 11, 4w - 23]$ | $-e^{9} - e^{8} + 14e^{7} + 9e^{6} - 64e^{5} - 16e^{4} + 106e^{3} - 6e^{2} - 43e + 6$ |
| 13 | $[13, 13, -2w + 11]$ | $-\frac{1}{2}e^{9} + \frac{1}{2}e^{8} + 9e^{7} - \frac{15}{2}e^{6} - \frac{105}{2}e^{5} + 36e^{4} + 107e^{3} - 59e^{2} - 52e + 19$ |
| 13 | $[13, 13, 2w + 9]$ | $-\frac{1}{2}e^{9} + \frac{1}{2}e^{8} + 9e^{7} - \frac{15}{2}e^{6} - \frac{105}{2}e^{5} + 36e^{4} + 107e^{3} - 59e^{2} - 52e + 19$ |
| 25 | $[25, 5, -5]$ | $\phantom{-}\frac{1}{4}e^{9} + \frac{1}{4}e^{8} - \frac{13}{4}e^{7} - \frac{7}{4}e^{6} + \frac{27}{2}e^{5} - e^{4} - \frac{39}{2}e^{3} + 14e^{2} + \frac{7}{2}e - 1$ |
| 31 | $[31, 31, 2w - 13]$ | $\phantom{-}\frac{1}{2}e^{7} + e^{6} - \frac{11}{2}e^{5} - 9e^{4} + 16e^{3} + 17e^{2} - 9e - 3$ |
| 31 | $[31, 31, -2w - 11]$ | $\phantom{-}\frac{1}{2}e^{7} + e^{6} - \frac{11}{2}e^{5} - 9e^{4} + 16e^{3} + 17e^{2} - 9e - 3$ |
| 41 | $[41, 41, -8w - 39]$ | $-\frac{3}{2}e^{9} + \frac{1}{2}e^{8} + \frac{49}{2}e^{7} - \frac{23}{2}e^{6} - 131e^{5} + 74e^{4} + 246e^{3} - 146e^{2} - 105e + 54$ |
| 41 | $[41, 41, 8w - 47]$ | $-\frac{3}{2}e^{9} + \frac{1}{2}e^{8} + \frac{49}{2}e^{7} - \frac{23}{2}e^{6} - 131e^{5} + 74e^{4} + 246e^{3} - 146e^{2} - 105e + 54$ |
| 53 | $[53, 53, -26w - 125]$ | $-\frac{1}{2}e^{9} - \frac{1}{2}e^{8} + \frac{13}{2}e^{7} + \frac{7}{2}e^{6} - 27e^{5} + 2e^{4} + 42e^{3} - 28e^{2} - 22e + 12$ |
| 53 | $[53, 53, 26w - 151]$ | $-\frac{1}{2}e^{9} - \frac{1}{2}e^{8} + \frac{13}{2}e^{7} + \frac{7}{2}e^{6} - 27e^{5} + 2e^{4} + 42e^{3} - 28e^{2} - 22e + 12$ |
| 61 | $[61, 61, -14w + 81]$ | $\phantom{-}\frac{3}{4}e^{9} - \frac{1}{4}e^{8} - \frac{47}{4}e^{7} + \frac{27}{4}e^{6} + \frac{121}{2}e^{5} - 46e^{4} - \frac{221}{2}e^{3} + 90e^{2} + \frac{97}{2}e - 31$ |
| 61 | $[61, 61, -14w - 67]$ | $\phantom{-}\frac{3}{4}e^{9} - \frac{1}{4}e^{8} - \frac{47}{4}e^{7} + \frac{27}{4}e^{6} + \frac{121}{2}e^{5} - 46e^{4} - \frac{221}{2}e^{3} + 90e^{2} + \frac{97}{2}e - 31$ |
| 83 | $[83, 83, 2w - 15]$ | $\phantom{-}\frac{3}{2}e^{9} - \frac{1}{2}e^{8} - \frac{49}{2}e^{7} + \frac{23}{2}e^{6} + 131e^{5} - 74e^{4} - 246e^{3} + 144e^{2} + 105e - 50$ |
| 83 | $[83, 83, -2w - 13]$ | $\phantom{-}\frac{3}{2}e^{9} - \frac{1}{2}e^{8} - \frac{49}{2}e^{7} + \frac{23}{2}e^{6} + 131e^{5} - 74e^{4} - 246e^{3} + 144e^{2} + 105e - 50$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $9$ | $[9, 3, 3]$ | $1$ |