Base field \(\Q(\sqrt{205}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 51\); narrow class number \(4\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[3,3,-w + 1]$ |
| Dimension: | $8$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $16$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{8} + 19x^{6} + 99x^{4} + 161x^{2} + 16\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 3 | $[3, 3, w + 2]$ | $-\frac{1}{8}e^{7} - 2e^{5} - \frac{53}{8}e^{3} - \frac{15}{4}e$ |
| 4 | $[4, 2, 2]$ | $-\frac{1}{16}e^{6} - \frac{7}{8}e^{4} - \frac{29}{16}e^{2}$ |
| 5 | $[5, 5, -w + 8]$ | $\phantom{-}\frac{5}{16}e^{6} + \frac{39}{8}e^{4} + \frac{233}{16}e^{2} + 3$ |
| 7 | $[7, 7, w + 1]$ | $-\frac{3}{16}e^{7} - \frac{21}{8}e^{5} - \frac{71}{16}e^{3} + 8e$ |
| 7 | $[7, 7, w + 5]$ | $-\frac{1}{16}e^{7} - \frac{5}{8}e^{5} + \frac{39}{16}e^{3} + 14e$ |
| 13 | $[13, 13, w + 3]$ | $\phantom{-}\frac{3}{16}e^{7} + \frac{21}{8}e^{5} + \frac{71}{16}e^{3} - 7e$ |
| 13 | $[13, 13, w + 9]$ | $\phantom{-}\frac{1}{4}e^{5} + \frac{17}{4}e^{3} + 14e$ |
| 17 | $[17, 17, w]$ | $-\frac{5}{16}e^{7} - \frac{39}{8}e^{5} - \frac{233}{16}e^{3} - 5e$ |
| 17 | $[17, 17, w + 16]$ | $\phantom{-}\frac{5}{16}e^{7} + \frac{39}{8}e^{5} + \frac{233}{16}e^{3} + 5e$ |
| 31 | $[31, 31, -w - 4]$ | $-\frac{7}{8}e^{6} - \frac{27}{2}e^{4} - \frac{309}{8}e^{2} - 2$ |
| 31 | $[31, 31, w - 5]$ | $\phantom{-}\frac{23}{16}e^{6} + \frac{177}{8}e^{4} + \frac{1003}{16}e^{2} + 3$ |
| 41 | $[41, 41, 3w - 22]$ | $-\frac{1}{16}e^{6} - \frac{5}{8}e^{4} + \frac{23}{16}e^{2} + 1$ |
| 47 | $[47, 47, w + 19]$ | $-\frac{3}{8}e^{7} - \frac{25}{4}e^{5} - \frac{183}{8}e^{3} - 15e$ |
| 47 | $[47, 47, w + 27]$ | $\phantom{-}\frac{1}{16}e^{7} + \frac{9}{8}e^{5} + \frac{81}{16}e^{3} + 8e$ |
| 53 | $[53, 53, w + 14]$ | $-\frac{3}{4}e^{7} - \frac{23}{2}e^{5} - \frac{127}{4}e^{3} + 2e$ |
| 53 | $[53, 53, w + 38]$ | $\phantom{-}\frac{11}{8}e^{7} + \frac{87}{4}e^{5} + \frac{547}{8}e^{3} + 28e$ |
| 59 | $[59, 59, -w - 10]$ | $\phantom{-}\frac{13}{16}e^{6} + \frac{101}{8}e^{4} + \frac{621}{16}e^{2} + 17$ |
| 59 | $[59, 59, w - 11]$ | $-\frac{3}{2}e^{6} - 23e^{4} - \frac{127}{2}e^{2} - 4$ |
| 61 | $[61, 61, 2w - 13]$ | $\phantom{-}\frac{5}{4}e^{6} + \frac{77}{4}e^{4} + 55e^{2} + 14$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3,3,-w + 1]$ | $\frac{1}{8}e^{7} + 2e^{5} + \frac{53}{8}e^{3} + \frac{15}{4}e$ |