Base field \(\Q(\sqrt{197}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 49\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[4, 2, 2]$ |
| Dimension: | $6$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $12$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{6} + 10x^{5} + 23x^{4} - 38x^{3} - 144x^{2} - 8x + 80\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, 2]$ | $-1$ |
| 7 | $[7, 7, w - 7]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w + 6]$ | $\phantom{-}e$ |
| 9 | $[9, 3, 3]$ | $\phantom{-}\frac{1}{16}e^{5} + \frac{1}{4}e^{4} - \frac{17}{16}e^{3} - 3e^{2} + 4e + \frac{7}{2}$ |
| 19 | $[19, 19, w + 5]$ | $\phantom{-}\frac{1}{4}e^{5} + \frac{7}{4}e^{4} + \frac{3}{4}e^{3} - \frac{43}{4}e^{2} - 6e + 11$ |
| 19 | $[19, 19, w - 6]$ | $\phantom{-}\frac{1}{4}e^{5} + \frac{7}{4}e^{4} + \frac{3}{4}e^{3} - \frac{43}{4}e^{2} - 6e + 11$ |
| 23 | $[23, 23, w + 8]$ | $-\frac{1}{2}e^{3} - \frac{3}{2}e^{2} + 4e + 4$ |
| 23 | $[23, 23, -w + 9]$ | $-\frac{1}{2}e^{3} - \frac{3}{2}e^{2} + 4e + 4$ |
| 25 | $[25, 5, 5]$ | $-\frac{1}{4}e^{4} - \frac{3}{2}e^{3} - \frac{5}{4}e^{2} + 3e + 9$ |
| 29 | $[29, 29, -w - 4]$ | $\phantom{-}\frac{1}{16}e^{5} + \frac{1}{4}e^{4} - \frac{9}{16}e^{3} - \frac{1}{2}e^{2} + 5e - \frac{9}{2}$ |
| 29 | $[29, 29, w - 5]$ | $\phantom{-}\frac{1}{16}e^{5} + \frac{1}{4}e^{4} - \frac{9}{16}e^{3} - \frac{1}{2}e^{2} + 5e - \frac{9}{2}$ |
| 37 | $[37, 37, -w - 3]$ | $-\frac{3}{16}e^{5} - e^{4} + \frac{19}{16}e^{3} + \frac{27}{4}e^{2} - \frac{11}{2}e - \frac{7}{2}$ |
| 37 | $[37, 37, w - 4]$ | $-\frac{3}{16}e^{5} - e^{4} + \frac{19}{16}e^{3} + \frac{27}{4}e^{2} - \frac{11}{2}e - \frac{7}{2}$ |
| 41 | $[41, 41, -w - 9]$ | $\phantom{-}\frac{1}{16}e^{5} + \frac{1}{2}e^{4} + \frac{15}{16}e^{3} - \frac{1}{4}e^{2} - 2e - \frac{9}{2}$ |
| 41 | $[41, 41, w - 10]$ | $\phantom{-}\frac{1}{16}e^{5} + \frac{1}{2}e^{4} + \frac{15}{16}e^{3} - \frac{1}{4}e^{2} - 2e - \frac{9}{2}$ |
| 43 | $[43, 43, -w - 2]$ | $-\frac{1}{8}e^{5} - e^{4} - \frac{11}{8}e^{3} + \frac{9}{2}e^{2} + \frac{19}{2}e$ |
| 43 | $[43, 43, w - 3]$ | $-\frac{1}{8}e^{5} - e^{4} - \frac{11}{8}e^{3} + \frac{9}{2}e^{2} + \frac{19}{2}e$ |
| 47 | $[47, 47, -w - 1]$ | $-e^{2} - 4e + 4$ |
| 47 | $[47, 47, w - 2]$ | $-e^{2} - 4e + 4$ |
| 53 | $[53, 53, 2w - 13]$ | $-\frac{1}{16}e^{5} - \frac{1}{2}e^{4} - \frac{7}{16}e^{3} + \frac{13}{4}e^{2} + \frac{3}{2}e - \frac{13}{2}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, 2]$ | $1$ |