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: | $[16, 4, 4]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $40$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} - x^{3} - 18x^{2} + 41x - 19\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, 2]$ | $\phantom{-}0$ |
| 7 | $[7, 7, w - 7]$ | $-\frac{1}{7}e^{3} - \frac{3}{7}e^{2} + \frac{6}{7}e + \frac{11}{7}$ |
| 7 | $[7, 7, w + 6]$ | $\phantom{-}e$ |
| 9 | $[9, 3, 3]$ | $\phantom{-}0$ |
| 19 | $[19, 19, w + 5]$ | $\phantom{-}\frac{1}{7}e^{3} - \frac{4}{7}e^{2} - \frac{27}{7}e + \frac{59}{7}$ |
| 19 | $[19, 19, w - 6]$ | $-\frac{4}{7}e^{3} - \frac{5}{7}e^{2} + \frac{66}{7}e - \frac{40}{7}$ |
| 23 | $[23, 23, w + 8]$ | $\phantom{-}\frac{2}{7}e^{3} - \frac{1}{7}e^{2} - \frac{26}{7}e + \frac{62}{7}$ |
| 23 | $[23, 23, -w + 9]$ | $-\frac{5}{7}e^{3} - \frac{8}{7}e^{2} + \frac{65}{7}e - \frac{15}{7}$ |
| 25 | $[25, 5, 5]$ | $\phantom{-}\frac{1}{7}e^{3} + \frac{3}{7}e^{2} - \frac{13}{7}e - \frac{39}{7}$ |
| 29 | $[29, 29, -w - 4]$ | $-\frac{1}{7}e^{3} - \frac{3}{7}e^{2} + \frac{6}{7}e + \frac{25}{7}$ |
| 29 | $[29, 29, w - 5]$ | $\phantom{-}e + 2$ |
| 37 | $[37, 37, -w - 3]$ | $\phantom{-}\frac{4}{7}e^{3} + \frac{5}{7}e^{2} - \frac{59}{7}e + \frac{47}{7}$ |
| 37 | $[37, 37, w - 4]$ | $-\frac{2}{7}e^{3} + \frac{1}{7}e^{2} + \frac{33}{7}e - \frac{41}{7}$ |
| 41 | $[41, 41, -w - 9]$ | $-\frac{8}{7}e^{3} - \frac{3}{7}e^{2} + \frac{139}{7}e - \frac{115}{7}$ |
| 41 | $[41, 41, w - 10]$ | $\phantom{-}\frac{8}{7}e^{3} + \frac{3}{7}e^{2} - \frac{139}{7}e + \frac{171}{7}$ |
| 43 | $[43, 43, -w - 2]$ | $-e^{3} - e^{2} + 15e - 16$ |
| 43 | $[43, 43, w - 3]$ | $\phantom{-}\frac{5}{7}e^{3} + \frac{1}{7}e^{2} - \frac{79}{7}e + \frac{64}{7}$ |
| 47 | $[47, 47, -w - 1]$ | $-\frac{9}{7}e^{3} - \frac{6}{7}e^{2} + \frac{152}{7}e - \frac{146}{7}$ |
| 47 | $[47, 47, w - 2]$ | $\phantom{-}e^{3} - 18e + 20$ |
| 53 | $[53, 53, 2w - 13]$ | $\phantom{-}\frac{5}{7}e^{3} + \frac{1}{7}e^{2} - \frac{79}{7}e + \frac{127}{7}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $4$ | $[4, 2, 2]$ | $1$ |