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: | $[9, 3, 3]$ |
| Dimension: | $4$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $30$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{4} - 3x^{3} - 23x^{2} + 41x + 139\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, 2]$ | $\phantom{-}\frac{1}{11}e^{3} - \frac{1}{11}e^{2} - \frac{14}{11}e - \frac{20}{11}$ |
| 7 | $[7, 7, w - 7]$ | $\phantom{-}\frac{1}{11}e^{3} - \frac{1}{11}e^{2} - \frac{25}{11}e + \frac{13}{11}$ |
| 7 | $[7, 7, w + 6]$ | $\phantom{-}e$ |
| 9 | $[9, 3, 3]$ | $-1$ |
| 19 | $[19, 19, w + 5]$ | $-\frac{2}{11}e^{3} - \frac{9}{11}e^{2} + \frac{39}{11}e + \frac{150}{11}$ |
| 19 | $[19, 19, w - 6]$ | $\phantom{-}e^{2} - e - 13$ |
| 23 | $[23, 23, w + 8]$ | $-\frac{2}{11}e^{3} + \frac{2}{11}e^{2} + \frac{39}{11}e + \frac{40}{11}$ |
| 23 | $[23, 23, -w + 9]$ | $-\frac{1}{11}e^{3} + \frac{1}{11}e^{2} + \frac{3}{11}e + \frac{53}{11}$ |
| 25 | $[25, 5, 5]$ | $-\frac{3}{11}e^{3} + \frac{3}{11}e^{2} + \frac{42}{11}e - \frac{50}{11}$ |
| 29 | $[29, 29, -w - 4]$ | $-\frac{1}{11}e^{3} + \frac{1}{11}e^{2} + \frac{14}{11}e - \frac{24}{11}$ |
| 29 | $[29, 29, w - 5]$ | $-\frac{1}{11}e^{3} + \frac{1}{11}e^{2} + \frac{14}{11}e - \frac{24}{11}$ |
| 37 | $[37, 37, -w - 3]$ | $\phantom{-}\frac{5}{11}e^{3} + \frac{6}{11}e^{2} - \frac{92}{11}e - \frac{155}{11}$ |
| 37 | $[37, 37, w - 4]$ | $\phantom{-}\frac{2}{11}e^{3} - \frac{13}{11}e^{2} - \frac{6}{11}e + \frac{125}{11}$ |
| 41 | $[41, 41, -w - 9]$ | $\phantom{-}\frac{5}{11}e^{3} - \frac{5}{11}e^{2} - \frac{59}{11}e - \frac{1}{11}$ |
| 41 | $[41, 41, w - 10]$ | $\phantom{-}\frac{6}{11}e^{3} - \frac{6}{11}e^{2} - \frac{95}{11}e + \frac{12}{11}$ |
| 43 | $[43, 43, -w - 2]$ | $-\frac{1}{11}e^{3} + \frac{12}{11}e^{2} - \frac{8}{11}e - \frac{145}{11}$ |
| 43 | $[43, 43, w - 3]$ | $-\frac{4}{11}e^{3} - \frac{7}{11}e^{2} + \frac{78}{11}e + \frac{135}{11}$ |
| 47 | $[47, 47, -w - 1]$ | $\phantom{-}\frac{1}{11}e^{3} - \frac{1}{11}e^{2} - \frac{14}{11}e - \frac{42}{11}$ |
| 47 | $[47, 47, w - 2]$ | $\phantom{-}\frac{1}{11}e^{3} - \frac{1}{11}e^{2} - \frac{14}{11}e - \frac{42}{11}$ |
| 53 | $[53, 53, 2w - 13]$ | $-\frac{2}{11}e^{3} - \frac{9}{11}e^{2} + \frac{61}{11}e + \frac{139}{11}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $9$ | $[9, 3, 3]$ | $1$ |