Base field \(\Q(\sqrt{85}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 21\); narrow class number \(2\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[15,15,-w + 3]$ |
| 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:
| \(x^{4} + 2x^{3} - 7x^{2} - 8x + 37\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}\frac{2}{25}e^{3} + \frac{3}{25}e^{2} - \frac{3}{25}e - \frac{2}{25}$ |
| 3 | $[3, 3, w + 2]$ | $-\frac{2}{25}e^{3} - \frac{3}{25}e^{2} + \frac{3}{25}e + \frac{2}{25}$ |
| 4 | $[4, 2, 2]$ | $-\frac{2}{25}e^{3} - \frac{3}{25}e^{2} + \frac{28}{25}e + \frac{2}{25}$ |
| 5 | $[5, 5, w + 2]$ | $-\frac{2}{25}e^{3} - \frac{3}{25}e^{2} + \frac{3}{25}e + \frac{2}{25}$ |
| 7 | $[7, 7, w]$ | $\phantom{-}\frac{1}{25}e^{3} - \frac{11}{25}e^{2} - \frac{14}{25}e + \frac{49}{25}$ |
| 7 | $[7, 7, w + 6]$ | $-\frac{1}{25}e^{3} + \frac{11}{25}e^{2} + \frac{14}{25}e - \frac{49}{25}$ |
| 17 | $[17, 17, w + 8]$ | $\phantom{-}\frac{6}{25}e^{3} + \frac{9}{25}e^{2} - \frac{9}{25}e - \frac{6}{25}$ |
| 19 | $[19, 19, w + 1]$ | $\phantom{-}\frac{2}{25}e^{3} + \frac{3}{25}e^{2} - \frac{28}{25}e + \frac{73}{25}$ |
| 19 | $[19, 19, w - 2]$ | $-\frac{4}{25}e^{3} - \frac{6}{25}e^{2} + \frac{56}{25}e + \frac{79}{25}$ |
| 23 | $[23, 23, w + 9]$ | $\phantom{-}\frac{6}{25}e^{3} + \frac{9}{25}e^{2} - \frac{9}{25}e - \frac{6}{25}$ |
| 23 | $[23, 23, w + 13]$ | $\phantom{-}\frac{3}{25}e^{3} + \frac{42}{25}e^{2} + \frac{33}{25}e - \frac{153}{25}$ |
| 37 | $[37, 37, w + 11]$ | $\phantom{-}\frac{2}{25}e^{3} - \frac{22}{25}e^{2} - \frac{28}{25}e + \frac{98}{25}$ |
| 37 | $[37, 37, w + 25]$ | $\phantom{-}\frac{4}{25}e^{3} - \frac{44}{25}e^{2} - \frac{56}{25}e + \frac{196}{25}$ |
| 59 | $[59, 59, 3w + 10]$ | $\phantom{-}\frac{6}{25}e^{3} + \frac{9}{25}e^{2} - \frac{84}{25}e - \frac{231}{25}$ |
| 59 | $[59, 59, 3w - 13]$ | $-\frac{6}{25}e^{3} - \frac{9}{25}e^{2} + \frac{84}{25}e + \frac{156}{25}$ |
| 73 | $[73, 73, w + 15]$ | $-\frac{7}{25}e^{3} + \frac{2}{25}e^{2} + \frac{23}{25}e - \frac{43}{25}$ |
| 73 | $[73, 73, w + 57]$ | $-\frac{8}{25}e^{3} + \frac{13}{25}e^{2} + \frac{37}{25}e - \frac{92}{25}$ |
| 89 | $[89, 89, -w - 10]$ | $\phantom{-}9$ |
| 89 | $[89, 89, w - 11]$ | $\phantom{-}\frac{6}{25}e^{3} + \frac{9}{25}e^{2} - \frac{84}{25}e - \frac{156}{25}$ |
| 97 | $[97, 97, w + 22]$ | $-\frac{11}{25}e^{3} + \frac{21}{25}e^{2} + \frac{54}{25}e - \frac{139}{25}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $3$ | $[3,3,-w + 3]$ | $-\frac{2}{25}e^{3} - \frac{3}{25}e^{2} + \frac{3}{25}e + \frac{2}{25}$ |
| $5$ | $[5,5,-w + 3]$ | $\frac{2}{25}e^{3} + \frac{3}{25}e^{2} - \frac{3}{25}e - \frac{2}{25}$ |