Base field \(\Q(\sqrt{2}, \sqrt{13})\)
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 9x^{2} + 10x - 1\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[9,3,\frac{2}{5}w^{3} - \frac{3}{5}w^{2} - \frac{22}{5}w + \frac{14}{5}]$ |
| Dimension: | $6$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $8$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{6} + 4x^{5} - 12x^{4} - 56x^{3} - 8x^{2} + 112x + 80\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{2}{5}w^{3} + \frac{3}{5}w^{2} + \frac{17}{5}w - \frac{9}{5}]$ | $\phantom{-}e$ |
| 9 | $[9, 3, -\frac{2}{5}w^{3} + \frac{3}{5}w^{2} + \frac{22}{5}w - \frac{9}{5}]$ | $\phantom{-}\frac{1}{4}e^{5} + \frac{3}{4}e^{4} - 4e^{3} - 10e^{2} + 12e + 20$ |
| 9 | $[9, 3, \frac{2}{5}w^{3} - \frac{3}{5}w^{2} - \frac{22}{5}w + \frac{14}{5}]$ | $-1$ |
| 17 | $[17, 17, w + 1]$ | $\phantom{-}\frac{1}{4}e^{4} - 4e^{2} + 8$ |
| 17 | $[17, 17, -\frac{4}{5}w^{3} + \frac{6}{5}w^{2} + \frac{39}{5}w - \frac{13}{5}]$ | $\phantom{-}\frac{1}{4}e^{4} - 4e^{2} + 8$ |
| 17 | $[17, 17, -\frac{4}{5}w^{3} + \frac{6}{5}w^{2} + \frac{39}{5}w - \frac{28}{5}]$ | $-\frac{7}{8}e^{5} - \frac{5}{2}e^{4} + \frac{27}{2}e^{3} + 34e^{2} - 33e - 62$ |
| 17 | $[17, 17, -w + 2]$ | $-\frac{7}{8}e^{5} - \frac{5}{2}e^{4} + \frac{27}{2}e^{3} + 34e^{2} - 33e - 62$ |
| 23 | $[23, 23, -\frac{1}{5}w^{3} + \frac{4}{5}w^{2} + \frac{6}{5}w - \frac{22}{5}]$ | $-\frac{5}{8}e^{5} - \frac{3}{2}e^{4} + 10e^{3} + 20e^{2} - 26e - 36$ |
| 23 | $[23, 23, -\frac{1}{5}w^{3} - \frac{1}{5}w^{2} + \frac{11}{5}w + \frac{3}{5}]$ | $\phantom{-}\frac{1}{2}e^{5} + \frac{5}{4}e^{4} - 8e^{3} - 17e^{2} + 22e + 34$ |
| 23 | $[23, 23, -\frac{1}{5}w^{3} + \frac{4}{5}w^{2} + \frac{6}{5}w - \frac{12}{5}]$ | $-\frac{5}{8}e^{5} - \frac{3}{2}e^{4} + 10e^{3} + 20e^{2} - 26e - 36$ |
| 23 | $[23, 23, -\frac{1}{5}w^{3} - \frac{1}{5}w^{2} + \frac{11}{5}w + \frac{13}{5}]$ | $\phantom{-}\frac{1}{2}e^{5} + \frac{5}{4}e^{4} - 8e^{3} - 17e^{2} + 22e + 34$ |
| 25 | $[25, 5, -\frac{4}{5}w^{3} + \frac{6}{5}w^{2} + \frac{39}{5}w - \frac{33}{5}]$ | $-\frac{1}{2}e^{5} - \frac{3}{2}e^{4} + \frac{15}{2}e^{3} + 20e^{2} - 18e - 34$ |
| 25 | $[25, 5, -w^{3} + w^{2} + 10w - 2]$ | $-\frac{1}{2}e^{5} - \frac{3}{2}e^{4} + \frac{15}{2}e^{3} + 20e^{2} - 18e - 34$ |
| 49 | $[49, 7, -\frac{4}{5}w^{3} + \frac{6}{5}w^{2} + \frac{34}{5}w - \frac{23}{5}]$ | $\phantom{-}\frac{1}{4}e^{5} + \frac{3}{4}e^{4} - 4e^{3} - 11e^{2} + 10e + 20$ |
| 49 | $[49, 7, \frac{4}{5}w^{3} - \frac{6}{5}w^{2} - \frac{34}{5}w + \frac{13}{5}]$ | $\phantom{-}\frac{1}{4}e^{5} + \frac{3}{4}e^{4} - 4e^{3} - 11e^{2} + 10e + 20$ |
| 79 | $[79, 79, \frac{3}{5}w^{3} - \frac{7}{5}w^{2} - \frac{28}{5}w + \frac{21}{5}]$ | $-\frac{5}{8}e^{5} - \frac{3}{2}e^{4} + \frac{19}{2}e^{3} + 20e^{2} - 20e - 40$ |
| 79 | $[79, 79, \frac{1}{5}w^{3} + \frac{1}{5}w^{2} - \frac{16}{5}w - \frac{13}{5}]$ | $-\frac{5}{8}e^{5} - \frac{3}{2}e^{4} + \frac{19}{2}e^{3} + 20e^{2} - 20e - 40$ |
| 79 | $[79, 79, -\frac{1}{5}w^{3} + \frac{4}{5}w^{2} + \frac{11}{5}w - \frac{27}{5}]$ | $\phantom{-}\frac{5}{4}e^{5} + \frac{7}{2}e^{4} - \frac{37}{2}e^{3} - 46e^{2} + 38e + 80$ |
| 79 | $[79, 79, \frac{3}{5}w^{3} - \frac{2}{5}w^{2} - \frac{33}{5}w + \frac{11}{5}]$ | $\phantom{-}\frac{5}{4}e^{5} + \frac{7}{2}e^{4} - \frac{37}{2}e^{3} - 46e^{2} + 38e + 80$ |
| 103 | $[103, 103, -\frac{1}{5}w^{3} + \frac{4}{5}w^{2} + \frac{1}{5}w - \frac{17}{5}]$ | $\phantom{-}\frac{1}{4}e^{5} + \frac{1}{2}e^{4} - 3e^{3} - 5e^{2} + 4$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $9$ | $[9,3,\frac{2}{5}w^{3} - \frac{3}{5}w^{2} - \frac{22}{5}w + \frac{14}{5}]$ | $1$ |