Base field \(\Q(\sqrt{2}, \sqrt{13})\)
Generator \(w\), with minimal polynomial \(x^4 - 2 x^3 - 9 x^2 + 10 x - 1\); 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:
| \(x^6 + 4 x^5 - 12 x^4 - 56 x^3 - 8 x^2 + 112 x + 80\) |
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 | $[9, 3, -\frac{2}{5} w^3 + \frac{3}{5} w^2 + \frac{22}{5} w - \frac{9}{5}]$ | $\phantom{-}\frac{1}{4} e^5 + \frac{3}{4} e^4 - 4 e^3 - 10 e^2 + 12 e + 20$ |
| 9 | $[9, 3, \frac{2}{5} w^3 - \frac{3}{5} w^2 - \frac{22}{5} w + \frac{14}{5}]$ | $-1$ |
| 17 | $[17, 17, w + 1]$ | $\phantom{-}\frac{1}{4} e^4 - 4 e^2 + 8$ |
| 17 | $[17, 17, -\frac{4}{5} w^3 + \frac{6}{5} w^2 + \frac{39}{5} w - \frac{13}{5}]$ | $\phantom{-}\frac{1}{4} e^4 - 4 e^2 + 8$ |
| 17 | $[17, 17, -\frac{4}{5} w^3 + \frac{6}{5} w^2 + \frac{39}{5} w - \frac{28}{5}]$ | $-\frac{7}{8} e^5 - \frac{5}{2} e^4 + \frac{27}{2} e^3 + 34 e^2 - 33 e - 62$ |
| 17 | $[17, 17, -w + 2]$ | $-\frac{7}{8} e^5 - \frac{5}{2} e^4 + \frac{27}{2} e^3 + 34 e^2 - 33 e - 62$ |
| 23 | $[23, 23, -\frac{1}{5} w^3 + \frac{4}{5} w^2 + \frac{6}{5} w - \frac{22}{5}]$ | $-\frac{5}{8} e^5 - \frac{3}{2} e^4 + 10 e^3 + 20 e^2 - 26 e - 36$ |
| 23 | $[23, 23, -\frac{1}{5} w^3 - \frac{1}{5} w^2 + \frac{11}{5} w + \frac{3}{5}]$ | $\phantom{-}\frac{1}{2} e^5 + \frac{5}{4} e^4 - 8 e^3 - 17 e^2 + 22 e + 34$ |
| 23 | $[23, 23, -\frac{1}{5} w^3 + \frac{4}{5} w^2 + \frac{6}{5} w - \frac{12}{5}]$ | $-\frac{5}{8} e^5 - \frac{3}{2} e^4 + 10 e^3 + 20 e^2 - 26 e - 36$ |
| 23 | $[23, 23, -\frac{1}{5} w^3 - \frac{1}{5} w^2 + \frac{11}{5} w + \frac{13}{5}]$ | $\phantom{-}\frac{1}{2} e^5 + \frac{5}{4} e^4 - 8 e^3 - 17 e^2 + 22 e + 34$ |
| 25 | $[25, 5, -\frac{4}{5} w^3 + \frac{6}{5} w^2 + \frac{39}{5} w - \frac{33}{5}]$ | $-\frac{1}{2} e^5 - \frac{3}{2} e^4 + \frac{15}{2} e^3 + 20 e^2 - 18 e - 34$ |
| 25 | $[25, 5, -w^3 + w^2 + 10 w - 2]$ | $-\frac{1}{2} e^5 - \frac{3}{2} e^4 + \frac{15}{2} e^3 + 20 e^2 - 18 e - 34$ |
| 49 | $[49, 7, -\frac{4}{5} w^3 + \frac{6}{5} w^2 + \frac{34}{5} w - \frac{23}{5}]$ | $\phantom{-}\frac{1}{4} e^5 + \frac{3}{4} e^4 - 4 e^3 - 11 e^2 + 10 e + 20$ |
| 49 | $[49, 7, \frac{4}{5} w^3 - \frac{6}{5} w^2 - \frac{34}{5} w + \frac{13}{5}]$ | $\phantom{-}\frac{1}{4} e^5 + \frac{3}{4} e^4 - 4 e^3 - 11 e^2 + 10 e + 20$ |
| 79 | $[79, 79, \frac{3}{5} w^3 - \frac{7}{5} w^2 - \frac{28}{5} w + \frac{21}{5}]$ | $-\frac{5}{8} e^5 - \frac{3}{2} e^4 + \frac{19}{2} e^3 + 20 e^2 - 20 e - 40$ |
| 79 | $[79, 79, \frac{1}{5} w^3 + \frac{1}{5} w^2 - \frac{16}{5} w - \frac{13}{5}]$ | $-\frac{5}{8} e^5 - \frac{3}{2} e^4 + \frac{19}{2} e^3 + 20 e^2 - 20 e - 40$ |
| 79 | $[79, 79, -\frac{1}{5} w^3 + \frac{4}{5} w^2 + \frac{11}{5} w - \frac{27}{5}]$ | $\phantom{-}\frac{5}{4} e^5 + \frac{7}{2} e^4 - \frac{37}{2} e^3 - 46 e^2 + 38 e + 80$ |
| 79 | $[79, 79, \frac{3}{5} w^3 - \frac{2}{5} w^2 - \frac{33}{5} w + \frac{11}{5}]$ | $\phantom{-}\frac{5}{4} e^5 + \frac{7}{2} e^4 - \frac{37}{2} e^3 - 46 e^2 + 38 e + 80$ |
| 103 | $[103, 103, -\frac{1}{5} w^3 + \frac{4}{5} w^2 + \frac{1}{5} w - \frac{17}{5}]$ | $\phantom{-}\frac{1}{4} e^5 + \frac{1}{2} e^4 - 3 e^3 - 5 e^2 + 4$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $9$ | $[9,3,\frac{2}{5} w^3 - \frac{3}{5} w^2 - \frac{22}{5} w + \frac{14}{5}]$ | $1$ |