Base field \(\Q(\sqrt{34}) \)
Generator \(w\), with minimal polynomial \(x^{2} - 34\); narrow class number \(4\) and class number \(2\).
Form
Weight: | $[2, 2]$ |
Level: | $[9, 3, 3]$ |
Dimension: | $9$ |
CM: | no |
Base change: | yes |
Newspace dimension: | $48$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{9} - 19x^{7} + 120x^{5} - 274x^{3} + 4x^{2} + 152x - 16\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
2 | $[2, 2, w - 6]$ | $\phantom{-}e$ |
3 | $[3, 3, w + 1]$ | $-1$ |
3 | $[3, 3, w + 2]$ | $-1$ |
5 | $[5, 5, w + 2]$ | $-\frac{1}{8}e^{8} - \frac{1}{4}e^{7} + \frac{11}{8}e^{6} + \frac{11}{4}e^{5} - 3e^{4} - 7e^{3} - \frac{11}{4}e^{2} + 2e + 3$ |
5 | $[5, 5, w + 3]$ | $-\frac{1}{8}e^{8} - \frac{1}{4}e^{7} + \frac{11}{8}e^{6} + \frac{11}{4}e^{5} - 3e^{4} - 7e^{3} - \frac{11}{4}e^{2} + 2e + 3$ |
11 | $[11, 11, w + 1]$ | $\phantom{-}\frac{1}{4}e^{8} + \frac{1}{2}e^{7} - \frac{13}{4}e^{6} - \frac{11}{2}e^{5} + \frac{25}{2}e^{4} + 14e^{3} - \frac{33}{2}e^{2} - 4e + 6$ |
11 | $[11, 11, w + 10]$ | $\phantom{-}\frac{1}{4}e^{8} + \frac{1}{2}e^{7} - \frac{13}{4}e^{6} - \frac{11}{2}e^{5} + \frac{25}{2}e^{4} + 14e^{3} - \frac{33}{2}e^{2} - 4e + 6$ |
17 | $[17, 17, -3w + 17]$ | $\phantom{-}\frac{1}{2}e^{7} + e^{6} - \frac{11}{2}e^{5} - 11e^{4} + 14e^{3} + 28e^{2} - 5e - 8$ |
29 | $[29, 29, w + 11]$ | $\phantom{-}\frac{1}{8}e^{8} + \frac{1}{4}e^{7} - \frac{15}{8}e^{6} - \frac{11}{4}e^{5} + \frac{17}{2}e^{4} + 7e^{3} - \frac{41}{4}e^{2} - 2e - 3$ |
29 | $[29, 29, w + 18]$ | $\phantom{-}\frac{1}{8}e^{8} + \frac{1}{4}e^{7} - \frac{15}{8}e^{6} - \frac{11}{4}e^{5} + \frac{17}{2}e^{4} + 7e^{3} - \frac{41}{4}e^{2} - 2e - 3$ |
37 | $[37, 37, w + 16]$ | $\phantom{-}\frac{1}{8}e^{8} + \frac{1}{4}e^{7} - \frac{15}{8}e^{6} - \frac{11}{4}e^{5} + \frac{17}{2}e^{4} + 7e^{3} - \frac{49}{4}e^{2} - 2e + 7$ |
37 | $[37, 37, w + 21]$ | $\phantom{-}\frac{1}{8}e^{8} + \frac{1}{4}e^{7} - \frac{15}{8}e^{6} - \frac{11}{4}e^{5} + \frac{17}{2}e^{4} + 7e^{3} - \frac{49}{4}e^{2} - 2e + 7$ |
47 | $[47, 47, -w - 9]$ | $-\frac{1}{2}e^{8} - e^{7} + \frac{13}{2}e^{6} + 12e^{5} - 25e^{4} - 39e^{3} + 33e^{2} + 34e - 12$ |
47 | $[47, 47, w - 9]$ | $-\frac{1}{2}e^{8} - e^{7} + \frac{13}{2}e^{6} + 12e^{5} - 25e^{4} - 39e^{3} + 33e^{2} + 34e - 12$ |
49 | $[49, 7, -7]$ | $\phantom{-}\frac{3}{4}e^{8} + 2e^{7} - \frac{37}{4}e^{6} - 24e^{5} + 31e^{4} + 76e^{3} - \frac{55}{2}e^{2} - 57e + 14$ |
61 | $[61, 61, w + 20]$ | $\phantom{-}\frac{1}{8}e^{8} + \frac{1}{4}e^{7} - \frac{15}{8}e^{6} - \frac{15}{4}e^{5} + \frac{17}{2}e^{4} + 18e^{3} - \frac{49}{4}e^{2} - 28e + 7$ |
61 | $[61, 61, w + 41]$ | $\phantom{-}\frac{1}{8}e^{8} + \frac{1}{4}e^{7} - \frac{15}{8}e^{6} - \frac{15}{4}e^{5} + \frac{17}{2}e^{4} + 18e^{3} - \frac{49}{4}e^{2} - 28e + 7$ |
89 | $[89, 89, 2w - 15]$ | $-\frac{1}{2}e^{7} + \frac{13}{2}e^{5} - 23e^{3} - 2e^{2} + 17e + 8$ |
89 | $[89, 89, -2w - 15]$ | $-\frac{1}{2}e^{7} + \frac{13}{2}e^{5} - 23e^{3} - 2e^{2} + 17e + 8$ |
103 | $[103, 103, -14w + 81]$ | $-\frac{1}{2}e^{6} - e^{5} + \frac{11}{2}e^{4} + 9e^{3} - 13e^{2} - 14e - 2$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$3$ | $[3, 3, w + 1]$ | $1$ |
$3$ | $[3, 3, w + 2]$ | $1$ |