Properties

Base field \(\Q(\sqrt{34}) \)
Weight [2, 2]
Level norm 9
Level $[9, 3, 3]$
Label 2.2.136.1-9.1-j
Dimension 9
CM no
Base change yes

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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 $[9, 3, 3]$
Label 2.2.136.1-9.1-j
Dimension 9
Is CM no
Is base change yes
Parent newspace dimension 48

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{9} \) \(\mathstrut -\mathstrut 19x^{7} \) \(\mathstrut +\mathstrut 120x^{5} \) \(\mathstrut -\mathstrut 274x^{3} \) \(\mathstrut +\mathstrut 4x^{2} \) \(\mathstrut +\mathstrut 152x \) \(\mathstrut -\mathstrut 16\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
2 $[2, 2, w - 6]$ $\phantom{-}e$
3 $[3, 3, w + 1]$ $\phantom{-}1$
3 $[3, 3, w + 2]$ $\phantom{-}1$
5 $[5, 5, w + 2]$ $\phantom{-}\frac{1}{8}e^{8} + \frac{1}{4}e^{7} - \frac{11}{8}e^{6} - \frac{11}{4}e^{5} + 3e^{4} + 7e^{3} + \frac{11}{4}e^{2} - 2e - 3$
5 $[5, 5, w + 3]$ $\phantom{-}\frac{1}{8}e^{8} + \frac{1}{4}e^{7} - \frac{11}{8}e^{6} - \frac{11}{4}e^{5} + 3e^{4} + 7e^{3} + \frac{11}{4}e^{2} - 2e - 3$
11 $[11, 11, w + 1]$ $-\frac{1}{4}e^{8} - \frac{1}{2}e^{7} + \frac{13}{4}e^{6} + \frac{11}{2}e^{5} - \frac{25}{2}e^{4} - 14e^{3} + \frac{33}{2}e^{2} + 4e - 6$
11 $[11, 11, w + 10]$ $-\frac{1}{4}e^{8} - \frac{1}{2}e^{7} + \frac{13}{4}e^{6} + \frac{11}{2}e^{5} - \frac{25}{2}e^{4} - 14e^{3} + \frac{33}{2}e^{2} + 4e - 6$
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 $[29, 29, w + 11]$ $-\frac{1}{8}e^{8} - \frac{1}{4}e^{7} + \frac{15}{8}e^{6} + \frac{11}{4}e^{5} - \frac{17}{2}e^{4} - 7e^{3} + \frac{41}{4}e^{2} + 2e + 3$
29 $[29, 29, w + 18]$ $-\frac{1}{8}e^{8} - \frac{1}{4}e^{7} + \frac{15}{8}e^{6} + \frac{11}{4}e^{5} - \frac{17}{2}e^{4} - 7e^{3} + \frac{41}{4}e^{2} + 2e + 3$
37 $[37, 37, w + 16]$ $-\frac{1}{8}e^{8} - \frac{1}{4}e^{7} + \frac{15}{8}e^{6} + \frac{11}{4}e^{5} - \frac{17}{2}e^{4} - 7e^{3} + \frac{49}{4}e^{2} + 2e - 7$
37 $[37, 37, w + 21]$ $-\frac{1}{8}e^{8} - \frac{1}{4}e^{7} + \frac{15}{8}e^{6} + \frac{11}{4}e^{5} - \frac{17}{2}e^{4} - 7e^{3} + \frac{49}{4}e^{2} + 2e - 7$
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 $[61, 61, w + 20]$ $-\frac{1}{8}e^{8} - \frac{1}{4}e^{7} + \frac{15}{8}e^{6} + \frac{15}{4}e^{5} - \frac{17}{2}e^{4} - 18e^{3} + \frac{49}{4}e^{2} + 28e - 7$
61 $[61, 61, w + 41]$ $-\frac{1}{8}e^{8} - \frac{1}{4}e^{7} + \frac{15}{8}e^{6} + \frac{15}{4}e^{5} - \frac{17}{2}e^{4} - 18e^{3} + \frac{49}{4}e^{2} + 28e - 7$
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 $[103, 103, -14w + 81]$ $-\frac{1}{2}e^{6} - e^{5} + \frac{11}{2}e^{4} + 9e^{3} - 13e^{2} - 14e - 2$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
3 $[3, 3, w + 1]$ $-1$
3 $[3, 3, w + 2]$ $-1$