Properties

Base field \(\Q(\sqrt{113}) \)
Weight [2, 2]
Level norm 9
Level $[9, 3, 3]$
Label 2.2.113.1-9.1-f
Dimension 10
CM no
Base change yes

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Base field \(\Q(\sqrt{113}) \)

Generator \(w\), with minimal polynomial \(x^{2} - x - 28\); narrow class number \(1\) and class number \(1\).

Form

Weight [2, 2]
Level $[9, 3, 3]$
Label 2.2.113.1-9.1-f
Dimension 10
Is CM no
Is base change yes
Parent newspace dimension 24

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{10} \) \(\mathstrut +\mathstrut x^{9} \) \(\mathstrut -\mathstrut 15x^{8} \) \(\mathstrut -\mathstrut 11x^{7} \) \(\mathstrut +\mathstrut 76x^{6} \) \(\mathstrut +\mathstrut 36x^{5} \) \(\mathstrut -\mathstrut 150x^{4} \) \(\mathstrut -\mathstrut 44x^{3} \) \(\mathstrut +\mathstrut 98x^{2} \) \(\mathstrut +\mathstrut 20x \) \(\mathstrut -\mathstrut 16\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
2 $[2, 2, -w + 6]$ $\phantom{-}e$
2 $[2, 2, w + 5]$ $\phantom{-}e$
7 $[7, 7, 6w - 35]$ $\phantom{-}\frac{3}{4}e^{9} + \frac{1}{4}e^{8} - \frac{45}{4}e^{7} - \frac{3}{4}e^{6} + 55e^{5} - 10e^{4} - \frac{189}{2}e^{3} + 33e^{2} + \frac{73}{2}e - 14$
7 $[7, 7, -6w - 29]$ $\phantom{-}\frac{3}{4}e^{9} + \frac{1}{4}e^{8} - \frac{45}{4}e^{7} - \frac{3}{4}e^{6} + 55e^{5} - 10e^{4} - \frac{189}{2}e^{3} + 33e^{2} + \frac{73}{2}e - 14$
9 $[9, 3, 3]$ $-1$
11 $[11, 11, 4w + 19]$ $-e^{9} - e^{8} + 14e^{7} + 9e^{6} - 64e^{5} - 16e^{4} + 106e^{3} - 6e^{2} - 43e + 6$
11 $[11, 11, 4w - 23]$ $-e^{9} - e^{8} + 14e^{7} + 9e^{6} - 64e^{5} - 16e^{4} + 106e^{3} - 6e^{2} - 43e + 6$
13 $[13, 13, -2w + 11]$ $-\frac{1}{2}e^{9} + \frac{1}{2}e^{8} + 9e^{7} - \frac{15}{2}e^{6} - \frac{105}{2}e^{5} + 36e^{4} + 107e^{3} - 59e^{2} - 52e + 19$
13 $[13, 13, 2w + 9]$ $-\frac{1}{2}e^{9} + \frac{1}{2}e^{8} + 9e^{7} - \frac{15}{2}e^{6} - \frac{105}{2}e^{5} + 36e^{4} + 107e^{3} - 59e^{2} - 52e + 19$
25 $[25, 5, -5]$ $\phantom{-}\frac{1}{4}e^{9} + \frac{1}{4}e^{8} - \frac{13}{4}e^{7} - \frac{7}{4}e^{6} + \frac{27}{2}e^{5} - e^{4} - \frac{39}{2}e^{3} + 14e^{2} + \frac{7}{2}e - 1$
31 $[31, 31, 2w - 13]$ $\phantom{-}\frac{1}{2}e^{7} + e^{6} - \frac{11}{2}e^{5} - 9e^{4} + 16e^{3} + 17e^{2} - 9e - 3$
31 $[31, 31, -2w - 11]$ $\phantom{-}\frac{1}{2}e^{7} + e^{6} - \frac{11}{2}e^{5} - 9e^{4} + 16e^{3} + 17e^{2} - 9e - 3$
41 $[41, 41, -8w - 39]$ $-\frac{3}{2}e^{9} + \frac{1}{2}e^{8} + \frac{49}{2}e^{7} - \frac{23}{2}e^{6} - 131e^{5} + 74e^{4} + 246e^{3} - 146e^{2} - 105e + 54$
41 $[41, 41, 8w - 47]$ $-\frac{3}{2}e^{9} + \frac{1}{2}e^{8} + \frac{49}{2}e^{7} - \frac{23}{2}e^{6} - 131e^{5} + 74e^{4} + 246e^{3} - 146e^{2} - 105e + 54$
53 $[53, 53, -26w - 125]$ $-\frac{1}{2}e^{9} - \frac{1}{2}e^{8} + \frac{13}{2}e^{7} + \frac{7}{2}e^{6} - 27e^{5} + 2e^{4} + 42e^{3} - 28e^{2} - 22e + 12$
53 $[53, 53, 26w - 151]$ $-\frac{1}{2}e^{9} - \frac{1}{2}e^{8} + \frac{13}{2}e^{7} + \frac{7}{2}e^{6} - 27e^{5} + 2e^{4} + 42e^{3} - 28e^{2} - 22e + 12$
61 $[61, 61, -14w + 81]$ $\phantom{-}\frac{3}{4}e^{9} - \frac{1}{4}e^{8} - \frac{47}{4}e^{7} + \frac{27}{4}e^{6} + \frac{121}{2}e^{5} - 46e^{4} - \frac{221}{2}e^{3} + 90e^{2} + \frac{97}{2}e - 31$
61 $[61, 61, -14w - 67]$ $\phantom{-}\frac{3}{4}e^{9} - \frac{1}{4}e^{8} - \frac{47}{4}e^{7} + \frac{27}{4}e^{6} + \frac{121}{2}e^{5} - 46e^{4} - \frac{221}{2}e^{3} + 90e^{2} + \frac{97}{2}e - 31$
83 $[83, 83, 2w - 15]$ $\phantom{-}\frac{3}{2}e^{9} - \frac{1}{2}e^{8} - \frac{49}{2}e^{7} + \frac{23}{2}e^{6} + 131e^{5} - 74e^{4} - 246e^{3} + 144e^{2} + 105e - 50$
83 $[83, 83, -2w - 13]$ $\phantom{-}\frac{3}{2}e^{9} - \frac{1}{2}e^{8} - \frac{49}{2}e^{7} + \frac{23}{2}e^{6} + 131e^{5} - 74e^{4} - 246e^{3} + 144e^{2} + 105e - 50$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
9 $[9, 3, 3]$ $1$