Properties

Base field \(\Q(\sqrt{113}) \)
Weight [2, 2]
Level norm 13
Level $[13, 13, -2w + 11]$
Label 2.2.113.1-13.1-a
Dimension 13
CM no
Base change no

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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 $[13, 13, -2w + 11]$
Label 2.2.113.1-13.1-a
Dimension 13
Is CM no
Is base change no
Parent newspace dimension 37

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{13} \) \(\mathstrut +\mathstrut x^{12} \) \(\mathstrut -\mathstrut 19x^{11} \) \(\mathstrut -\mathstrut 15x^{10} \) \(\mathstrut +\mathstrut 134x^{9} \) \(\mathstrut +\mathstrut 73x^{8} \) \(\mathstrut -\mathstrut 433x^{7} \) \(\mathstrut -\mathstrut 125x^{6} \) \(\mathstrut +\mathstrut 631x^{5} \) \(\mathstrut +\mathstrut 54x^{4} \) \(\mathstrut -\mathstrut 337x^{3} \) \(\mathstrut -\mathstrut 35x^{2} \) \(\mathstrut +\mathstrut 58x \) \(\mathstrut +\mathstrut 11\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
2 $[2, 2, -w + 6]$ $...$
2 $[2, 2, w + 5]$ $\phantom{-}e$
7 $[7, 7, 6w - 35]$ $-\frac{424}{47}e^{12} - \frac{900}{47}e^{11} + \frac{7074}{47}e^{10} + \frac{14353}{47}e^{9} - \frac{41198}{47}e^{8} - \frac{78014}{47}e^{7} + \frac{99122}{47}e^{6} + \frac{168567}{47}e^{5} - \frac{86867}{47}e^{4} - \frac{128994}{47}e^{3} + \frac{7795}{47}e^{2} + \frac{28757}{47}e + \frac{4976}{47}$
7 $[7, 7, -6w - 29]$ $...$
9 $[9, 3, 3]$ $...$
11 $[11, 11, 4w + 19]$ $...$
11 $[11, 11, 4w - 23]$ $...$
13 $[13, 13, -2w + 11]$ $\phantom{-}1$
13 $[13, 13, 2w + 9]$ $...$
25 $[25, 5, -5]$ $...$
31 $[31, 31, 2w - 13]$ $...$
31 $[31, 31, -2w - 11]$ $...$
41 $[41, 41, -8w - 39]$ $...$
41 $[41, 41, 8w - 47]$ $...$
53 $[53, 53, -26w - 125]$ $...$
53 $[53, 53, 26w - 151]$ $...$
61 $[61, 61, -14w + 81]$ $...$
61 $[61, 61, -14w - 67]$ $...$
83 $[83, 83, 2w - 15]$ $...$
83 $[83, 83, -2w - 13]$ $...$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
13 $[13, 13, -2w + 11]$ $-1$