Properties

Base field \(\Q(\sqrt{157}) \)
Weight [2, 2]
Level norm 13
Level $[13,13,-2w - 11]$
Label 2.2.157.1-13.2-b
Dimension 23
CM no
Base change no

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Base field \(\Q(\sqrt{157}) \)

Generator \(w\), with minimal polynomial \(x^{2} - x - 39\); narrow class number \(1\) and class number \(1\).

Form

Weight [2, 2]
Level $[13,13,-2w - 11]$
Label 2.2.157.1-13.2-b
Dimension 23
Is CM no
Is base change no
Parent newspace dimension 44

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{23} \) \(\mathstrut -\mathstrut 2x^{22} \) \(\mathstrut -\mathstrut 53x^{21} \) \(\mathstrut +\mathstrut 102x^{20} \) \(\mathstrut +\mathstrut 1211x^{19} \) \(\mathstrut -\mathstrut 2207x^{18} \) \(\mathstrut -\mathstrut 15684x^{17} \) \(\mathstrut +\mathstrut 26463x^{16} \) \(\mathstrut +\mathstrut 127465x^{15} \) \(\mathstrut -\mathstrut 192791x^{14} \) \(\mathstrut -\mathstrut 680274x^{13} \) \(\mathstrut +\mathstrut 880397x^{12} \) \(\mathstrut +\mathstrut 2423734x^{11} \) \(\mathstrut -\mathstrut 2506491x^{10} \) \(\mathstrut -\mathstrut 5731973x^{9} \) \(\mathstrut +\mathstrut 4259090x^{8} \) \(\mathstrut +\mathstrut 8733548x^{7} \) \(\mathstrut -\mathstrut 3850250x^{6} \) \(\mathstrut -\mathstrut 8028594x^{5} \) \(\mathstrut +\mathstrut 1235189x^{4} \) \(\mathstrut +\mathstrut 3856635x^{3} \) \(\mathstrut +\mathstrut 339195x^{2} \) \(\mathstrut -\mathstrut 655130x \) \(\mathstrut -\mathstrut 154628\)

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Norm Prime Eigenvalue
3 $[3, 3, w + 6]$ $...$
3 $[3, 3, -w + 7]$ $\phantom{-}e$
4 $[4, 2, 2]$ $...$
11 $[11, 11, -3w - 17]$ $...$
11 $[11, 11, 3w - 20]$ $...$
13 $[13, 13, 2w - 13]$ $...$
13 $[13, 13, 2w + 11]$ $-1$
17 $[17, 17, w + 7]$ $...$
17 $[17, 17, -w + 8]$ $...$
19 $[19, 19, -w - 4]$ $...$
19 $[19, 19, -w + 5]$ $...$
25 $[25, 5, 5]$ $...$
31 $[31, 31, -6w - 35]$ $...$
31 $[31, 31, -6w + 41]$ $...$
37 $[37, 37, -w - 1]$ $...$
37 $[37, 37, w - 2]$ $...$
47 $[47, 47, 3w + 16]$ $...$
47 $[47, 47, -3w + 19]$ $...$
49 $[49, 7, -7]$ $...$
67 $[67, 67, 3w - 22]$ $...$
Display number of eigenvalues

Atkin-Lehner eigenvalues

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