Properties

Base field \(\Q(\sqrt{197}) \)
Weight [2, 2]
Level norm 19
Level $[19,19,-w + 6]$
Label 2.2.197.1-19.2-a
Dimension 33
CM no
Base change no

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Base field \(\Q(\sqrt{197}) \)

Generator \(w\), with minimal polynomial \(x^{2} - x - 49\); narrow class number \(1\) and class number \(1\).

Form

Weight [2, 2]
Level $[19,19,-w + 6]$
Label 2.2.197.1-19.2-a
Dimension 33
Is CM no
Is base change no
Parent newspace dimension 72

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{33} \) \(\mathstrut +\mathstrut 14x^{32} \) \(\mathstrut +\mathstrut 19x^{31} \) \(\mathstrut -\mathstrut 602x^{30} \) \(\mathstrut -\mathstrut 2697x^{29} \) \(\mathstrut +\mathstrut 8646x^{28} \) \(\mathstrut +\mathstrut 74602x^{27} \) \(\mathstrut -\mathstrut 5947x^{26} \) \(\mathstrut -\mathstrut 1026307x^{25} \) \(\mathstrut -\mathstrut 1351051x^{24} \) \(\mathstrut +\mathstrut 8117420x^{23} \) \(\mathstrut +\mathstrut 19619818x^{22} \) \(\mathstrut -\mathstrut 36637524x^{21} \) \(\mathstrut -\mathstrut 146030461x^{20} \) \(\mathstrut +\mathstrut 70082299x^{19} \) \(\mathstrut +\mathstrut 673278148x^{18} \) \(\mathstrut +\mathstrut 163882435x^{17} \) \(\mathstrut -\mathstrut 2020460543x^{16} \) \(\mathstrut -\mathstrut 1458370712x^{15} \) \(\mathstrut +\mathstrut 3957086135x^{14} \) \(\mathstrut +\mathstrut 4418395775x^{13} \) \(\mathstrut -\mathstrut 4897863060x^{12} \) \(\mathstrut -\mathstrut 7587620519x^{11} \) \(\mathstrut +\mathstrut 3524241168x^{10} \) \(\mathstrut +\mathstrut 7923621173x^{9} \) \(\mathstrut -\mathstrut 1170611350x^{8} \) \(\mathstrut -\mathstrut 5016700788x^{7} \) \(\mathstrut -\mathstrut 13154880x^{6} \) \(\mathstrut +\mathstrut 1866514983x^{5} \) \(\mathstrut +\mathstrut 92029467x^{4} \) \(\mathstrut -\mathstrut 376004169x^{3} \) \(\mathstrut -\mathstrut 9448781x^{2} \) \(\mathstrut +\mathstrut 31622528x \) \(\mathstrut -\mathstrut 1348627\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
4 $[4, 2, 2]$ $\phantom{-}e$
7 $[7, 7, w - 7]$ $...$
7 $[7, 7, w + 6]$ $...$
9 $[9, 3, 3]$ $...$
19 $[19, 19, w + 5]$ $...$
19 $[19, 19, w - 6]$ $\phantom{-}1$
23 $[23, 23, w + 8]$ $...$
23 $[23, 23, -w + 9]$ $...$
25 $[25, 5, 5]$ $...$
29 $[29, 29, -w - 4]$ $...$
29 $[29, 29, w - 5]$ $...$
37 $[37, 37, -w - 3]$ $...$
37 $[37, 37, w - 4]$ $...$
41 $[41, 41, -w - 9]$ $...$
41 $[41, 41, w - 10]$ $...$
43 $[43, 43, -w - 2]$ $...$
43 $[43, 43, w - 3]$ $...$
47 $[47, 47, -w - 1]$ $...$
47 $[47, 47, w - 2]$ $...$
53 $[53, 53, 2w - 13]$ $...$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
19 $[19,19,-w + 6]$ $-1$