Properties

Base field \(\Q(\sqrt{197}) \)
Weight [2, 2]
Level norm 19
Level $[19,19,-w + 6]$
Label 2.2.197.1-19.2-b
Dimension 39
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-b
Dimension 39
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:
\(x^{39} \) \(\mathstrut +\mathstrut 2x^{38} \) \(\mathstrut -\mathstrut 96x^{37} \) \(\mathstrut -\mathstrut 180x^{36} \) \(\mathstrut +\mathstrut 4201x^{35} \) \(\mathstrut +\mathstrut 7320x^{34} \) \(\mathstrut -\mathstrut 111083x^{33} \) \(\mathstrut -\mathstrut 178179x^{32} \) \(\mathstrut +\mathstrut 1982867x^{31} \) \(\mathstrut +\mathstrut 2898584x^{30} \) \(\mathstrut -\mathstrut 25277156x^{29} \) \(\mathstrut -\mathstrut 33324178x^{28} \) \(\mathstrut +\mathstrut 237349425x^{27} \) \(\mathstrut +\mathstrut 279257347x^{26} \) \(\mathstrut -\mathstrut 1668614198x^{25} \) \(\mathstrut -\mathstrut 1735007118x^{24} \) \(\mathstrut +\mathstrut 8838368734x^{23} \) \(\mathstrut +\mathstrut 8057692004x^{22} \) \(\mathstrut -\mathstrut 35216286172x^{21} \) \(\mathstrut -\mathstrut 28034041673x^{20} \) \(\mathstrut +\mathstrut 104638293843x^{19} \) \(\mathstrut +\mathstrut 72901624152x^{18} \) \(\mathstrut -\mathstrut 228220201161x^{17} \) \(\mathstrut -\mathstrut 140764790322x^{16} \) \(\mathstrut +\mathstrut 356901679915x^{15} \) \(\mathstrut +\mathstrut 199120624825x^{14} \) \(\mathstrut -\mathstrut 387505342619x^{13} \) \(\mathstrut -\mathstrut 200854317166x^{12} \) \(\mathstrut +\mathstrut 279891060515x^{11} \) \(\mathstrut +\mathstrut 137389816085x^{10} \) \(\mathstrut -\mathstrut 127281070207x^{9} \) \(\mathstrut -\mathstrut 58929466554x^{8} \) \(\mathstrut +\mathstrut 34089090706x^{7} \) \(\mathstrut +\mathstrut 14351771879x^{6} \) \(\mathstrut -\mathstrut 4864407338x^{5} \) \(\mathstrut -\mathstrut 1694988485x^{4} \) \(\mathstrut +\mathstrut 306641353x^{3} \) \(\mathstrut +\mathstrut 68873322x^{2} \) \(\mathstrut -\mathstrut 5245834x \) \(\mathstrut -\mathstrut 625625\)

  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 $[19, 19, w - 6]$ $-1$
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 $[19,19,-w + 6]$ $1$