Properties

Label 2.2.197.1-19.2-b
Base field \(\Q(\sqrt{197}) \)
Weight $[2, 2]$
Level norm $19$
Level $[19,19,-w + 6]$
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: $[19,19,-w + 6]$
Dimension: $39$
CM: no
Base change: no
Newspace dimension: $72$

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:

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

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