Properties

Base field \(\Q(\sqrt{197}) \)
Weight [2, 2]
Level norm 4
Level $[4, 2, 2]$
Label 2.2.197.1-4.1-b
Dimension 4
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 $[4, 2, 2]$
Label 2.2.197.1-4.1-b
Dimension 4
Is CM no
Is base change no
Parent newspace dimension 12

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{4} \) \(\mathstrut +\mathstrut 2x^{3} \) \(\mathstrut -\mathstrut 18x^{2} \) \(\mathstrut -\mathstrut 34x \) \(\mathstrut +\mathstrut 29\)

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Norm Prime Eigenvalue
4 $[4, 2, 2]$ $\phantom{-}1$
7 $[7, 7, w - 7]$ $\phantom{-}\frac{1}{5}e^{3} - \frac{18}{5}e - \frac{8}{5}$
7 $[7, 7, w + 6]$ $\phantom{-}e$
9 $[9, 3, 3]$ $-3$
19 $[19, 19, w + 5]$ $-\frac{1}{2}e^{3} + \frac{1}{2}e^{2} + \frac{15}{2}e - \frac{5}{2}$
19 $[19, 19, w - 6]$ $-\frac{1}{10}e^{3} - \frac{1}{2}e^{2} + \frac{3}{10}e + \frac{53}{10}$
23 $[23, 23, w + 8]$ $-\frac{2}{5}e^{3} + \frac{1}{2}e^{2} + \frac{26}{5}e - \frac{73}{10}$
23 $[23, 23, -w + 9]$ $-\frac{1}{5}e^{3} - \frac{1}{2}e^{2} + \frac{13}{5}e + \frac{21}{10}$
25 $[25, 5, 5]$ $\phantom{-}\frac{1}{10}e^{3} - \frac{13}{10}e - \frac{24}{5}$
29 $[29, 29, -w - 4]$ $\phantom{-}\frac{1}{2}e^{3} - \frac{15}{2}e - 2$
29 $[29, 29, w - 5]$ $\phantom{-}\frac{3}{10}e^{3} - \frac{29}{10}e - \frac{2}{5}$
37 $[37, 37, -w - 3]$ $\phantom{-}\frac{1}{10}e^{3} - e^{2} - \frac{23}{10}e + \frac{61}{5}$
37 $[37, 37, w - 4]$ $-\frac{1}{2}e^{3} + e^{2} + \frac{15}{2}e - 5$
41 $[41, 41, -w - 9]$ $-\frac{1}{5}e^{3} + \frac{23}{5}e - \frac{17}{5}$
41 $[41, 41, w - 10]$ $\phantom{-}\frac{1}{5}e^{3} - \frac{23}{5}e - \frac{33}{5}$
43 $[43, 43, -w - 2]$ $-\frac{1}{10}e^{3} + \frac{1}{2}e^{2} + \frac{3}{10}e - \frac{97}{10}$
43 $[43, 43, w - 3]$ $-\frac{1}{10}e^{3} - \frac{1}{2}e^{2} + \frac{23}{10}e + \frac{13}{10}$
47 $[47, 47, -w - 1]$ $-e - 2$
47 $[47, 47, w - 2]$ $-\frac{1}{5}e^{3} + \frac{18}{5}e - \frac{2}{5}$
53 $[53, 53, 2w - 13]$ $-\frac{1}{10}e^{3} - \frac{1}{2}e^{2} - \frac{7}{10}e + \frac{43}{10}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
4 $[4, 2, 2]$ $-1$