Properties

Base field \(\Q(\sqrt{197}) \)
Weight [2, 2]
Level norm 4
Level $[4, 2, 2]$
Label 2.2.197.1-4.1-c
Dimension 6
CM no
Base change yes

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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-c
Dimension 6
Is CM no
Is base change yes
Parent newspace dimension 12

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{6} \) \(\mathstrut +\mathstrut 10x^{5} \) \(\mathstrut +\mathstrut 23x^{4} \) \(\mathstrut -\mathstrut 38x^{3} \) \(\mathstrut -\mathstrut 144x^{2} \) \(\mathstrut -\mathstrut 8x \) \(\mathstrut +\mathstrut 80\)

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Norm Prime Eigenvalue
4 $[4, 2, 2]$ $-1$
7 $[7, 7, w - 7]$ $\phantom{-}e$
7 $[7, 7, w + 6]$ $\phantom{-}e$
9 $[9, 3, 3]$ $\phantom{-}\frac{1}{16}e^{5} + \frac{1}{4}e^{4} - \frac{17}{16}e^{3} - 3e^{2} + 4e + \frac{7}{2}$
19 $[19, 19, w + 5]$ $\phantom{-}\frac{1}{4}e^{5} + \frac{7}{4}e^{4} + \frac{3}{4}e^{3} - \frac{43}{4}e^{2} - 6e + 11$
19 $[19, 19, w - 6]$ $\phantom{-}\frac{1}{4}e^{5} + \frac{7}{4}e^{4} + \frac{3}{4}e^{3} - \frac{43}{4}e^{2} - 6e + 11$
23 $[23, 23, w + 8]$ $-\frac{1}{2}e^{3} - \frac{3}{2}e^{2} + 4e + 4$
23 $[23, 23, -w + 9]$ $-\frac{1}{2}e^{3} - \frac{3}{2}e^{2} + 4e + 4$
25 $[25, 5, 5]$ $-\frac{1}{4}e^{4} - \frac{3}{2}e^{3} - \frac{5}{4}e^{2} + 3e + 9$
29 $[29, 29, -w - 4]$ $\phantom{-}\frac{1}{16}e^{5} + \frac{1}{4}e^{4} - \frac{9}{16}e^{3} - \frac{1}{2}e^{2} + 5e - \frac{9}{2}$
29 $[29, 29, w - 5]$ $\phantom{-}\frac{1}{16}e^{5} + \frac{1}{4}e^{4} - \frac{9}{16}e^{3} - \frac{1}{2}e^{2} + 5e - \frac{9}{2}$
37 $[37, 37, -w - 3]$ $-\frac{3}{16}e^{5} - e^{4} + \frac{19}{16}e^{3} + \frac{27}{4}e^{2} - \frac{11}{2}e - \frac{7}{2}$
37 $[37, 37, w - 4]$ $-\frac{3}{16}e^{5} - e^{4} + \frac{19}{16}e^{3} + \frac{27}{4}e^{2} - \frac{11}{2}e - \frac{7}{2}$
41 $[41, 41, -w - 9]$ $\phantom{-}\frac{1}{16}e^{5} + \frac{1}{2}e^{4} + \frac{15}{16}e^{3} - \frac{1}{4}e^{2} - 2e - \frac{9}{2}$
41 $[41, 41, w - 10]$ $\phantom{-}\frac{1}{16}e^{5} + \frac{1}{2}e^{4} + \frac{15}{16}e^{3} - \frac{1}{4}e^{2} - 2e - \frac{9}{2}$
43 $[43, 43, -w - 2]$ $-\frac{1}{8}e^{5} - e^{4} - \frac{11}{8}e^{3} + \frac{9}{2}e^{2} + \frac{19}{2}e$
43 $[43, 43, w - 3]$ $-\frac{1}{8}e^{5} - e^{4} - \frac{11}{8}e^{3} + \frac{9}{2}e^{2} + \frac{19}{2}e$
47 $[47, 47, -w - 1]$ $-e^{2} - 4e + 4$
47 $[47, 47, w - 2]$ $-e^{2} - 4e + 4$
53 $[53, 53, 2w - 13]$ $-\frac{1}{16}e^{5} - \frac{1}{2}e^{4} - \frac{7}{16}e^{3} + \frac{13}{4}e^{2} + \frac{3}{2}e - \frac{13}{2}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

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