Properties

Base field \(\Q(\sqrt{197}) \)
Weight [2, 2]
Level norm 16
Level $[16, 4, 4]$
Label 2.2.197.1-16.1-d
Dimension 6
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 $[16, 4, 4]$
Label 2.2.197.1-16.1-d
Dimension 6
Is CM no
Is base change no
Parent newspace dimension 40

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{6} \) \(\mathstrut -\mathstrut 4x^{5} \) \(\mathstrut -\mathstrut 8x^{4} \) \(\mathstrut +\mathstrut 28x^{3} \) \(\mathstrut +\mathstrut 34x^{2} \) \(\mathstrut -\mathstrut 39x \) \(\mathstrut -\mathstrut 41\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
4 $[4, 2, 2]$ $\phantom{-}0$
7 $[7, 7, w - 7]$ $-\frac{4}{7}e^{5} + \frac{22}{7}e^{4} - \frac{1}{7}e^{3} - \frac{107}{7}e^{2} + 2e + \frac{114}{7}$
7 $[7, 7, w + 6]$ $\phantom{-}e$
9 $[9, 3, 3]$ $-\frac{12}{7}e^{5} + \frac{66}{7}e^{4} - \frac{10}{7}e^{3} - \frac{300}{7}e^{2} + 12e + \frac{265}{7}$
19 $[19, 19, w + 5]$ $-\frac{13}{7}e^{5} + \frac{75}{7}e^{4} - \frac{19}{7}e^{3} - \frac{353}{7}e^{2} + 16e + \frac{360}{7}$
19 $[19, 19, w - 6]$ $\phantom{-}\frac{11}{7}e^{5} - \frac{64}{7}e^{4} + \frac{22}{7}e^{3} + \frac{289}{7}e^{2} - 16e - \frac{282}{7}$
23 $[23, 23, w + 8]$ $-3e^{5} + 17e^{4} - 4e^{3} - 78e^{2} + 27e + 75$
23 $[23, 23, -w + 9]$ $-\frac{5}{7}e^{5} + \frac{24}{7}e^{4} + \frac{11}{7}e^{3} - \frac{118}{7}e^{2} - 3e + \frac{111}{7}$
25 $[25, 5, 5]$ $\phantom{-}\frac{6}{7}e^{5} - \frac{33}{7}e^{4} + \frac{12}{7}e^{3} + \frac{129}{7}e^{2} - 9e - \frac{115}{7}$
29 $[29, 29, -w - 4]$ $\phantom{-}\frac{27}{7}e^{5} - \frac{152}{7}e^{4} + \frac{33}{7}e^{3} + \frac{703}{7}e^{2} - 33e - \frac{710}{7}$
29 $[29, 29, w - 5]$ $-e^{5} + 6e^{4} - 2e^{3} - 30e^{2} + 12e + 33$
37 $[37, 37, -w - 3]$ $\phantom{-}\frac{15}{7}e^{5} - \frac{86}{7}e^{4} + \frac{30}{7}e^{3} + \frac{375}{7}e^{2} - 24e - \frac{354}{7}$
37 $[37, 37, w - 4]$ $-\frac{1}{7}e^{5} + \frac{9}{7}e^{4} - \frac{9}{7}e^{3} - \frac{53}{7}e^{2} + 6e + \frac{60}{7}$
41 $[41, 41, -w - 9]$ $-\frac{5}{7}e^{5} + \frac{24}{7}e^{4} + \frac{4}{7}e^{3} - \frac{90}{7}e^{2} - 2e + \frac{48}{7}$
41 $[41, 41, w - 10]$ $-\frac{31}{7}e^{5} + \frac{174}{7}e^{4} - \frac{41}{7}e^{3} - \frac{789}{7}e^{2} + 41e + \frac{761}{7}$
43 $[43, 43, -w - 2]$ $-3e^{5} + 17e^{4} - 4e^{3} - 79e^{2} + 27e + 82$
43 $[43, 43, w - 3]$ $\phantom{-}\frac{13}{7}e^{5} - \frac{75}{7}e^{4} + \frac{19}{7}e^{3} + \frac{360}{7}e^{2} - 18e - \frac{367}{7}$
47 $[47, 47, -w - 1]$ $-\frac{4}{7}e^{5} + \frac{22}{7}e^{4} - \frac{15}{7}e^{3} - \frac{72}{7}e^{2} + 12e + \frac{65}{7}$
47 $[47, 47, w - 2]$ $\phantom{-}\frac{2}{7}e^{5} - \frac{11}{7}e^{4} - \frac{10}{7}e^{3} + \frac{92}{7}e^{2} - \frac{120}{7}$
53 $[53, 53, 2w - 13]$ $\phantom{-}\frac{13}{7}e^{5} - \frac{75}{7}e^{4} + \frac{26}{7}e^{3} + \frac{339}{7}e^{2} - 21e - \frac{353}{7}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

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