# Properties

 Base field $$\Q(\sqrt{197})$$ Weight [2, 2] Level norm 9 Level $[9, 3, 3]$ Label 2.2.197.1-9.1-d Dimension 4 CM no Base change no

# Related objects

• L-function not available

## 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 $[9, 3, 3]$ Label 2.2.197.1-9.1-d Dimension 4 Is CM no Is base change no Parent newspace dimension 30

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$$x^{4}$$ $$\mathstrut -\mathstrut 3x^{3}$$ $$\mathstrut -\mathstrut 23x^{2}$$ $$\mathstrut +\mathstrut 41x$$ $$\mathstrut +\mathstrut 139$$
Norm Prime Eigenvalue
4 $[4, 2, 2]$ $\phantom{-}\frac{1}{11}e^{3} - \frac{1}{11}e^{2} - \frac{14}{11}e - \frac{20}{11}$
7 $[7, 7, w - 7]$ $\phantom{-}\frac{1}{11}e^{3} - \frac{1}{11}e^{2} - \frac{25}{11}e + \frac{13}{11}$
7 $[7, 7, w + 6]$ $\phantom{-}e$
9 $[9, 3, 3]$ $-1$
19 $[19, 19, w + 5]$ $-\frac{2}{11}e^{3} - \frac{9}{11}e^{2} + \frac{39}{11}e + \frac{150}{11}$
19 $[19, 19, w - 6]$ $\phantom{-}e^{2} - e - 13$
23 $[23, 23, w + 8]$ $-\frac{2}{11}e^{3} + \frac{2}{11}e^{2} + \frac{39}{11}e + \frac{40}{11}$
23 $[23, 23, -w + 9]$ $-\frac{1}{11}e^{3} + \frac{1}{11}e^{2} + \frac{3}{11}e + \frac{53}{11}$
25 $[25, 5, 5]$ $-\frac{3}{11}e^{3} + \frac{3}{11}e^{2} + \frac{42}{11}e - \frac{50}{11}$
29 $[29, 29, -w - 4]$ $-\frac{1}{11}e^{3} + \frac{1}{11}e^{2} + \frac{14}{11}e - \frac{24}{11}$
29 $[29, 29, w - 5]$ $-\frac{1}{11}e^{3} + \frac{1}{11}e^{2} + \frac{14}{11}e - \frac{24}{11}$
37 $[37, 37, -w - 3]$ $\phantom{-}\frac{5}{11}e^{3} + \frac{6}{11}e^{2} - \frac{92}{11}e - \frac{155}{11}$
37 $[37, 37, w - 4]$ $\phantom{-}\frac{2}{11}e^{3} - \frac{13}{11}e^{2} - \frac{6}{11}e + \frac{125}{11}$
41 $[41, 41, -w - 9]$ $\phantom{-}\frac{5}{11}e^{3} - \frac{5}{11}e^{2} - \frac{59}{11}e - \frac{1}{11}$
41 $[41, 41, w - 10]$ $\phantom{-}\frac{6}{11}e^{3} - \frac{6}{11}e^{2} - \frac{95}{11}e + \frac{12}{11}$
43 $[43, 43, -w - 2]$ $-\frac{1}{11}e^{3} + \frac{12}{11}e^{2} - \frac{8}{11}e - \frac{145}{11}$
43 $[43, 43, w - 3]$ $-\frac{4}{11}e^{3} - \frac{7}{11}e^{2} + \frac{78}{11}e + \frac{135}{11}$
47 $[47, 47, -w - 1]$ $\phantom{-}\frac{1}{11}e^{3} - \frac{1}{11}e^{2} - \frac{14}{11}e - \frac{42}{11}$
47 $[47, 47, w - 2]$ $\phantom{-}\frac{1}{11}e^{3} - \frac{1}{11}e^{2} - \frac{14}{11}e - \frac{42}{11}$
53 $[53, 53, 2w - 13]$ $-\frac{2}{11}e^{3} - \frac{9}{11}e^{2} + \frac{61}{11}e + \frac{139}{11}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
9 $[9, 3, 3]$ $1$