# Properties

 Base field $$\Q(\sqrt{197})$$ Weight [2, 2] Level norm 16 Level $[16, 4, 4]$ Label 2.2.197.1-16.1-c Dimension 4 CM no Base change no

# Related objects

• L-function not available

# Learn more about

## 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-c Dimension 4 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:
$$x^{4}$$ $$\mathstrut -\mathstrut x^{3}$$ $$\mathstrut -\mathstrut 18x^{2}$$ $$\mathstrut +\mathstrut 41x$$ $$\mathstrut -\mathstrut 19$$
Norm Prime Eigenvalue
4 $[4, 2, 2]$ $\phantom{-}0$
7 $[7, 7, w - 7]$ $-\frac{1}{7}e^{3} - \frac{3}{7}e^{2} + \frac{6}{7}e + \frac{11}{7}$
7 $[7, 7, w + 6]$ $\phantom{-}e$
9 $[9, 3, 3]$ $\phantom{-}0$
19 $[19, 19, w + 5]$ $\phantom{-}\frac{1}{7}e^{3} - \frac{4}{7}e^{2} - \frac{27}{7}e + \frac{59}{7}$
19 $[19, 19, w - 6]$ $-\frac{4}{7}e^{3} - \frac{5}{7}e^{2} + \frac{66}{7}e - \frac{40}{7}$
23 $[23, 23, w + 8]$ $\phantom{-}\frac{2}{7}e^{3} - \frac{1}{7}e^{2} - \frac{26}{7}e + \frac{62}{7}$
23 $[23, 23, -w + 9]$ $-\frac{5}{7}e^{3} - \frac{8}{7}e^{2} + \frac{65}{7}e - \frac{15}{7}$
25 $[25, 5, 5]$ $\phantom{-}\frac{1}{7}e^{3} + \frac{3}{7}e^{2} - \frac{13}{7}e - \frac{39}{7}$
29 $[29, 29, -w - 4]$ $-\frac{1}{7}e^{3} - \frac{3}{7}e^{2} + \frac{6}{7}e + \frac{25}{7}$
29 $[29, 29, w - 5]$ $\phantom{-}e + 2$
37 $[37, 37, -w - 3]$ $\phantom{-}\frac{4}{7}e^{3} + \frac{5}{7}e^{2} - \frac{59}{7}e + \frac{47}{7}$
37 $[37, 37, w - 4]$ $-\frac{2}{7}e^{3} + \frac{1}{7}e^{2} + \frac{33}{7}e - \frac{41}{7}$
41 $[41, 41, -w - 9]$ $-\frac{8}{7}e^{3} - \frac{3}{7}e^{2} + \frac{139}{7}e - \frac{115}{7}$
41 $[41, 41, w - 10]$ $\phantom{-}\frac{8}{7}e^{3} + \frac{3}{7}e^{2} - \frac{139}{7}e + \frac{171}{7}$
43 $[43, 43, -w - 2]$ $-e^{3} - e^{2} + 15e - 16$
43 $[43, 43, w - 3]$ $\phantom{-}\frac{5}{7}e^{3} + \frac{1}{7}e^{2} - \frac{79}{7}e + \frac{64}{7}$
47 $[47, 47, -w - 1]$ $-\frac{9}{7}e^{3} - \frac{6}{7}e^{2} + \frac{152}{7}e - \frac{146}{7}$
47 $[47, 47, w - 2]$ $\phantom{-}e^{3} - 18e + 20$
53 $[53, 53, 2w - 13]$ $\phantom{-}\frac{5}{7}e^{3} + \frac{1}{7}e^{2} - \frac{79}{7}e + \frac{127}{7}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

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