# 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

# 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 $[4, 2, 2]$ 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:
$$x^{6}$$ $$\mathstrut +\mathstrut 10x^{5}$$ $$\mathstrut +\mathstrut 23x^{4}$$ $$\mathstrut -\mathstrut 38x^{3}$$ $$\mathstrut -\mathstrut 144x^{2}$$ $$\mathstrut -\mathstrut 8x$$ $$\mathstrut +\mathstrut 80$$
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 $[4, 2, 2]$ $1$