# Properties

 Base field 3.3.316.1 Weight [2, 2, 2] Level norm 23 Level $[23, 23, 2w - 3]$ Label 3.3.316.1-23.1-a Dimension 9 CM no Base change no

# Related objects

• L-function not available

## Base field 3.3.316.1

Generator $$w$$, with minimal polynomial $$x^{3} - x^{2} - 4x + 2$$; narrow class number $$1$$ and class number $$1$$.

## Form

 Weight [2, 2, 2] Level $[23, 23, 2w - 3]$ Label 3.3.316.1-23.1-a Dimension 9 Is CM no Is base change no Parent newspace dimension 9

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$$x^{9}$$ $$\mathstrut -\mathstrut x^{8}$$ $$\mathstrut -\mathstrut 16x^{7}$$ $$\mathstrut +\mathstrut 16x^{6}$$ $$\mathstrut +\mathstrut 82x^{5}$$ $$\mathstrut -\mathstrut 76x^{4}$$ $$\mathstrut -\mathstrut 148x^{3}$$ $$\mathstrut +\mathstrut 108x^{2}$$ $$\mathstrut +\mathstrut 80x$$ $$\mathstrut -\mathstrut 32$$
Norm Prime Eigenvalue
2 $[2, 2, -w]$ $\phantom{-}e$
2 $[2, 2, w - 1]$ $-\frac{1}{8}e^{8} - \frac{1}{8}e^{7} + \frac{7}{4}e^{6} + \frac{3}{2}e^{5} - \frac{31}{4}e^{4} - \frac{11}{2}e^{3} + \frac{23}{2}e^{2} + \frac{11}{2}e - 3$
11 $[11, 11, w^{2} - w - 1]$ $\phantom{-}\frac{1}{4}e^{8} - \frac{13}{4}e^{6} + e^{5} + \frac{25}{2}e^{4} - \frac{15}{2}e^{3} - 14e^{2} + 10e + 4$
17 $[17, 17, -w^{2} - w + 3]$ $-\frac{1}{4}e^{8} + \frac{13}{4}e^{6} - e^{5} - \frac{25}{2}e^{4} + \frac{13}{2}e^{3} + 13e^{2} - 4e$
19 $[19, 19, w^{2} - w + 1]$ $-\frac{1}{2}e^{7} - \frac{1}{2}e^{6} + 6e^{5} + 4e^{4} - 20e^{3} - 6e^{2} + 16e$
23 $[23, 23, 2w - 3]$ $\phantom{-}1$
27 $[27, 3, 3]$ $\phantom{-}\frac{1}{4}e^{8} + \frac{1}{2}e^{7} - \frac{11}{4}e^{6} - 5e^{5} + \frac{15}{2}e^{4} + \frac{27}{2}e^{3} - 12e - 4$
29 $[29, 29, 2w + 1]$ $\phantom{-}\frac{1}{4}e^{8} + \frac{1}{4}e^{7} - \frac{7}{2}e^{6} - 3e^{5} + \frac{29}{2}e^{4} + 12e^{3} - 15e^{2} - 17e - 2$
31 $[31, 31, 2w^{2} - 2w - 9]$ $-\frac{1}{4}e^{8} - \frac{1}{4}e^{7} + \frac{7}{2}e^{6} + 2e^{5} - \frac{31}{2}e^{4} - 2e^{3} + 21e^{2} - 5e - 2$
37 $[37, 37, 2w^{2} - 2w - 5]$ $\phantom{-}\frac{1}{4}e^{8} + \frac{1}{4}e^{7} - \frac{7}{2}e^{6} - 2e^{5} + \frac{33}{2}e^{4} + 3e^{3} - 29e^{2} - e + 10$
41 $[41, 41, 2w^{2} - 9]$ $-\frac{1}{2}e^{8} + \frac{15}{2}e^{6} - e^{5} - 35e^{4} + 7e^{3} + 52e^{2} - 6e - 14$
43 $[43, 43, w^{2} + w - 5]$ $-\frac{1}{4}e^{7} + \frac{1}{4}e^{6} + 4e^{5} - 3e^{4} - \frac{39}{2}e^{3} + 8e^{2} + 27e - 2$
43 $[43, 43, -3w^{2} + w + 15]$ $-\frac{3}{4}e^{7} - \frac{5}{4}e^{6} + 8e^{5} + 11e^{4} - \frac{45}{2}e^{3} - 22e^{2} + 13e + 10$
43 $[43, 43, -2w^{2} + 2w + 11]$ $\phantom{-}\frac{1}{2}e^{8} + \frac{1}{2}e^{7} - 7e^{6} - 5e^{5} + 31e^{4} + 13e^{3} - 44e^{2} - 8e + 8$
53 $[53, 53, w^{2} - w - 7]$ $\phantom{-}\frac{1}{4}e^{8} - \frac{17}{4}e^{6} - e^{5} + \frac{43}{2}e^{4} + \frac{19}{2}e^{3} - 31e^{2} - 20e + 8$
61 $[61, 61, 4w^{2} - 2w - 15]$ $\phantom{-}\frac{1}{2}e^{7} + \frac{1}{2}e^{6} - 6e^{5} - 3e^{4} + 21e^{3} - 2e^{2} - 18e + 8$
67 $[67, 67, -5w^{2} + 3w + 23]$ $-\frac{1}{2}e^{7} + \frac{1}{2}e^{6} + 7e^{5} - 6e^{4} - 28e^{3} + 18e^{2} + 26e - 12$
73 $[73, 73, 2w^{2} - 3]$ $\phantom{-}\frac{1}{4}e^{8} + \frac{1}{4}e^{7} - \frac{9}{2}e^{6} - 4e^{5} + \frac{53}{2}e^{4} + 18e^{3} - 55e^{2} - 19e + 26$
73 $[73, 73, -3w^{2} - w + 7]$ $\phantom{-}\frac{1}{4}e^{8} + \frac{1}{2}e^{7} - \frac{11}{4}e^{6} - 5e^{5} + \frac{17}{2}e^{4} + \frac{25}{2}e^{3} - 10e^{2} - 4e + 10$
73 $[73, 73, -6w^{2} + 4w + 25]$ $\phantom{-}\frac{1}{4}e^{8} + \frac{3}{4}e^{7} - 2e^{6} - 8e^{5} + \frac{1}{2}e^{4} + 24e^{3} + 15e^{2} - 19e - 10$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
23 $[23, 23, 2w - 3]$ $-1$