# Properties

 Base field 3.3.321.1 Weight [2, 2, 2] Level norm 47 Level $[47, 47, w^{2} + w - 4]$ Label 3.3.321.1-47.1-c Dimension 9 CM no Base change no

# Related objects

• L-function not available

## Base field 3.3.321.1

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

## Form

 Weight [2, 2, 2] Level $[47, 47, w^{2} + w - 4]$ Label 3.3.321.1-47.1-c Dimension 9 Is CM no Is base change no Parent newspace dimension 13

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$$x^{9} - 2x^{8} - 15x^{7} + 24x^{6} + 81x^{5} - 86x^{4} - 181x^{3} + 84x^{2} + 122x - 16$$
Norm Prime Eigenvalue
3 $[3, 3, w + 1]$ $\phantom{-}e$
3 $[3, 3, w - 1]$ $-\frac{1}{2}e^{7} + \frac{1}{2}e^{6} + 6e^{5} - 4e^{4} - \frac{43}{2}e^{3} + \frac{15}{2}e^{2} + 20e - 4$
7 $[7, 7, w^{2} - 2]$ $\phantom{-}\frac{1}{2}e^{7} - \frac{13}{2}e^{5} - \frac{1}{2}e^{4} + 24e^{3} + \frac{5}{2}e^{2} - 20e + 2$
8 $[8, 2, 2]$ $\phantom{-}e^{2} - e - 3$
11 $[11, 11, -w^{2} + w + 1]$ $\phantom{-}\frac{1}{2}e^{8} - \frac{1}{2}e^{7} - \frac{13}{2}e^{6} + 5e^{5} + \frac{51}{2}e^{4} - \frac{29}{2}e^{3} - \frac{55}{2}e^{2} + 14e + 4$
23 $[23, 23, -w - 3]$ $\phantom{-}\frac{1}{2}e^{8} - \frac{1}{2}e^{7} - 6e^{6} + 4e^{5} + \frac{41}{2}e^{4} - \frac{13}{2}e^{3} - 13e^{2} + e - 6$
29 $[29, 29, -w^{2} + 2w + 4]$ $-\frac{1}{2}e^{8} + \frac{13}{2}e^{6} + e^{5} - \frac{51}{2}e^{4} - 6e^{3} + \frac{57}{2}e^{2} + 5e - 6$
31 $[31, 31, 2w - 3]$ $\phantom{-}e^{3} - e^{2} - 5e + 4$
41 $[41, 41, -2w^{2} + 3w + 6]$ $-\frac{1}{2}e^{6} + \frac{1}{2}e^{5} + \frac{9}{2}e^{4} - \frac{9}{2}e^{3} - 10e^{2} + 10e + 6$
43 $[43, 43, w^{2} - 3w + 3]$ $\phantom{-}\frac{1}{2}e^{8} - \frac{1}{2}e^{7} - 6e^{6} + \frac{9}{2}e^{5} + 22e^{4} - 11e^{3} - \frac{53}{2}e^{2} + 7e + 12$
47 $[47, 47, w^{2} + w - 4]$ $\phantom{-}1$
49 $[49, 7, 2w^{2} - 3w - 3]$ $-\frac{1}{2}e^{7} + \frac{1}{2}e^{6} + \frac{15}{2}e^{5} - \frac{11}{2}e^{4} - 33e^{3} + 15e^{2} + 36e - 6$
53 $[53, 53, w^{2} - 3w - 2]$ $-\frac{1}{2}e^{8} + e^{7} + \frac{11}{2}e^{6} - 10e^{5} - \frac{35}{2}e^{4} + 30e^{3} + \frac{25}{2}e^{2} - 29e + 2$
59 $[59, 59, 2w^{2} - w - 5]$ $\phantom{-}e^{8} - e^{7} - 13e^{6} + 10e^{5} + 51e^{4} - 28e^{3} - 55e^{2} + 21e + 8$
59 $[59, 59, w^{2} - w - 7]$ $-\frac{1}{2}e^{5} - \frac{1}{2}e^{4} + \frac{7}{2}e^{3} + \frac{9}{2}e^{2} - e - 6$
59 $[59, 59, -w^{2} - w + 7]$ $\phantom{-}\frac{1}{2}e^{8} + e^{7} - \frac{15}{2}e^{6} - 14e^{5} + \frac{67}{2}e^{4} + 55e^{3} - \frac{81}{2}e^{2} - 44e + 10$
67 $[67, 67, 2w^{2} - 3w - 7]$ $-\frac{1}{2}e^{7} + 6e^{5} - \frac{41}{2}e^{3} + 4e^{2} + 15e - 8$
73 $[73, 73, -w^{2} + 4w - 5]$ $-e^{7} + 12e^{5} + 3e^{4} - 42e^{3} - 17e^{2} + 35e + 12$
79 $[79, 79, w^{2} - 8]$ $\phantom{-}e^{8} + \frac{1}{2}e^{7} - \frac{27}{2}e^{6} - 9e^{5} + 53e^{4} + \frac{83}{2}e^{3} - \frac{101}{2}e^{2} - 33e + 6$
79 $[79, 79, w^{2} - 5w + 5]$ $\phantom{-}2e^{3} - 12e$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
47 $[47, 47, w^{2} + w - 4]$ $-1$