# Properties

 Base field 3.3.321.1 Weight [2, 2, 2] Level norm 49 Level $[49, 7, 2w^{2} - 3w - 3]$ Label 3.3.321.1-49.1-b Dimension 8 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 $[49, 7, 2w^{2} - 3w - 3]$ Label 3.3.321.1-49.1-b Dimension 8 Is CM no Is base change no Parent newspace dimension 11

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$$x^{8} - 20x^{6} + 8x^{5} + 120x^{4} - 96x^{3} - 192x^{2} + 240x - 64$$
Norm Prime Eigenvalue
3 $[3, 3, w + 1]$ $\phantom{-}e$
3 $[3, 3, w - 1]$ $\phantom{-}\frac{1}{16}e^{7} - \frac{5}{4}e^{5} + \frac{1}{4}e^{4} + \frac{15}{2}e^{3} - 3e^{2} - 13e + 8$
7 $[7, 7, w^{2} - 2]$ $\phantom{-}\frac{1}{8}e^{7} + \frac{1}{8}e^{6} - \frac{5}{2}e^{5} - \frac{5}{4}e^{4} + \frac{31}{2}e^{3} - 28e + 12$
8 $[8, 2, 2]$ $\phantom{-}\frac{1}{4}e^{7} + \frac{1}{4}e^{6} - \frac{19}{4}e^{5} - \frac{11}{4}e^{4} + 27e^{3} + 3e^{2} - 43e + 17$
11 $[11, 11, -w^{2} + w + 1]$ $\phantom{-}\frac{1}{4}e^{7} + \frac{1}{4}e^{6} - \frac{19}{4}e^{5} - \frac{5}{2}e^{4} + \frac{55}{2}e^{3} - 47e + 24$
23 $[23, 23, -w - 3]$ $-\frac{3}{16}e^{7} - \frac{1}{8}e^{6} + \frac{15}{4}e^{5} + e^{4} - 23e^{3} + \frac{7}{2}e^{2} + 42e - 20$
29 $[29, 29, -w^{2} + 2w + 4]$ $-\frac{5}{16}e^{7} - \frac{3}{8}e^{6} + \frac{23}{4}e^{5} + 4e^{4} - 32e^{3} - \frac{7}{2}e^{2} + 52e - 22$
31 $[31, 31, 2w - 3]$ $\phantom{-}\frac{5}{16}e^{7} + \frac{1}{2}e^{6} - \frac{11}{2}e^{5} - \frac{25}{4}e^{4} + 29e^{3} + \frac{31}{2}e^{2} - 45e + 8$
41 $[41, 41, -2w^{2} + 3w + 6]$ $\phantom{-}\frac{5}{16}e^{7} + \frac{1}{4}e^{6} - \frac{25}{4}e^{5} - \frac{9}{4}e^{4} + \frac{77}{2}e^{3} - 3e^{2} - 70e + 34$
43 $[43, 43, w^{2} - 3w + 3]$ $-\frac{1}{4}e^{7} - \frac{1}{4}e^{6} + \frac{9}{2}e^{5} + 2e^{4} - \frac{49}{2}e^{3} + 4e^{2} + 39e - 24$
47 $[47, 47, w^{2} + w - 4]$ $-\frac{3}{16}e^{7} + \frac{15}{4}e^{5} - \frac{3}{2}e^{4} - \frac{47}{2}e^{3} + 16e^{2} + 44e - 32$
49 $[49, 7, 2w^{2} - 3w - 3]$ $\phantom{-}1$
53 $[53, 53, w^{2} - 3w - 2]$ $-\frac{3}{16}e^{7} - \frac{1}{4}e^{6} + \frac{7}{2}e^{5} + \frac{11}{4}e^{4} - \frac{39}{2}e^{3} - \frac{5}{2}e^{2} + 30e - 14$
59 $[59, 59, 2w^{2} - w - 5]$ $-\frac{3}{8}e^{7} - \frac{1}{4}e^{6} + \frac{29}{4}e^{5} + \frac{5}{2}e^{4} - 42e^{3} - 2e^{2} + 68e - 20$
59 $[59, 59, w^{2} - w - 7]$ $-\frac{1}{16}e^{7} - \frac{1}{4}e^{6} + e^{5} + \frac{17}{4}e^{4} - \frac{9}{2}e^{3} - \frac{37}{2}e^{2} + 6e + 16$
59 $[59, 59, -w^{2} - w + 7]$ $\phantom{-}\frac{1}{4}e^{7} + \frac{1}{8}e^{6} - \frac{19}{4}e^{5} - \frac{3}{4}e^{4} + \frac{55}{2}e^{3} - 6e^{2} - 48e + 32$
67 $[67, 67, 2w^{2} - 3w - 7]$ $\phantom{-}\frac{7}{16}e^{7} + \frac{1}{2}e^{6} - \frac{31}{4}e^{5} - \frac{21}{4}e^{4} + \frac{81}{2}e^{3} + 5e^{2} - 60e + 28$
73 $[73, 73, -w^{2} + 4w - 5]$ $\phantom{-}\frac{1}{16}e^{7} + \frac{1}{4}e^{6} - \frac{3}{4}e^{5} - \frac{15}{4}e^{4} + \frac{1}{2}e^{3} + \frac{27}{2}e^{2} + 8e - 10$
79 $[79, 79, w^{2} - 8]$ $-\frac{7}{16}e^{7} - \frac{3}{8}e^{6} + \frac{33}{4}e^{5} + 3e^{4} - \frac{95}{2}e^{3} + \frac{15}{2}e^{2} + 79e - 48$
79 $[79, 79, w^{2} - 5w + 5]$ $-\frac{1}{2}e^{7} - \frac{1}{2}e^{6} + \frac{37}{4}e^{5} + 5e^{4} - \frac{105}{2}e^{3} - 2e^{2} + 89e - 36$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
49 $[49, 7, 2w^{2} - 3w - 3]$ $-1$