# Properties

 Label 6.6.371293.1-79.3-e Base field $$\Q(\zeta_{13})^+$$ Weight $[2, 2, 2, 2, 2, 2]$ Level norm $79$ Level $[79,79,-w^{3} + w^{2} + 4w - 1]$ Dimension $5$ CM no Base change no

# Related objects

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## Base field $$\Q(\zeta_{13})^+$$

Generator $$w$$, with minimal polynomial $$x^{6} - x^{5} - 5x^{4} + 4x^{3} + 6x^{2} - 3x - 1$$; narrow class number $$1$$ and class number $$1$$.

## Form

 Weight: $[2, 2, 2, 2, 2, 2]$ Level: $[79,79,-w^{3} + w^{2} + 4w - 1]$ Dimension: $5$ CM: no Base change: no Newspace dimension: $9$

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:

 $$x^{5} - 7x^{4} - 8x^{3} + 106x^{2} - 19x - 361$$
Norm Prime Eigenvalue
13 $[13, 13, w^{5} - 5w^{3} + 4w]$ $\phantom{-}e$
25 $[25, 5, w^{5} - 5w^{3} + 6w - 1]$ $-\frac{7}{38}e^{4} + \frac{15}{19}e^{3} + \frac{47}{19}e^{2} - \frac{124}{19}e - \frac{17}{2}$
25 $[25, 5, -w^{3} + w^{2} + 3w - 1]$ $\phantom{-}\frac{17}{38}e^{4} - \frac{31}{19}e^{3} - \frac{163}{19}e^{2} + \frac{312}{19}e + \frac{91}{2}$
25 $[25, 5, w^{5} - 4w^{3} - w^{2} + 3w + 2]$ $\phantom{-}\frac{5}{19}e^{4} - \frac{16}{19}e^{3} - \frac{116}{19}e^{2} + \frac{188}{19}e + 30$
27 $[27, 3, w^{4} - w^{3} - 4w^{2} + 2w + 2]$ $\phantom{-}\frac{4}{19}e^{4} - \frac{9}{19}e^{3} - \frac{108}{19}e^{2} + \frac{101}{19}e + 32$
27 $[27, 3, w^{4} - w^{3} - 4w^{2} + 2w + 3]$ $-\frac{7}{38}e^{4} + \frac{15}{19}e^{3} + \frac{66}{19}e^{2} - \frac{162}{19}e - \frac{37}{2}$
53 $[53, 53, -w^{4} + w^{3} + 3w^{2} - 2w + 1]$ $-\frac{17}{38}e^{4} + \frac{31}{19}e^{3} + \frac{163}{19}e^{2} - \frac{312}{19}e - \frac{83}{2}$
53 $[53, 53, -w^{4} + w^{3} + 4w^{2} - 3w - 4]$ $\phantom{-}\frac{3}{19}e^{4} - \frac{21}{19}e^{3} - \frac{24}{19}e^{2} + \frac{204}{19}e + 9$
53 $[53, 53, -w^{5} + w^{4} + 4w^{3} - 4w^{2} - 3w + 1]$ $-\frac{17}{38}e^{4} + \frac{31}{19}e^{3} + \frac{163}{19}e^{2} - \frac{312}{19}e - \frac{83}{2}$
53 $[53, 53, w^{3} - 2w - 2]$ $-\frac{17}{38}e^{4} + \frac{31}{19}e^{3} + \frac{163}{19}e^{2} - \frac{312}{19}e - \frac{83}{2}$
53 $[53, 53, w^{5} - 5w^{3} - w^{2} + 5w]$ $\phantom{-}\frac{3}{38}e^{4} - \frac{1}{19}e^{3} - \frac{31}{19}e^{2} - \frac{50}{19}e + \frac{17}{2}$
53 $[53, 53, -w^{4} + 4w^{2} + w - 4]$ $\phantom{-}\frac{7}{38}e^{4} - \frac{15}{19}e^{3} - \frac{47}{19}e^{2} + \frac{124}{19}e + \frac{25}{2}$
64 $[64, 2, -2]$ $\phantom{-}\frac{12}{19}e^{4} - \frac{46}{19}e^{3} - \frac{229}{19}e^{2} + \frac{436}{19}e + 68$
79 $[79, 79, -2w^{5} + w^{4} + 9w^{3} - 3w^{2} - 9w + 2]$ $\phantom{-}\frac{5}{19}e^{4} - \frac{16}{19}e^{3} - \frac{116}{19}e^{2} + \frac{226}{19}e + 37$
79 $[79, 79, -w^{5} - w^{4} + 5w^{3} + 4w^{2} - 6w - 1]$ $-\frac{21}{38}e^{4} + \frac{45}{19}e^{3} + \frac{160}{19}e^{2} - \frac{410}{19}e - \frac{71}{2}$
79 $[79, 79, w^{3} - w^{2} - 4w + 1]$ $-1$
79 $[79, 79, -2w^{5} + 2w^{4} + 9w^{3} - 7w^{2} - 9w + 3]$ $-\frac{10}{19}e^{4} + \frac{32}{19}e^{3} + \frac{232}{19}e^{2} - \frac{338}{19}e - 74$
79 $[79, 79, -2w^{4} + w^{3} + 7w^{2} - 3w - 3]$ $-\frac{2}{19}e^{4} + \frac{14}{19}e^{3} + \frac{16}{19}e^{2} - \frac{174}{19}e$
79 $[79, 79, -w^{5} + 6w^{3} - w^{2} - 8w + 1]$ $-\frac{5}{38}e^{4} + \frac{8}{19}e^{3} + \frac{39}{19}e^{2} - \frac{94}{19}e - \frac{7}{2}$
103 $[103, 103, 2w^{4} - 7w^{2} - w + 3]$ $-\frac{5}{19}e^{4} + \frac{16}{19}e^{3} + \frac{116}{19}e^{2} - \frac{188}{19}e - 33$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$79$ $[79,79,-w^{3} + w^{2} + 4w - 1]$ $1$