Properties

Label 3.3.1257.1-13.3-e
Base field 3.3.1257.1
Weight $[2, 2, 2]$
Level norm $13$
Level $[13, 13, -w^{2} - w + 4]$
Dimension $8$
CM no
Base change no

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Base field 3.3.1257.1

Generator \(w\), with minimal polynomial \(x^{3} - x^{2} - 8x + 9\); narrow class number \(2\) and class number \(1\).

Form

Weight: $[2, 2, 2]$
Level: $[13, 13, -w^{2} - w + 4]$
Dimension: $8$
CM: no
Base change: no
Newspace dimension: $22$

Hecke eigenvalues ($q$-expansion)

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

\(x^{8} + 6x^{7} + 3x^{6} - 40x^{5} - 67x^{4} + 24x^{3} + 95x^{2} + 42x + 1\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
3 $[3, 3, -w + 3]$ $\phantom{-}e$
3 $[3, 3, w - 2]$ $-e^{7} - 4e^{6} + \frac{9}{2}e^{5} + \frac{59}{2}e^{4} + \frac{23}{2}e^{3} - \frac{73}{2}e^{2} - \frac{49}{2}e - 1$
5 $[5, 5, w^{2} + w - 5]$ $\phantom{-}e^{7} + 4e^{6} - \frac{9}{2}e^{5} - 30e^{4} - 12e^{3} + 40e^{2} + 27e + \frac{1}{2}$
8 $[8, 2, 2]$ $-\frac{1}{2}e^{7} - \frac{1}{2}e^{6} + \frac{13}{2}e^{5} + \frac{7}{2}e^{4} - \frac{49}{2}e^{3} - 4e^{2} + 21e + 5$
13 $[13, 13, w + 2]$ $-e^{7} - \frac{7}{2}e^{6} + 6e^{5} + 26e^{4} + e^{3} - 34e^{2} - \frac{31}{2}e + 4$
13 $[13, 13, -2w + 5]$ $\phantom{-}\frac{3}{2}e^{7} + \frac{13}{2}e^{6} - \frac{11}{2}e^{5} - 49e^{4} - 27e^{3} + \frac{133}{2}e^{2} + \frac{103}{2}e - \frac{1}{2}$
13 $[13, 13, -w^{2} - w + 4]$ $\phantom{-}1$
19 $[19, 19, -w^{2} + w + 4]$ $\phantom{-}\frac{3}{2}e^{7} + \frac{11}{2}e^{6} - \frac{15}{2}e^{5} - 39e^{4} - 12e^{3} + \frac{81}{2}e^{2} + \frac{63}{2}e + \frac{15}{2}$
23 $[23, 23, -w^{2} - w + 7]$ $\phantom{-}e^{7} + 4e^{6} - 4e^{5} - \frac{57}{2}e^{4} - \frac{31}{2}e^{3} + \frac{61}{2}e^{2} + \frac{65}{2}e + \frac{19}{2}$
25 $[25, 5, w^{2} - 2w - 1]$ $-\frac{1}{2}e^{7} - 2e^{6} + 2e^{5} + \frac{29}{2}e^{4} + \frac{19}{2}e^{3} - 16e^{2} - \frac{49}{2}e - \frac{7}{2}$
29 $[29, 29, w^{2} - 5]$ $-\frac{1}{2}e^{7} - \frac{3}{2}e^{6} + 4e^{5} + \frac{23}{2}e^{4} - \frac{15}{2}e^{3} - 17e^{2} + 3e + \frac{11}{2}$
31 $[31, 31, w^{2} + w - 8]$ $\phantom{-}\frac{5}{2}e^{7} + 11e^{6} - 8e^{5} - \frac{161}{2}e^{4} - \frac{107}{2}e^{3} + 96e^{2} + \frac{195}{2}e + \frac{23}{2}$
41 $[41, 41, w^{2} + w - 11]$ $\phantom{-}3e^{7} + 11e^{6} - \frac{31}{2}e^{5} - 80e^{4} - 22e^{3} + 94e^{2} + 68e + \frac{13}{2}$
47 $[47, 47, w^{2} + 2w - 1]$ $-e^{7} - 4e^{6} + 5e^{5} + 30e^{4} + 8e^{3} - 38e^{2} - 23e - 6$
59 $[59, 59, w^{2} + w - 1]$ $-2e^{6} - 6e^{5} + 14e^{4} + 43e^{3} - 13e^{2} - 50e - 13$
61 $[61, 61, -w^{2} + 5w - 7]$ $\phantom{-}\frac{1}{2}e^{7} + 3e^{6} + e^{5} - 22e^{4} - 32e^{3} + \frac{51}{2}e^{2} + 52e + 6$
67 $[67, 67, 2w^{2} - 13]$ $-\frac{5}{2}e^{7} - \frac{17}{2}e^{6} + \frac{29}{2}e^{5} + 60e^{4} + 6e^{3} - \frac{121}{2}e^{2} - \frac{77}{2}e - \frac{27}{2}$
89 $[89, 89, -2w^{2} + w + 20]$ $-e^{7} - 6e^{6} - \frac{5}{2}e^{5} + \frac{85}{2}e^{4} + \frac{127}{2}e^{3} - \frac{83}{2}e^{2} - \frac{173}{2}e - 20$
89 $[89, 89, 5w^{2} + 8w - 19]$ $-\frac{1}{2}e^{7} - 3e^{6} - e^{5} + \frac{45}{2}e^{4} + \frac{63}{2}e^{3} - 30e^{2} - \frac{99}{2}e - \frac{1}{2}$
89 $[89, 89, w^{2} + 2w - 7]$ $\phantom{-}\frac{1}{2}e^{6} + 2e^{5} - \frac{3}{2}e^{4} - \frac{25}{2}e^{3} - \frac{15}{2}e^{2} + 8e + \frac{21}{2}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$13$ $[13, 13, -w^{2} - w + 4]$ $-1$