Properties

Label 3.3.1937.1-13.2-b
Base field 3.3.1937.1
Weight $[2, 2, 2]$
Level norm $13$
Level $[13, 13, -w + 2]$
Dimension $4$
CM no
Base change no

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Base field 3.3.1937.1

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

Form

Weight: $[2, 2, 2]$
Level: $[13, 13, -w + 2]$
Dimension: $4$
CM: no
Base change: no
Newspace dimension: $44$

Hecke eigenvalues ($q$-expansion)

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

\(x^{4} - 8x^{3} + 6x^{2} + 40x - 31\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
3 $[3, 3, -w - 2]$ $\phantom{-}\frac{1}{10}e^{3} - \frac{3}{5}e^{2} - \frac{1}{10}e + \frac{9}{5}$
5 $[5, 5, w + 1]$ $-\frac{1}{5}e^{3} + \frac{6}{5}e^{2} + \frac{1}{5}e - \frac{18}{5}$
7 $[7, 7, w - 3]$ $\phantom{-}1$
8 $[8, 2, 2]$ $-\frac{1}{10}e^{3} + \frac{3}{5}e^{2} + \frac{11}{10}e - \frac{9}{5}$
9 $[9, 3, w^{2} - 3w - 2]$ $-\frac{1}{10}e^{3} + \frac{11}{10}e^{2} - \frac{19}{10}e - \frac{43}{10}$
13 $[13, 13, w + 3]$ $\phantom{-}\frac{1}{5}e^{3} - \frac{6}{5}e^{2} - \frac{11}{5}e + \frac{38}{5}$
13 $[13, 13, -w + 2]$ $\phantom{-}1$
19 $[19, 19, -w^{2} + 2w + 4]$ $-\frac{1}{5}e^{3} + \frac{6}{5}e^{2} + \frac{1}{5}e - \frac{18}{5}$
23 $[23, 23, w^{2} - 4w + 1]$ $-\frac{1}{10}e^{3} + \frac{3}{5}e^{2} + \frac{11}{10}e - \frac{39}{5}$
25 $[25, 5, w^{2} - 2w - 1]$ $-\frac{1}{20}e^{3} + \frac{1}{20}e^{2} + \frac{21}{20}e + \frac{7}{20}$
31 $[31, 31, w^{2} - 2w - 9]$ $\phantom{-}\frac{1}{10}e^{3} - \frac{1}{10}e^{2} - \frac{21}{10}e - \frac{7}{10}$
37 $[37, 37, -w^{2} + 3w + 3]$ $-\frac{1}{10}e^{3} + \frac{1}{10}e^{2} + \frac{21}{10}e + \frac{7}{10}$
41 $[41, 41, w^{2} - w - 1]$ $\phantom{-}e^{2} - 4e - 5$
41 $[41, 41, w^{2} - w - 5]$ $-e^{2} + 4e + 5$
41 $[41, 41, w^{2} - w - 10]$ $-\frac{1}{5}e^{3} + \frac{6}{5}e^{2} + \frac{11}{5}e - \frac{38}{5}$
43 $[43, 43, -2w - 3]$ $\phantom{-}\frac{2}{5}e^{3} - \frac{12}{5}e^{2} - \frac{2}{5}e + \frac{36}{5}$
47 $[47, 47, -w^{2} - w + 4]$ $\phantom{-}\frac{1}{10}e^{3} - \frac{3}{5}e^{2} - \frac{11}{10}e + \frac{49}{5}$
49 $[49, 7, -w^{2} + 5w - 5]$ $-\frac{1}{10}e^{3} + \frac{3}{5}e^{2} + \frac{11}{10}e + \frac{21}{5}$
59 $[59, 59, w^{2} - w - 4]$ $\phantom{-}\frac{1}{10}e^{3} - \frac{11}{10}e^{2} + \frac{19}{10}e + \frac{43}{10}$
59 $[59, 59, w^{2} - 3]$ $-\frac{1}{4}e^{3} + \frac{7}{4}e^{2} - \frac{3}{4}e - \frac{23}{4}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$13$ $[13, 13, -w + 2]$ $-1$