Base field \(\Q(\sqrt{5}, \sqrt{7})\)
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 15x^{2} + 16x + 29\); narrow class number \(2\) and class number \(1\).
Form
Weight: | $[2, 2, 2, 2]$ |
Level: | $[9, 3, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w + \frac{24}{23}]$ |
Dimension: | $6$ |
CM: | no |
Base change: | no |
Newspace dimension: | $18$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{6} - 17x^{4} + 40x^{2} - 21\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
4 | $[4, 2, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} - \frac{3}{23}w + \frac{1}{23}]$ | $\phantom{-}\frac{1}{11}e^{4} - \frac{12}{11}e^{2} + \frac{2}{11}$ |
9 | $[9, 3, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w + \frac{24}{23}]$ | $\phantom{-}1$ |
9 | $[9, 3, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w - \frac{68}{23}]$ | $\phantom{-}\frac{2}{11}e^{4} - \frac{35}{11}e^{2} + \frac{26}{11}$ |
19 | $[19, 19, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} - \frac{3}{23}w + \frac{24}{23}]$ | $\phantom{-}e$ |
19 | $[19, 19, -\frac{6}{23}w^{3} + \frac{9}{23}w^{2} + \frac{83}{23}w - \frac{20}{23}]$ | $-\frac{1}{11}e^{5} + \frac{12}{11}e^{3} + \frac{31}{11}e$ |
19 | $[19, 19, \frac{6}{23}w^{3} - \frac{9}{23}w^{2} - \frac{83}{23}w + \frac{66}{23}]$ | $-e$ |
19 | $[19, 19, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} - \frac{3}{23}w - \frac{22}{23}]$ | $\phantom{-}\frac{1}{11}e^{5} - \frac{12}{11}e^{3} - \frac{31}{11}e$ |
25 | $[25, 5, -\frac{4}{23}w^{3} + \frac{6}{23}w^{2} + \frac{40}{23}w - \frac{21}{23}]$ | $-\frac{6}{11}e^{4} + \frac{94}{11}e^{2} - \frac{166}{11}$ |
29 | $[29, 29, w]$ | $\phantom{-}\frac{5}{11}e^{4} - \frac{82}{11}e^{2} + \frac{87}{11}$ |
29 | $[29, 29, -\frac{4}{23}w^{3} + \frac{6}{23}w^{2} + \frac{63}{23}w - \frac{21}{23}]$ | $\phantom{-}\frac{5}{11}e^{4} - \frac{82}{11}e^{2} + \frac{87}{11}$ |
29 | $[29, 29, \frac{4}{23}w^{3} - \frac{6}{23}w^{2} - \frac{63}{23}w + \frac{44}{23}]$ | $-\frac{7}{11}e^{4} + \frac{106}{11}e^{2} - \frac{102}{11}$ |
29 | $[29, 29, -w + 1]$ | $-\frac{7}{11}e^{4} + \frac{106}{11}e^{2} - \frac{102}{11}$ |
31 | $[31, 31, w + 2]$ | $-\frac{7}{11}e^{5} + \frac{117}{11}e^{3} - \frac{223}{11}e$ |
31 | $[31, 31, -\frac{4}{23}w^{3} + \frac{6}{23}w^{2} + \frac{63}{23}w + \frac{25}{23}]$ | $\phantom{-}\frac{7}{11}e^{5} - \frac{117}{11}e^{3} + \frac{223}{11}e$ |
31 | $[31, 31, \frac{4}{23}w^{3} - \frac{6}{23}w^{2} - \frac{63}{23}w + \frac{90}{23}]$ | $-\frac{1}{11}e^{5} + \frac{12}{11}e^{3} + \frac{9}{11}e$ |
31 | $[31, 31, -w + 3]$ | $\phantom{-}\frac{1}{11}e^{5} - \frac{12}{11}e^{3} - \frac{9}{11}e$ |
49 | $[49, 7, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w - \frac{22}{23}]$ | $-\frac{5}{11}e^{4} + \frac{71}{11}e^{2} - \frac{131}{11}$ |
59 | $[59, 59, -\frac{20}{23}w^{3} + \frac{30}{23}w^{2} + \frac{269}{23}w - \frac{128}{23}]$ | $\phantom{-}\frac{3}{11}e^{5} - \frac{47}{11}e^{3} + \frac{17}{11}e$ |
59 | $[59, 59, -\frac{8}{23}w^{3} + \frac{12}{23}w^{2} + \frac{103}{23}w - \frac{88}{23}]$ | $-\frac{3}{11}e^{5} + \frac{47}{11}e^{3} - \frac{17}{11}e$ |
59 | $[59, 59, -\frac{8}{23}w^{3} + \frac{12}{23}w^{2} + \frac{103}{23}w - \frac{19}{23}]$ | $\phantom{-}\frac{9}{11}e^{5} - \frac{141}{11}e^{3} + \frac{172}{11}e$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$9$ | $[9,3,-\frac{2}{23}w^{3}+\frac{3}{23}w^{2}+\frac{43}{23}w+\frac{24}{23}]$ | $-1$ |