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: | $8$ |
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:
\(x^{8} - 149x^{6} + 7288x^{4} - 119797x^{2} + 105644\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
4 | $[4, 2, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} - \frac{3}{23}w + \frac{1}{23}]$ | $\phantom{-}\frac{2597}{1674161}e^{6} - \frac{275874}{1674161}e^{4} + \frac{7110480}{1674161}e^{2} - \frac{2404723}{1674161}$ |
9 | $[9, 3, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w + \frac{24}{23}]$ | $-1$ |
9 | $[9, 3, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w - \frac{68}{23}]$ | $\phantom{-}\frac{4840}{1674161}e^{6} - \frac{512854}{1674161}e^{4} + \frac{13285889}{1674161}e^{2} - \frac{9962486}{1674161}$ |
19 | $[19, 19, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} - \frac{3}{23}w + \frac{24}{23}]$ | $\phantom{-}e$ |
19 | $[19, 19, -\frac{6}{23}w^{3} + \frac{9}{23}w^{2} + \frac{83}{23}w - \frac{20}{23}]$ | $-\frac{39706}{82033889}e^{7} + \frac{4371959}{82033889}e^{5} - \frac{123883356}{82033889}e^{3} + \frac{403045207}{82033889}e$ |
19 | $[19, 19, \frac{6}{23}w^{3} - \frac{9}{23}w^{2} - \frac{83}{23}w + \frac{66}{23}]$ | $\phantom{-}e$ |
19 | $[19, 19, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} - \frac{3}{23}w - \frac{22}{23}]$ | $-\frac{39706}{82033889}e^{7} + \frac{4371959}{82033889}e^{5} - \frac{123883356}{82033889}e^{3} + \frac{403045207}{82033889}e$ |
25 | $[25, 5, -\frac{4}{23}w^{3} + \frac{6}{23}w^{2} + \frac{40}{23}w - \frac{21}{23}]$ | $-\frac{116}{1674161}e^{6} + \frac{31662}{1674161}e^{4} - \frac{1479266}{1674161}e^{2} + \frac{4951326}{1674161}$ |
29 | $[29, 29, w]$ | $\phantom{-}\frac{2481}{1674161}e^{6} - \frac{244212}{1674161}e^{4} + \frac{5631214}{1674161}e^{2} + \frac{872442}{1674161}$ |
29 | $[29, 29, -\frac{4}{23}w^{3} + \frac{6}{23}w^{2} + \frac{63}{23}w - \frac{21}{23}]$ | $\phantom{-}\frac{2481}{1674161}e^{6} - \frac{244212}{1674161}e^{4} + \frac{5631214}{1674161}e^{2} + \frac{872442}{1674161}$ |
29 | $[29, 29, \frac{4}{23}w^{3} - \frac{6}{23}w^{2} - \frac{63}{23}w + \frac{44}{23}]$ | $\phantom{-}\frac{1120}{1674161}e^{6} - \frac{132513}{1674161}e^{4} + \frac{3544846}{1674161}e^{2} + \frac{7960974}{1674161}$ |
29 | $[29, 29, -w + 1]$ | $\phantom{-}\frac{1120}{1674161}e^{6} - \frac{132513}{1674161}e^{4} + \frac{3544846}{1674161}e^{2} + \frac{7960974}{1674161}$ |
31 | $[31, 31, w + 2]$ | $\phantom{-}\frac{5557}{82033889}e^{7} - \frac{506504}{82033889}e^{5} + \frac{10739021}{82033889}e^{3} - \frac{38286480}{82033889}e$ |
31 | $[31, 31, -\frac{4}{23}w^{3} + \frac{6}{23}w^{2} + \frac{63}{23}w + \frac{25}{23}]$ | $\phantom{-}\frac{5557}{82033889}e^{7} - \frac{506504}{82033889}e^{5} + \frac{10739021}{82033889}e^{3} - \frac{38286480}{82033889}e$ |
31 | $[31, 31, \frac{4}{23}w^{3} - \frac{6}{23}w^{2} - \frac{63}{23}w + \frac{90}{23}]$ | $-\frac{47841}{82033889}e^{7} + \frac{4773908}{82033889}e^{5} - \frac{100646808}{82033889}e^{3} - \frac{606225098}{82033889}e$ |
31 | $[31, 31, -w + 3]$ | $-\frac{47841}{82033889}e^{7} + \frac{4773908}{82033889}e^{5} - \frac{100646808}{82033889}e^{3} - \frac{606225098}{82033889}e$ |
49 | $[49, 7, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w - \frac{22}{23}]$ | $\phantom{-}\frac{1535}{1674161}e^{6} - \frac{159192}{1674161}e^{4} + \frac{4305267}{1674161}e^{2} - \frac{2796394}{1674161}$ |
59 | $[59, 59, -\frac{20}{23}w^{3} + \frac{30}{23}w^{2} + \frac{269}{23}w - \frac{128}{23}]$ | $\phantom{-}\frac{36184}{82033889}e^{7} - \frac{3814743}{82033889}e^{5} + \frac{100676143}{82033889}e^{3} - \frac{186958589}{82033889}e$ |
59 | $[59, 59, -\frac{8}{23}w^{3} + \frac{12}{23}w^{2} + \frac{103}{23}w - \frac{88}{23}]$ | $\phantom{-}\frac{36184}{82033889}e^{7} - \frac{3814743}{82033889}e^{5} + \frac{100676143}{82033889}e^{3} - \frac{186958589}{82033889}e$ |
59 | $[59, 59, -\frac{8}{23}w^{3} + \frac{12}{23}w^{2} + \frac{103}{23}w - \frac{19}{23}]$ | $\phantom{-}\frac{73855}{82033889}e^{7} - \frac{8237414}{82033889}e^{5} + \frac{237027691}{82033889}e^{3} - \frac{849837823}{82033889}e$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$9$ | $[9,3,-\frac{2}{23}w^{3}+\frac{3}{23}w^{2}+\frac{43}{23}w+\frac{24}{23}]$ | $1$ |