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: | $[16, 2, 2]$ |
Dimension: | $8$ |
CM: | no |
Base change: | yes |
Newspace dimension: | $30$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{8} - 148x^{6} + 7716x^{4} - 168784x^{2} + 1317904\) |
Show full eigenvalues Hide large eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
4 | $[4, 2, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} - \frac{3}{23}w + \frac{1}{23}]$ | $\phantom{-}0$ |
9 | $[9, 3, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w + \frac{24}{23}]$ | $-\frac{1}{1368}e^{6} + \frac{11}{114}e^{4} - \frac{224}{57}e^{2} + \frac{8066}{171}$ |
9 | $[9, 3, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w - \frac{68}{23}]$ | $-\frac{1}{1368}e^{6} + \frac{11}{114}e^{4} - \frac{224}{57}e^{2} + \frac{8066}{171}$ |
19 | $[19, 19, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} - \frac{3}{23}w + \frac{24}{23}]$ | $\phantom{-}e$ |
19 | $[19, 19, -\frac{6}{23}w^{3} + \frac{9}{23}w^{2} + \frac{83}{23}w - \frac{20}{23}]$ | $\phantom{-}e$ |
19 | $[19, 19, \frac{6}{23}w^{3} - \frac{9}{23}w^{2} - \frac{83}{23}w + \frac{66}{23}]$ | $\phantom{-}\frac{115}{130872}e^{7} - \frac{1227}{10906}e^{5} + \frac{48689}{10906}e^{3} - \frac{126194}{2337}e$ |
19 | $[19, 19, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} - \frac{3}{23}w - \frac{22}{23}]$ | $\phantom{-}\frac{115}{130872}e^{7} - \frac{1227}{10906}e^{5} + \frac{48689}{10906}e^{3} - \frac{126194}{2337}e$ |
25 | $[25, 5, -\frac{4}{23}w^{3} + \frac{6}{23}w^{2} + \frac{40}{23}w - \frac{21}{23}]$ | $\phantom{-}\frac{1}{684}e^{6} - \frac{11}{57}e^{4} + \frac{448}{57}e^{2} - \frac{15790}{171}$ |
29 | $[29, 29, w]$ | $-\frac{13}{4788}e^{6} + \frac{89}{266}e^{4} - \frac{1650}{133}e^{2} + \frac{23323}{171}$ |
29 | $[29, 29, -\frac{4}{23}w^{3} + \frac{6}{23}w^{2} + \frac{63}{23}w - \frac{21}{23}]$ | $-\frac{1}{4788}e^{6} + \frac{41}{798}e^{4} - \frac{1322}{399}e^{2} + \frac{9625}{171}$ |
29 | $[29, 29, \frac{4}{23}w^{3} - \frac{6}{23}w^{2} - \frac{63}{23}w + \frac{44}{23}]$ | $-\frac{13}{4788}e^{6} + \frac{89}{266}e^{4} - \frac{1650}{133}e^{2} + \frac{23323}{171}$ |
29 | $[29, 29, -w + 1]$ | $-\frac{1}{4788}e^{6} + \frac{41}{798}e^{4} - \frac{1322}{399}e^{2} + \frac{9625}{171}$ |
31 | $[31, 31, w + 2]$ | $-\frac{11}{10332}e^{7} + \frac{149}{1148}e^{5} - \frac{1401}{287}e^{3} + \frac{21044}{369}e$ |
31 | $[31, 31, -\frac{4}{23}w^{3} + \frac{6}{23}w^{2} + \frac{63}{23}w + \frac{25}{23}]$ | $\phantom{-}\frac{317}{392616}e^{7} - \frac{540}{5453}e^{5} + \frac{20148}{5453}e^{3} - \frac{291484}{7011}e$ |
31 | $[31, 31, \frac{4}{23}w^{3} - \frac{6}{23}w^{2} - \frac{63}{23}w + \frac{90}{23}]$ | $-\frac{11}{10332}e^{7} + \frac{149}{1148}e^{5} - \frac{1401}{287}e^{3} + \frac{21044}{369}e$ |
31 | $[31, 31, -w + 3]$ | $\phantom{-}\frac{317}{392616}e^{7} - \frac{540}{5453}e^{5} + \frac{20148}{5453}e^{3} - \frac{291484}{7011}e$ |
49 | $[49, 7, -\frac{2}{23}w^{3} + \frac{3}{23}w^{2} + \frac{43}{23}w - \frac{22}{23}]$ | $-\frac{1}{684}e^{6} + \frac{11}{57}e^{4} - \frac{448}{57}e^{2} + \frac{17500}{171}$ |
59 | $[59, 59, -\frac{20}{23}w^{3} + \frac{30}{23}w^{2} + \frac{269}{23}w - \frac{128}{23}]$ | $-\frac{43}{196308}e^{7} + \frac{965}{32718}e^{5} - \frac{21335}{16359}e^{3} + \frac{134974}{7011}e$ |
59 | $[59, 59, -\frac{8}{23}w^{3} + \frac{12}{23}w^{2} + \frac{103}{23}w - \frac{88}{23}]$ | $-\frac{29}{98154}e^{7} + \frac{524}{16359}e^{5} - \frac{17491}{16359}e^{3} + \frac{81730}{7011}e$ |
59 | $[59, 59, -\frac{8}{23}w^{3} + \frac{12}{23}w^{2} + \frac{103}{23}w - \frac{19}{23}]$ | $-\frac{43}{196308}e^{7} + \frac{965}{32718}e^{5} - \frac{21335}{16359}e^{3} + \frac{134974}{7011}e$ |
Atkin-Lehner eigenvalues
Norm | Prime | Eigenvalue |
---|---|---|
$4$ | $[4,2,-\frac{2}{23}w^{3}+\frac{3}{23}w^{2}-\frac{3}{23}w+\frac{1}{23}]$ | $1$ |