Properties

Base field \(\Q(\sqrt{5}, \sqrt{7})\)
Weight [2, 2, 2, 2]
Level norm 16
Level $[16, 2, 2]$
Label 4.4.19600.1-16.1-j
Dimension 8
CM no
Base change yes

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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 $[16, 2, 2]$
Label 4.4.19600.1-16.1-j
Dimension 8
Is CM no
Is base change yes
Parent newspace dimension 30

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{8} \) \(\mathstrut -\mathstrut 148x^{6} \) \(\mathstrut +\mathstrut 7716x^{4} \) \(\mathstrut -\mathstrut 168784x^{2} \) \(\mathstrut +\mathstrut 1317904\)

  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 $[59, 59, -\frac{8}{23}w^{3} + \frac{12}{23}w^{2} + \frac{103}{23}w - \frac{19}{23}]$ $-\frac{43}{196308}e^{7} + \frac{965}{32718}e^{5} - \frac{21335}{16359}e^{3} + \frac{134974}{7011}e$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
4 $[4,2,-\frac{2}{23}w^{3}+\frac{3}{23}w^{2}-\frac{3}{23}w+\frac{1}{23}]$ $1$