[N,k,chi] = [363,8,Mod(1,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Refresh table
\( p \)
Sign
\(3\)
\(-1\)
\(11\)
\(1\)
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 269T_{2}^{2} + 8100 \)
T2^4 - 269*T2^2 + 8100
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(363))\).
$p$
$F_p(T)$
$2$
\( T^{4} - 269T^{2} + 8100 \)
T^4 - 269*T^2 + 8100
$3$
\( (T - 27)^{4} \)
(T - 27)^4
$5$
\( (T^{2} - 98 T - 37560)^{2} \)
(T^2 - 98*T - 37560)^2
$7$
\( T^{4} - 1207124 T^{2} + \cdots + 204123240000 \)
T^4 - 1207124*T^2 + 204123240000
$11$
\( T^{4} \)
T^4
$13$
\( T^{4} - 19331316 T^{2} + \cdots + 53728313601600 \)
T^4 - 19331316*T^2 + 53728313601600
$17$
\( T^{4} - 576262856 T^{2} + \cdots + 75\!\cdots\!00 \)
T^4 - 576262856*T^2 + 7502736824355600
$19$
\( T^{4} - 3411058184 T^{2} + \cdots + 27\!\cdots\!00 \)
T^4 - 3411058184*T^2 + 2723485139102250000
$23$
\( (T^{2} + 27766 T - 1031067936)^{2} \)
(T^2 + 27766*T - 1031067936)^2
$29$
\( T^{4} - 21975535016 T^{2} + \cdots + 65\!\cdots\!00 \)
T^4 - 21975535016*T^2 + 65049095383367763600
$31$
\( (T^{2} + 3908 T - 47822945280)^{2} \)
(T^2 + 3908*T - 47822945280)^2
$37$
\( (T^{2} + 381648 T + 18475306076)^{2} \)
(T^2 + 381648*T + 18475306076)^2
$41$
\( T^{4} - 265364846984 T^{2} + \cdots + 58\!\cdots\!00 \)
T^4 - 265364846984*T^2 + 5850337475739501392400
$43$
\( T^{4} - 83697885896 T^{2} + \cdots + 74\!\cdots\!00 \)
T^4 - 83697885896*T^2 + 7470760628999408400
$47$
\( (T^{2} + 570994 T + 80050719768)^{2} \)
(T^2 + 570994*T + 80050719768)^2
$53$
\( (T^{2} + 517590 T - 934849376064)^{2} \)
(T^2 + 517590*T - 934849376064)^2
$59$
\( (T^{2} + 1945932 T + 857468446560)^{2} \)
(T^2 + 1945932*T + 857468446560)^2
$61$
\( T^{4} - 9615216642500 T^{2} + \cdots + 14\!\cdots\!00 \)
T^4 - 9615216642500*T^2 + 14736375535897664100000000
$67$
\( (T^{2} - 694816 T - 1559358273552)^{2} \)
(T^2 - 694816*T - 1559358273552)^2
$71$
\( (T^{2} + 4154510 T + 2044949023416)^{2} \)
(T^2 + 4154510*T + 2044949023416)^2
$73$
\( T^{4} - 26031180027296 T^{2} + \cdots + 16\!\cdots\!00 \)
T^4 - 26031180027296*T^2 + 160678797339659679844358400
$79$
\( T^{4} - 57456944294724 T^{2} + \cdots + 51\!\cdots\!00 \)
T^4 - 57456944294724*T^2 + 518720697184299825490560000
$83$
\( T^{4} - 129351300110864 T^{2} + \cdots + 37\!\cdots\!00 \)
T^4 - 129351300110864*T^2 + 3757545568191403795991040000
$89$
\( (T^{2} + 2648372 T - 61992462497004)^{2} \)
(T^2 + 2648372*T - 61992462497004)^2
$97$
\( (T^{2} + 2409800 T - 219354462031316)^{2} \)
(T^2 + 2409800*T - 219354462031316)^2
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