[N,k,chi] = [400,6,Mod(1,400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.1");
S:= CuspForms(chi, 6);
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
sage: f.q_expansion() # note that sage often uses an isomorphic number field
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{241}\).
We also show the integral \(q\)-expansion of the trace form .
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
\(2\)
\(-1\)
\(5\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 20T_{3} - 141 \)
T3^2 - 20*T3 - 141
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).
$p$
$F_p(T)$
$2$
\( T^{2} \)
T^2
$3$
\( T^{2} - 20T - 141 \)
T^2 - 20*T - 141
$5$
\( T^{2} \)
T^2
$7$
\( T^{2} - 200T + 9036 \)
T^2 - 200*T + 9036
$11$
\( T^{2} - 196T - 141021 \)
T^2 - 196*T - 141021
$13$
\( T^{2} - 360T - 29296 \)
T^2 - 360*T - 29296
$17$
\( T^{2} + 1490 T - 559359 \)
T^2 + 1490*T - 559359
$19$
\( T^{2} - 3180 T + 2232875 \)
T^2 - 3180*T + 2232875
$23$
\( T^{2} - 1560 T + 599724 \)
T^2 - 1560*T + 599724
$29$
\( T^{2} + 3920 T + 3456000 \)
T^2 + 3920*T + 3456000
$31$
\( T^{2} - 1096 T - 72602196 \)
T^2 - 1096*T - 72602196
$37$
\( T^{2} + 2020 T - 7864124 \)
T^2 + 2020*T - 7864124
$41$
\( T^{2} - 27754 T + 182931129 \)
T^2 - 27754*T + 182931129
$43$
\( T^{2} + 3000 T - 270585136 \)
T^2 + 3000*T - 270585136
$47$
\( T^{2} - 25760 T + 22262256 \)
T^2 - 25760*T + 22262256
$53$
\( T^{2} - 26980 T + 147908484 \)
T^2 - 26980*T + 147908484
$59$
\( T^{2} + 11960 T - 195696000 \)
T^2 + 11960*T - 195696000
$61$
\( T^{2} + 24396 T - 92208796 \)
T^2 + 24396*T - 92208796
$67$
\( T^{2} + 40060 T + 249648291 \)
T^2 + 40060*T + 249648291
$71$
\( T^{2} - 87296 T + 1844897904 \)
T^2 - 87296*T + 1844897904
$73$
\( T^{2} - 70290 T + 1160669249 \)
T^2 - 70290*T + 1160669249
$79$
\( T^{2} + 65480 T - 446416500 \)
T^2 + 65480*T - 446416500
$83$
\( T^{2} - 92580 T + 2098410219 \)
T^2 - 92580*T + 2098410219
$89$
\( T^{2} + 72810 T - 5241540375 \)
T^2 + 72810*T - 5241540375
$97$
\( T^{2} - 126140 T - 3238386044 \)
T^2 - 126140*T - 3238386044
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