[N,k,chi] = [210,8,Mod(1,210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("210.1");
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
sage: f.q_expansion() # note that sage often uses an isomorphic number field
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{536866}\).
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
\(2\)
\(-1\)
\(3\)
\(1\)
\(5\)
\(-1\)
\(7\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{2} - 2736T_{11} - 32488000 \)
T11^2 - 2736*T11 - 32488000
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(210))\).
$p$
$F_p(T)$
$2$
\( (T - 8)^{2} \)
(T - 8)^2
$3$
\( (T + 27)^{2} \)
(T + 27)^2
$5$
\( (T - 125)^{2} \)
(T - 125)^2
$7$
\( (T + 343)^{2} \)
(T + 343)^2
$11$
\( T^{2} - 2736 T - 32488000 \)
T^2 - 2736*T - 32488000
$13$
\( T^{2} + 8092 T - 60938588 \)
T^2 + 8092*T - 60938588
$17$
\( T^{2} - 23284 T + 126946308 \)
T^2 - 23284*T + 126946308
$19$
\( T^{2} - 27136 T + 46652928 \)
T^2 - 27136*T + 46652928
$23$
\( T^{2} + 58464 T - 3689524000 \)
T^2 + 58464*T - 3689524000
$29$
\( T^{2} + 17844 T - 7144466812 \)
T^2 + 17844*T - 7144466812
$31$
\( T^{2} - 156376 T - 53062154640 \)
T^2 - 156376*T - 53062154640
$37$
\( T^{2} - 224300 T + 7208962500 \)
T^2 - 224300*T + 7208962500
$41$
\( T^{2} - 649348 T + 50438127876 \)
T^2 - 649348*T + 50438127876
$43$
\( T^{2} - 813400 T + 100388269936 \)
T^2 - 813400*T + 100388269936
$47$
\( T^{2} + 751760 T + 27684928800 \)
T^2 + 751760*T + 27684928800
$53$
\( T^{2} + 339916 T - 744201318236 \)
T^2 + 339916*T - 744201318236
$59$
\( T^{2} - 1398088 T - 1816202007408 \)
T^2 - 1398088*T - 1816202007408
$61$
\( T^{2} - 3603836 T + 3231025834980 \)
T^2 - 3603836*T + 3231025834980
$67$
\( T^{2} - 528024 T + 52307877744 \)
T^2 - 528024*T + 52307877744
$71$
\( T^{2} - 8153960 T + 16382203426416 \)
T^2 - 8153960*T + 16382203426416
$73$
\( T^{2} - 10691020 T + 28460876314500 \)
T^2 - 10691020*T + 28460876314500
$79$
\( T^{2} - 2146832 T - 10486139625920 \)
T^2 - 2146832*T - 10486139625920
$83$
\( T^{2} - 10477208 T + 21917942130192 \)
T^2 - 10477208*T + 21917942130192
$89$
\( T^{2} + \cdots - 108733766668604 \)
T^2 - 3219172*T - 108733766668604
$97$
\( T^{2} - 13140748 T + 3868465956100 \)
T^2 - 13140748*T + 3868465956100
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