[N,k,chi] = [462,4,Mod(1,462)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(462, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("462.1");
S:= CuspForms(chi, 4);
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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\).
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
\(2\)
\(1\)
\(3\)
\(1\)
\(7\)
\(-1\)
\(11\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 3T_{5} - 42 \)
T5^2 - 3*T5 - 42
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(462))\).
$p$
$F_p(T)$
$2$
\( (T + 2)^{2} \)
(T + 2)^2
$3$
\( (T + 3)^{2} \)
(T + 3)^2
$5$
\( T^{2} - 3T - 42 \)
T^2 - 3*T - 42
$7$
\( (T - 7)^{2} \)
(T - 7)^2
$11$
\( (T - 11)^{2} \)
(T - 11)^2
$13$
\( T^{2} - T - 398 \)
T^2 - T - 398
$17$
\( T^{2} - 114T + 3072 \)
T^2 - 114*T + 3072
$19$
\( T^{2} + 77T - 5996 \)
T^2 + 77*T - 5996
$23$
\( T^{2} - 30T - 8448 \)
T^2 - 30*T - 8448
$29$
\( T^{2} - 105T - 29502 \)
T^2 - 105*T - 29502
$31$
\( T^{2} + 80T - 43712 \)
T^2 + 80*T - 43712
$37$
\( T^{2} + 149T - 22106 \)
T^2 + 149*T - 22106
$41$
\( T^{2} + 54T - 216096 \)
T^2 + 54*T - 216096
$43$
\( T^{2} + 170T - 43928 \)
T^2 + 170*T - 43928
$47$
\( T^{2} - 231T - 29184 \)
T^2 - 231*T - 29184
$53$
\( T^{2} - 684T + 71652 \)
T^2 - 684*T + 71652
$59$
\( T^{2} - 447T - 366396 \)
T^2 - 447*T - 366396
$61$
\( T^{2} - 1240 T + 378028 \)
T^2 - 1240*T + 378028
$67$
\( T^{2} - 241T - 91724 \)
T^2 - 241*T - 91724
$71$
\( (T - 528)^{2} \)
(T - 528)^2
$73$
\( T^{2} - 187T - 390614 \)
T^2 - 187*T - 390614
$79$
\( T^{2} - 64T - 44288 \)
T^2 - 64*T - 44288
$83$
\( T^{2} - 1284 T + 354816 \)
T^2 - 1284*T + 354816
$89$
\( T^{2} - 1932 T + 794388 \)
T^2 - 1932*T + 794388
$97$
\( T^{2} - 1024 T - 508868 \)
T^2 - 1024*T - 508868
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