[N,k,chi] = [138,4,Mod(1,138)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(138, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("138.1");
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 \(\beta = \sqrt{277}\).
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
\(2\)
\(1\)
\(3\)
\(-1\)
\(23\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 2T_{5} - 276 \)
T5^2 - 2*T5 - 276
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(138))\).
$p$
$F_p(T)$
$2$
\( (T + 2)^{2} \)
(T + 2)^2
$3$
\( (T - 3)^{2} \)
(T - 3)^2
$5$
\( T^{2} - 2T - 276 \)
T^2 - 2*T - 276
$7$
\( (T - 14)^{2} \)
(T - 14)^2
$11$
\( T^{2} + 2T - 276 \)
T^2 + 2*T - 276
$13$
\( T^{2} - 48T - 532 \)
T^2 - 48*T - 532
$17$
\( T^{2} + 4T - 1104 \)
T^2 + 4*T - 1104
$19$
\( T^{2} - 78T + 1244 \)
T^2 - 78*T + 1244
$23$
\( (T - 23)^{2} \)
(T - 23)^2
$29$
\( T^{2} - 184T - 19236 \)
T^2 - 184*T - 19236
$31$
\( T^{2} - 308T + 22608 \)
T^2 - 308*T + 22608
$37$
\( T^{2} - 370T + 31732 \)
T^2 - 370*T + 31732
$41$
\( T^{2} - 12T - 159516 \)
T^2 - 12*T - 159516
$43$
\( T^{2} - 186T - 4924 \)
T^2 - 186*T - 4924
$47$
\( T^{2} + 528T + 29808 \)
T^2 + 528*T + 29808
$53$
\( T^{2} + 926T + 207444 \)
T^2 + 926*T + 207444
$59$
\( (T + 396)^{2} \)
(T + 396)^2
$61$
\( T^{2} + 42T - 378772 \)
T^2 + 42*T - 378772
$67$
\( T^{2} + 434T - 673388 \)
T^2 + 434*T - 673388
$71$
\( T^{2} - 624T - 261648 \)
T^2 - 624*T - 261648
$73$
\( T^{2} + 352T - 555156 \)
T^2 + 352*T - 555156
$79$
\( T^{2} + 508T - 6396 \)
T^2 + 508*T - 6396
$83$
\( T^{2} - 994T - 590916 \)
T^2 - 994*T - 590916
$89$
\( T^{2} - 358992 \)
T^2 - 358992
$97$
\( T^{2} + 784T - 33588 \)
T^2 + 784*T - 33588
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