[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 \(\beta = 8\sqrt{2}\).
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
\( T_{5}^{2} - 8T_{5} - 112 \)
T5^2 - 8*T5 - 112
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
$5$
\( T^{2} - 8T - 112 \)
T^2 - 8*T - 112
$7$
\( T^{2} - 12T - 92 \)
T^2 - 12*T - 92
$11$
\( T^{2} - 48T - 1472 \)
T^2 - 48*T - 1472
$13$
\( T^{2} - 84T - 284 \)
T^2 - 84*T - 284
$17$
\( T^{2} - 104T + 1552 \)
T^2 - 104*T + 1552
$19$
\( T^{2} - 28T - 956 \)
T^2 - 28*T - 956
$23$
\( (T - 23)^{2} \)
(T - 23)^2
$29$
\( T^{2} - 132T - 8444 \)
T^2 - 132*T - 8444
$31$
\( T^{2} + 8T - 12784 \)
T^2 + 8*T - 12784
$37$
\( T^{2} + 324T - 35708 \)
T^2 + 324*T - 35708
$41$
\( T^{2} + 140T - 13532 \)
T^2 + 140*T - 13532
$43$
\( T^{2} + 548T + 59588 \)
T^2 + 548*T + 59588
$47$
\( T^{2} + 464T + 49216 \)
T^2 + 464*T + 49216
$53$
\( T^{2} - 336T - 186944 \)
T^2 - 336*T - 186944
$59$
\( T^{2} - 72T - 490736 \)
T^2 - 72*T - 490736
$61$
\( T^{2} - 60T - 269948 \)
T^2 - 60*T - 269948
$67$
\( T^{2} + 444T - 44028 \)
T^2 + 444*T - 44028
$71$
\( T^{2} - 672T - 288512 \)
T^2 - 672*T - 288512
$73$
\( T^{2} + 252T - 330236 \)
T^2 + 252*T - 330236
$79$
\( T^{2} + 1692 T + 715588 \)
T^2 + 1692*T + 715588
$83$
\( T^{2} + 400T - 951232 \)
T^2 + 400*T - 951232
$89$
\( T^{2} + 336T - 17984 \)
T^2 + 336*T - 17984
$97$
\( T^{2} - 220T - 848572 \)
T^2 - 220*T - 848572
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