[N,k,chi] = [138,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("138.1");
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 \(\beta = \sqrt{514}\).
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
\( T_{5}^{2} - 40T_{5} - 4226 \)
T5^2 - 40*T5 - 4226
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(138))\).
$p$
$F_p(T)$
$2$
\( (T - 4)^{2} \)
(T - 4)^2
$3$
\( (T + 9)^{2} \)
(T + 9)^2
$5$
\( T^{2} - 40T - 4226 \)
T^2 - 40*T - 4226
$7$
\( T^{2} + 100T - 10350 \)
T^2 + 100*T - 10350
$11$
\( T^{2} - 160T - 26496 \)
T^2 - 160*T - 26496
$13$
\( T^{2} - 632T - 196208 \)
T^2 - 632*T - 196208
$17$
\( T^{2} - 1216 T + 365038 \)
T^2 - 1216*T + 365038
$19$
\( T^{2} - 2924 T + 1910770 \)
T^2 - 2924*T + 1910770
$23$
\( (T + 529)^{2} \)
(T + 529)^2
$29$
\( T^{2} - 5300 T + 1674844 \)
T^2 - 5300*T + 1674844
$31$
\( T^{2} - 2512 T - 15448200 \)
T^2 - 2512*T - 15448200
$37$
\( T^{2} - 21868 T + 103990492 \)
T^2 - 21868*T + 103990492
$41$
\( T^{2} - 9532 T + 14293380 \)
T^2 - 9532*T + 14293380
$43$
\( T^{2} - 6612 T - 23549998 \)
T^2 - 6612*T - 23549998
$47$
\( T^{2} - 3860 T - 18087204 \)
T^2 - 3860*T - 18087204
$53$
\( T^{2} - 27744 T + 21306878 \)
T^2 - 27744*T + 21306878
$59$
\( T^{2} + 59140 T + 873099900 \)
T^2 + 59140*T + 873099900
$61$
\( T^{2} + 5724 T + 7103420 \)
T^2 + 5724*T + 7103420
$67$
\( T^{2} + 46124 T - 1139059982 \)
T^2 + 46124*T - 1139059982
$71$
\( T^{2} + 16320 T - 275664384 \)
T^2 + 16320*T - 275664384
$73$
\( T^{2} - 9756 T - 2090176092 \)
T^2 - 9756*T - 2090176092
$79$
\( T^{2} + 3028 T - 1683484398 \)
T^2 + 3028*T - 1683484398
$83$
\( T^{2} - 61560 T - 106612336 \)
T^2 - 61560*T - 106612336
$89$
\( T^{2} - 65592 T - 10138998258 \)
T^2 - 65592*T - 10138998258
$97$
\( T^{2} + 106724 T - 11623465860 \)
T^2 + 106724*T - 11623465860
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