[N,k,chi] = [38,8,Mod(1,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.1");
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 \(\beta = \frac{1}{2}(1 + \sqrt{2737})\).
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
\(2\)
\(1\)
\(19\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 61T_{3} + 246 \)
T3^2 + 61*T3 + 246
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(38))\).
$p$
$F_p(T)$
$2$
\( (T + 8)^{2} \)
(T + 8)^2
$3$
\( T^{2} + 61T + 246 \)
T^2 + 61*T + 246
$5$
\( T^{2} - 175T - 146300 \)
T^2 - 175*T - 146300
$7$
\( T^{2} + 2592 T + 1654983 \)
T^2 + 2592*T + 1654983
$11$
\( T^{2} - 1045 T - 50723462 \)
T^2 - 1045*T - 50723462
$13$
\( T^{2} - 14647 T + 48454564 \)
T^2 - 14647*T + 48454564
$17$
\( T^{2} - 29616 T + 213734439 \)
T^2 - 29616*T + 213734439
$19$
\( (T + 6859)^{2} \)
(T + 6859)^2
$23$
\( T^{2} - 69985 T - 84282392 \)
T^2 - 69985*T - 84282392
$29$
\( T^{2} - 138821 T - 18767350094 \)
T^2 - 138821*T - 18767350094
$31$
\( T^{2} + 199396 T - 6382036064 \)
T^2 + 199396*T - 6382036064
$37$
\( T^{2} + 67840 T - 142395679268 \)
T^2 + 67840*T - 142395679268
$41$
\( T^{2} + 539350 T - 64948688 \)
T^2 + 539350*T - 64948688
$43$
\( T^{2} - 602639 T + 90498085252 \)
T^2 - 602639*T + 90498085252
$47$
\( T^{2} + 1031975 T - 263524205000 \)
T^2 + 1031975*T - 263524205000
$53$
\( T^{2} - 2138263 T + 147117109708 \)
T^2 - 2138263*T + 147117109708
$59$
\( T^{2} + 3936369 T + 3864174091866 \)
T^2 + 3936369*T + 3864174091866
$61$
\( T^{2} + 1027655 T - 4064856437738 \)
T^2 + 1027655*T - 4064856437738
$67$
\( T^{2} - 764949 T - 2495267011548 \)
T^2 - 764949*T - 2495267011548
$71$
\( T^{2} + 3572084 T - 8985205484828 \)
T^2 + 3572084*T - 8985205484828
$73$
\( T^{2} - 9069522 T + 20561142356433 \)
T^2 - 9069522*T + 20561142356433
$79$
\( T^{2} + 2753414 T - 50903350780448 \)
T^2 + 2753414*T - 50903350780448
$83$
\( T^{2} + 7643046 T + 14334319474056 \)
T^2 + 7643046*T + 14334319474056
$89$
\( T^{2} - 1393620 T - 513986601600 \)
T^2 - 1393620*T - 513986601600
$97$
\( T^{2} + 6921466 T - 48822541081424 \)
T^2 + 6921466*T - 48822541081424
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