[N,k,chi] = [33,6,Mod(1,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, 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("33.1");
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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\).
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
\(3\)
\(1\)
\(11\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 5T_{2} - 38 \)
T2^2 + 5*T2 - 38
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(33))\).
$p$
$F_p(T)$
$2$
\( T^{2} + 5T - 38 \)
T^2 + 5*T - 38
$3$
\( (T + 9)^{2} \)
(T + 9)^2
$5$
\( T^{2} - 58T - 3584 \)
T^2 - 58*T - 3584
$7$
\( T^{2} + 286T + 16024 \)
T^2 + 286*T + 16024
$11$
\( (T + 121)^{2} \)
(T + 121)^2
$13$
\( T^{2} + 166T - 57008 \)
T^2 + 166*T - 57008
$17$
\( T^{2} + 800T + 700 \)
T^2 + 800*T + 700
$19$
\( T^{2} + 1476 T + 538272 \)
T^2 + 1476*T + 538272
$23$
\( T^{2} + 3370 T - 3088328 \)
T^2 + 3370*T - 3088328
$29$
\( T^{2} - 6600 T + 3378828 \)
T^2 - 6600*T + 3378828
$31$
\( T^{2} + 7528 T + 8931328 \)
T^2 + 7528*T + 8931328
$37$
\( T^{2} + 29916 T + 222923316 \)
T^2 + 29916*T + 222923316
$41$
\( T^{2} + 5780 T - 14080172 \)
T^2 + 5780*T - 14080172
$43$
\( T^{2} + 16656 T - 29676624 \)
T^2 + 16656*T - 29676624
$47$
\( T^{2} - 7850 T - 266939288 \)
T^2 - 7850*T - 266939288
$53$
\( T^{2} - 14178 T - 165085872 \)
T^2 - 14178*T - 165085872
$59$
\( T^{2} - 17300 T - 21579488 \)
T^2 - 17300*T - 21579488
$61$
\( T^{2} + 2946 T - 238059096 \)
T^2 + 2946*T - 238059096
$67$
\( T^{2} - 31336 T - 207633776 \)
T^2 - 31336*T - 207633776
$71$
\( T^{2} + 33810 T - 138914952 \)
T^2 + 33810*T - 138914952
$73$
\( T^{2} - 60644 T - 1593591164 \)
T^2 - 60644*T - 1593591164
$79$
\( T^{2} - 1870 T - 3977569112 \)
T^2 - 1870*T - 3977569112
$83$
\( T^{2} + 58296 T + 552313872 \)
T^2 + 58296*T + 552313872
$89$
\( T^{2} - 92388 T - 7896843132 \)
T^2 - 92388*T - 7896843132
$97$
\( T^{2} - 7120 T - 20063108900 \)
T^2 - 7120*T - 20063108900
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