[N,k,chi] = [363,8,Mod(1,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.1");
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 \(\beta = \frac{1}{2}(1 + 3\sqrt{97})\).
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
\( T_{2}^{2} + T_{2} - 218 \)
T2^2 + T2 - 218
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(363))\).
$p$
$F_p(T)$
$2$
\( T^{2} + T - 218 \)
T^2 + T - 218
$3$
\( (T + 27)^{2} \)
(T + 27)^2
$5$
\( T^{2} + 194T - 33368 \)
T^2 + 194*T - 33368
$7$
\( T^{2} - 418T - 341312 \)
T^2 - 418*T - 341312
$11$
\( T^{2} \)
T^2
$13$
\( T^{2} - 13246 T + 43548976 \)
T^2 - 13246*T + 43548976
$17$
\( T^{2} - 10256 T + 19225084 \)
T^2 - 10256*T + 19225084
$19$
\( T^{2} + 14196 T - 1177576704 \)
T^2 + 14196*T - 1177576704
$23$
\( T^{2} + 13666 T - 870505736 \)
T^2 + 13666*T - 870505736
$29$
\( T^{2} + 15312 T - 4487554116 \)
T^2 + 15312*T - 4487554116
$31$
\( T^{2} + 48040 T + 569571328 \)
T^2 + 48040*T + 569571328
$37$
\( T^{2} - 274092 T - 18695152476 \)
T^2 - 274092*T - 18695152476
$41$
\( T^{2} + 755836 T - 49849727804 \)
T^2 + 755836*T - 49849727804
$43$
\( T^{2} - 1704096 T + 714621373632 \)
T^2 - 1704096*T + 714621373632
$47$
\( T^{2} + 1182094 T - 47516184584 \)
T^2 + 1182094*T - 47516184584
$53$
\( T^{2} + 2156394 T + 1006243830216 \)
T^2 + 2156394*T + 1006243830216
$59$
\( T^{2} - 927332 T - 1058263475744 \)
T^2 - 927332*T - 1058263475744
$61$
\( T^{2} - 1061994 T - 6224547468744 \)
T^2 - 1061994*T - 6224547468744
$67$
\( T^{2} + 3259952 T + 2655573007408 \)
T^2 + 3259952*T + 2655573007408
$71$
\( T^{2} + 5495514 T + 7547380719912 \)
T^2 + 5495514*T + 7547380719912
$73$
\( T^{2} - 5450812 T + 4341768952708 \)
T^2 - 5450812*T + 4341768952708
$79$
\( T^{2} - 1536590 T + 177407563168 \)
T^2 - 1536590*T + 177407563168
$83$
\( T^{2} + 8850888 T + 5354532740304 \)
T^2 + 8850888*T + 5354532740304
$89$
\( T^{2} - 6810132 T + 10431675116388 \)
T^2 - 6810132*T + 10431675116388
$97$
\( T^{2} - 9897376 T - 26135282253956 \)
T^2 - 9897376*T - 26135282253956
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