[N,k,chi] = [105,8,Mod(1,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("105.1");
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 \(\beta = 4\sqrt{2}\).
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
\(3\)
\(-1\)
\(5\)
\(-1\)
\(7\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 12T_{2} - 92 \)
T2^2 + 12*T2 - 92
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(105))\).
$p$
$F_p(T)$
$2$
\( T^{2} + 12T - 92 \)
T^2 + 12*T - 92
$3$
\( (T - 27)^{2} \)
(T - 27)^2
$5$
\( (T - 125)^{2} \)
(T - 125)^2
$7$
\( (T - 343)^{2} \)
(T - 343)^2
$11$
\( T^{2} + 10976 T + 29829344 \)
T^2 + 10976*T + 29829344
$13$
\( T^{2} + 2796 T + 1362532 \)
T^2 + 2796*T + 1362532
$17$
\( T^{2} + 8284 T - 11058908 \)
T^2 + 8284*T - 11058908
$19$
\( T^{2} + 8096 T + 15731936 \)
T^2 + 8096*T + 15731936
$23$
\( T^{2} + 90976 T + 1167902176 \)
T^2 + 90976*T + 1167902176
$29$
\( T^{2} + 12532 T - 2533764092 \)
T^2 + 12532*T - 2533764092
$31$
\( T^{2} + 117960 T - 45934979312 \)
T^2 + 117960*T - 45934979312
$37$
\( T^{2} + 174212 T - 83604374812 \)
T^2 + 174212*T - 83604374812
$41$
\( T^{2} + 492700 T - 31383664700 \)
T^2 + 492700*T - 31383664700
$43$
\( T^{2} + 661176 T - 41087241488 \)
T^2 + 661176*T - 41087241488
$47$
\( T^{2} + 1675408 T + 701642111264 \)
T^2 + 1675408*T + 701642111264
$53$
\( T^{2} + 555436 T - 3142721699644 \)
T^2 + 555436*T - 3142721699644
$59$
\( T^{2} + 3121176 T + 1671072643216 \)
T^2 + 3121176*T + 1671072643216
$61$
\( T^{2} + 511588 T - 4537899892316 \)
T^2 + 511588*T - 4537899892316
$67$
\( T^{2} + 252728 T + 2164616944 \)
T^2 + 252728*T + 2164616944
$71$
\( T^{2} + 1099336 T - 33999492976 \)
T^2 + 1099336*T - 33999492976
$73$
\( T^{2} - 5012588 T + 6268678876004 \)
T^2 - 5012588*T + 6268678876004
$79$
\( T^{2} - 83648 T - 28634836096 \)
T^2 - 83648*T - 28634836096
$83$
\( T^{2} + 4404184 T - 23626557722224 \)
T^2 + 4404184*T - 23626557722224
$89$
\( T^{2} + 4381564 T + 4683494838532 \)
T^2 + 4381564*T + 4683494838532
$97$
\( T^{2} + \cdots - 157167991209052 \)
T^2 + 8539828*T - 157167991209052
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