[N,k,chi] = [105,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.1");
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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\).
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
\( T_{2}^{2} + T_{2} - 18 \)
T2^2 + T2 - 18
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(105))\).
$p$
$F_p(T)$
$2$
\( T^{2} + T - 18 \)
T^2 + T - 18
$3$
\( (T - 9)^{2} \)
(T - 9)^2
$5$
\( (T + 25)^{2} \)
(T + 25)^2
$7$
\( (T - 49)^{2} \)
(T - 49)^2
$11$
\( T^{2} + 396T - 26496 \)
T^2 + 396*T - 26496
$13$
\( T^{2} + 232T - 70932 \)
T^2 + 232*T - 70932
$17$
\( T^{2} + 2372 T + 1265268 \)
T^2 + 2372*T + 1265268
$19$
\( T^{2} + 1044 T - 5698624 \)
T^2 + 1044*T - 5698624
$23$
\( T^{2} + 4112 T + 3661824 \)
T^2 + 4112*T + 3661824
$29$
\( T^{2} + 3148 T - 10155612 \)
T^2 + 3148*T - 10155612
$31$
\( T^{2} + 4380 T - 16648672 \)
T^2 + 4380*T - 16648672
$37$
\( T^{2} + 4716 T - 11824348 \)
T^2 + 4716*T - 11824348
$41$
\( T^{2} + 15780 T - 115400700 \)
T^2 + 15780*T - 115400700
$43$
\( T^{2} + 13752 T + 9014528 \)
T^2 + 13752*T + 9014528
$47$
\( T^{2} - 2896 T - 158667984 \)
T^2 - 2896*T - 158667984
$53$
\( T^{2} + 36112 T + 274264764 \)
T^2 + 36112*T + 274264764
$59$
\( T^{2} + 11584 T - 116985744 \)
T^2 + 11584*T - 116985744
$61$
\( T^{2} - 40932 T - 200164156 \)
T^2 - 40932*T - 200164156
$67$
\( T^{2} - 89856 T + 2005401536 \)
T^2 - 89856*T + 2005401536
$71$
\( T^{2} + 16476 T - 1677718656 \)
T^2 + 16476*T - 1677718656
$73$
\( T^{2} + 2424 T - 1704362644 \)
T^2 + 2424*T - 1704362644
$79$
\( T^{2} - 60952 T - 2262667136 \)
T^2 - 60952*T - 2262667136
$83$
\( T^{2} + 81248 T + 1176371184 \)
T^2 + 81248*T + 1176371184
$89$
\( T^{2} + 25476 T - 5196718908 \)
T^2 + 25476*T - 5196718908
$97$
\( T^{2} - 123296 T - 3560972868 \)
T^2 - 123296*T - 3560972868
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