[N,k,chi] = [7,12,Mod(1,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.1");
S:= CuspForms(chi, 12);
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 \(\beta = \sqrt{3369}\).
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
\(7\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 54T_{2} - 2640 \)
T2^2 + 54*T2 - 2640
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(7))\).
$p$
$F_p(T)$
$2$
\( T^{2} + 54T - 2640 \)
T^2 + 54*T - 2640
$3$
\( T^{2} - 120T - 117684 \)
T^2 - 120*T - 117684
$5$
\( T^{2} + 13500 T + 45225600 \)
T^2 + 13500*T + 45225600
$7$
\( (T - 16807)^{2} \)
(T - 16807)^2
$11$
\( T^{2} + 750816 T + 82628600064 \)
T^2 + 750816*T + 82628600064
$13$
\( T^{2} + 9548 T - 3004246048160 \)
T^2 + 9548*T - 3004246048160
$17$
\( T^{2} - 4160052 T + 4294132070676 \)
T^2 - 4160052*T + 4294132070676
$19$
\( T^{2} + 17998712 T + 60418820423500 \)
T^2 + 17998712*T + 60418820423500
$23$
\( T^{2} + 66161016 T + 10\!\cdots\!64 \)
T^2 + 66161016*T + 1089063527288064
$29$
\( T^{2} - 61515612 T - 11\!\cdots\!40 \)
T^2 - 61515612*T - 11577825800104140
$31$
\( T^{2} + 15281552 T - 62\!\cdots\!08 \)
T^2 + 15281552*T - 6272688056617808
$37$
\( T^{2} + 527218340 T - 39\!\cdots\!36 \)
T^2 + 527218340*T - 39899433794297036
$41$
\( T^{2} + 178276140 T + 28\!\cdots\!16 \)
T^2 + 178276140*T + 2837391053183316
$43$
\( T^{2} - 1826745232 T + 54\!\cdots\!20 \)
T^2 - 1826745232*T + 549908118448121920
$47$
\( T^{2} - 568240704 T - 48\!\cdots\!32 \)
T^2 - 568240704*T - 4819095623733362832
$53$
\( T^{2} + 4185816372 T - 98\!\cdots\!88 \)
T^2 + 4185816372*T - 9829745180584088988
$59$
\( T^{2} - 3111345000 T - 34\!\cdots\!00 \)
T^2 - 3111345000*T - 34349908314521774100
$61$
\( T^{2} - 15042595060 T + 45\!\cdots\!56 \)
T^2 - 15042595060*T + 45659772847609730656
$67$
\( T^{2} - 9856523968 T - 11\!\cdots\!48 \)
T^2 - 9856523968*T - 114557670887448005648
$71$
\( T^{2} + 24312011328 T + 13\!\cdots\!60 \)
T^2 + 24312011328*T + 135260805034035855360
$73$
\( T^{2} + 30890001932 T + 10\!\cdots\!92 \)
T^2 + 30890001932*T + 104884626434894673892
$79$
\( T^{2} - 1992804256 T - 63\!\cdots\!20 \)
T^2 - 1992804256*T - 636469558726940275520
$83$
\( T^{2} - 5277014568 T - 37\!\cdots\!88 \)
T^2 - 5277014568*T - 371087002249402072788
$89$
\( T^{2} + 101541312828 T + 20\!\cdots\!00 \)
T^2 + 101541312828*T + 2064140707157901298500
$97$
\( T^{2} + 192228621116 T + 90\!\cdots\!08 \)
T^2 + 192228621116*T + 9098626134582595352308
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