[N,k,chi] = [7,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.1");
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 \(\beta = \frac{1}{2}(1 + \sqrt{865})\).
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
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 3T_{2} - 214 \)
T2^2 + 3*T2 - 214
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(7))\).
$p$
$F_p(T)$
$2$
\( T^{2} + 3T - 214 \)
T^2 + 3*T - 214
$3$
\( T^{2} - 94T + 1344 \)
T^2 - 94*T + 1344
$5$
\( T^{2} - 330T + 5600 \)
T^2 - 330*T + 5600
$7$
\( (T + 343)^{2} \)
(T + 343)^2
$11$
\( T^{2} - 2844 T - 887776 \)
T^2 - 2844*T - 887776
$13$
\( T^{2} - 2534 T - 166620776 \)
T^2 - 2534*T - 166620776
$17$
\( T^{2} + 1488 T - 22147524 \)
T^2 + 1488*T - 22147524
$19$
\( T^{2} - 32810 T + 109928560 \)
T^2 - 32810*T + 109928560
$23$
\( T^{2} + 6576 T + 10312704 \)
T^2 + 6576*T + 10312704
$29$
\( T^{2} - 20640 T - 18920124100 \)
T^2 - 20640*T - 18920124100
$31$
\( T^{2} + 391836 T + 37023636384 \)
T^2 + 391836*T + 37023636384
$37$
\( T^{2} - 367392 T - 126010986084 \)
T^2 - 367392*T - 126010986084
$41$
\( T^{2} - 734664 T + 13303276364 \)
T^2 - 734664*T + 13303276364
$43$
\( T^{2} + 480476 T + 50864711104 \)
T^2 + 480476*T + 50864711104
$47$
\( T^{2} + 1089108 T + 2090896416 \)
T^2 + 1089108*T + 2090896416
$53$
\( T^{2} - 2858844 T + 2037435782724 \)
T^2 - 2858844*T + 2037435782724
$59$
\( T^{2} - 160170 T - 615374101440 \)
T^2 - 160170*T - 615374101440
$61$
\( T^{2} + 864646 T - 529516501136 \)
T^2 + 864646*T - 529516501136
$67$
\( T^{2} + 328648 T - 533876854064 \)
T^2 + 328648*T - 533876854064
$71$
\( T^{2} + 7500216 T + 10359492378624 \)
T^2 + 7500216*T + 10359492378624
$73$
\( T^{2} - 4301244 T - 3340687254156 \)
T^2 - 4301244*T - 3340687254156
$79$
\( T^{2} + 6408440 T - 6335206025600 \)
T^2 + 6408440*T - 6335206025600
$83$
\( T^{2} - 11659074 T + 30181573873584 \)
T^2 - 11659074*T + 30181573873584
$89$
\( T^{2} - 9772260 T - 4649674734460 \)
T^2 - 9772260*T - 4649674734460
$97$
\( T^{2} - 10762752 T + 27021168617436 \)
T^2 - 10762752*T + 27021168617436
show more
show less