[N,k,chi] = [22,8,Mod(1,22)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22, 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("22.1");
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 \(\beta = \frac{1}{2}(1 + \sqrt{14881})\).
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
\(2\)
\(-1\)
\(11\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 23T_{3} - 3588 \)
T3^2 + 23*T3 - 3588
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(22))\).
$p$
$F_p(T)$
$2$
\( (T - 8)^{2} \)
(T - 8)^2
$3$
\( T^{2} + 23T - 3588 \)
T^2 + 23*T - 3588
$5$
\( T^{2} - 331T + 23670 \)
T^2 - 331*T + 23670
$7$
\( T^{2} - 1794T + 75440 \)
T^2 - 1794*T + 75440
$11$
\( (T - 1331)^{2} \)
(T - 1331)^2
$13$
\( T^{2} + 5406 T - 44494552 \)
T^2 + 5406*T - 44494552
$17$
\( T^{2} - 15032 T - 150712788 \)
T^2 - 15032*T - 150712788
$19$
\( T^{2} - 16916 T - 799953120 \)
T^2 - 16916*T - 799953120
$23$
\( T^{2} + 51351 T - 536788152 \)
T^2 + 51351*T - 536788152
$29$
\( T^{2} + 207130 T + 8743188000 \)
T^2 + 207130*T + 8743188000
$31$
\( T^{2} + 19071 T - 6148026496 \)
T^2 + 19071*T - 6148026496
$37$
\( T^{2} - 351333 T - 11768742334 \)
T^2 - 351333*T - 11768742334
$41$
\( T^{2} - 123610 T - 234633419904 \)
T^2 - 123610*T - 234633419904
$43$
\( T^{2} + 159822 T - 591953661640 \)
T^2 + 159822*T - 591953661640
$47$
\( T^{2} - 451160 T - 72885726336 \)
T^2 - 451160*T - 72885726336
$53$
\( T^{2} + 1260832 T + 75106099260 \)
T^2 + 1260832*T + 75106099260
$59$
\( T^{2} - 887547 T - 564563979540 \)
T^2 - 887547*T - 564563979540
$61$
\( T^{2} + 597918 T - 456927065080 \)
T^2 + 597918*T - 456927065080
$67$
\( T^{2} - 2864711 T - 2490832261212 \)
T^2 - 2864711*T - 2490832261212
$71$
\( T^{2} - 1306267 T - 5755066505400 \)
T^2 - 1306267*T - 5755066505400
$73$
\( T^{2} + 4577530 T + 2038977114936 \)
T^2 + 4577530*T + 2038977114936
$79$
\( T^{2} + 2946342 T - 9189351784480 \)
T^2 + 2946342*T - 9189351784480
$83$
\( T^{2} - 9965450 T + 22547926115976 \)
T^2 - 9965450*T + 22547926115976
$89$
\( T^{2} - 10185377 T + 8815411816710 \)
T^2 - 10185377*T + 8815411816710
$97$
\( T^{2} + \cdots + 192724568772626 \)
T^2 + 27765477*T + 192724568772626
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