[N,k,chi] = [21,14,Mod(1,21)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.1");
S:= CuspForms(chi, 14);
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 \(\beta = 28\sqrt{3}\).
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
\(3\)
\(-1\)
\(7\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 144T_{2} + 2832 \)
T2^2 - 144*T2 + 2832
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(21))\).
$p$
$F_p(T)$
$2$
\( T^{2} - 144T + 2832 \)
T^2 - 144*T + 2832
$3$
\( (T - 729)^{2} \)
(T - 729)^2
$5$
\( T^{2} + 52560 T - 597316800 \)
T^2 + 52560*T - 597316800
$7$
\( (T - 117649)^{2} \)
(T - 117649)^2
$11$
\( T^{2} + 8593596 T - 2293276854396 \)
T^2 + 8593596*T - 2293276854396
$13$
\( T^{2} + \cdots + 248421977380228 \)
T^2 + 37982012*T + 248421977380228
$17$
\( T^{2} - 16643088 T - 15\!\cdots\!52 \)
T^2 - 16643088*T - 15698604078236352
$19$
\( T^{2} + 211195304 T - 46\!\cdots\!04 \)
T^2 + 211195304*T - 46799673266056304
$23$
\( T^{2} + 194997852 T - 36\!\cdots\!36 \)
T^2 + 194997852*T - 367159465799362236
$29$
\( T^{2} + 3481642548 T - 14\!\cdots\!72 \)
T^2 + 3481642548*T - 14198914849778126172
$31$
\( T^{2} + 8393797520 T - 14\!\cdots\!92 \)
T^2 + 8393797520*T - 1460505601522016192
$37$
\( T^{2} + 22337946716 T + 76\!\cdots\!92 \)
T^2 + 22337946716*T + 76223406498380877892
$41$
\( T^{2} + 12507403320 T - 56\!\cdots\!00 \)
T^2 + 12507403320*T - 560923308691487881200
$43$
\( T^{2} - 26062746328 T - 16\!\cdots\!32 \)
T^2 - 26062746328*T - 169308591389372562032
$47$
\( T^{2} - 29222658936 T - 59\!\cdots\!24 \)
T^2 - 29222658936*T - 5961656829671027223024
$53$
\( T^{2} - 523911868308 T + 68\!\cdots\!44 \)
T^2 - 523911868308*T + 68598759314907576806244
$59$
\( T^{2} + 94444581024 T + 17\!\cdots\!56 \)
T^2 + 94444581024*T + 1789567769269355801856
$61$
\( T^{2} - 461720554252 T - 45\!\cdots\!36 \)
T^2 - 461720554252*T - 4562907637247589059036
$67$
\( T^{2} + 1237879942064 T + 36\!\cdots\!56 \)
T^2 + 1237879942064*T + 369641704643024445307456
$71$
\( T^{2} - 1961126619804 T + 95\!\cdots\!04 \)
T^2 - 1961126619804*T + 956380286910742761156804
$73$
\( T^{2} + 2407791783284 T + 13\!\cdots\!96 \)
T^2 + 2407791783284*T + 1393544381247733334536996
$79$
\( T^{2} + 2622494223848 T + 16\!\cdots\!64 \)
T^2 + 2622494223848*T + 1696895234756139706117264
$83$
\( T^{2} - 4489561815912 T + 23\!\cdots\!64 \)
T^2 - 4489561815912*T + 2388136026953618267051664
$89$
\( T^{2} - 520927840584 T - 19\!\cdots\!88 \)
T^2 - 520927840584*T - 193588577813963732437488
$97$
\( T^{2} + 13027956631604 T + 42\!\cdots\!32 \)
T^2 + 13027956631604*T + 42404723336841600764692132
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