[N,k,chi] = [21,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.1");
S:= CuspForms(chi, 10);
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 \(\beta = \sqrt{345}\).
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
\( T_{2}^{2} - 30T_{2} - 120 \)
T2^2 - 30*T2 - 120
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(21))\).
$p$
$F_p(T)$
$2$
\( T^{2} - 30T - 120 \)
T^2 - 30*T - 120
$3$
\( (T + 81)^{2} \)
(T + 81)^2
$5$
\( T^{2} - 1128 T - 1372404 \)
T^2 - 1128*T - 1372404
$7$
\( (T - 2401)^{2} \)
(T - 2401)^2
$11$
\( T^{2} - 73284 T + 324360384 \)
T^2 - 73284*T + 324360384
$13$
\( T^{2} - 141100 T + 4881122020 \)
T^2 - 141100*T + 4881122020
$17$
\( T^{2} + 101784 T - 73285474836 \)
T^2 + 101784*T - 73285474836
$19$
\( T^{2} - 481744 T + 55133856304 \)
T^2 - 481744*T + 55133856304
$23$
\( T^{2} + 982212 T - 2917833651264 \)
T^2 + 982212*T - 2917833651264
$29$
\( T^{2} + 2550924 T - 5355274808556 \)
T^2 + 2550924*T - 5355274808556
$31$
\( T^{2} + 4935848 T - 7175316130304 \)
T^2 + 4935848*T - 7175316130304
$37$
\( T^{2} + 16256516 T - 1422306192956 \)
T^2 + 16256516*T - 1422306192956
$41$
\( T^{2} + \cdots + 579780377334684 \)
T^2 - 48707856*T + 579780377334684
$43$
\( T^{2} - 7989640 T + 6134871382480 \)
T^2 - 7989640*T + 6134871382480
$47$
\( T^{2} - 85572408 T + 16\!\cdots\!96 \)
T^2 - 85572408*T + 1627083056570496
$53$
\( T^{2} - 26565324 T - 48\!\cdots\!36 \)
T^2 - 26565324*T - 4813638861804636
$59$
\( T^{2} - 115200960 T + 23\!\cdots\!80 \)
T^2 - 115200960*T + 2380787479463280
$61$
\( T^{2} + 142820204 T - 89\!\cdots\!96 \)
T^2 + 142820204*T - 8917126591287596
$67$
\( T^{2} - 27521392 T - 11\!\cdots\!84 \)
T^2 - 27521392*T - 11963574198357584
$71$
\( T^{2} - 38070180 T - 12\!\cdots\!80 \)
T^2 - 38070180*T - 125949323289847680
$73$
\( T^{2} + 2095316 T - 17\!\cdots\!16 \)
T^2 + 2095316*T - 17273814922601516
$79$
\( T^{2} - 435097048 T - 99\!\cdots\!44 \)
T^2 - 435097048*T - 99418231347666944
$83$
\( T^{2} - 264288744 T - 39\!\cdots\!36 \)
T^2 - 264288744*T - 393242104842799536
$89$
\( T^{2} - 642673776 T + 54\!\cdots\!44 \)
T^2 - 642673776*T + 54201803181262044
$97$
\( T^{2} - 345361228 T - 49\!\cdots\!84 \)
T^2 - 345361228*T - 491161799172939884
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