[N,k,chi] = [729,2,Mod(1,729)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(729, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("729.1");
S:= CuspForms(chi, 2);
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
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
\(3\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 3T_{2}^{5} - 6T_{2}^{4} - 21T_{2}^{3} + 18T_{2} - 3 \)
T2^6 + 3*T2^5 - 6*T2^4 - 21*T2^3 + 18*T2 - 3
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(729))\).
$p$
$F_p(T)$
$2$
\( T^{6} + 3 T^{5} - 6 T^{4} - 21 T^{3} + \cdots - 3 \)
T^6 + 3*T^5 - 6*T^4 - 21*T^3 + 18*T - 3
$3$
\( T^{6} \)
T^6
$5$
\( T^{6} - 3 T^{5} - 15 T^{4} + 57 T^{3} + \cdots + 159 \)
T^6 - 3*T^5 - 15*T^4 + 57*T^3 + 9*T^2 - 189*T + 159
$7$
\( T^{6} - 6 T^{5} - 9 T^{4} + 83 T^{3} + \cdots + 136 \)
T^6 - 6*T^5 - 9*T^4 + 83*T^3 - 6*T^2 - 288*T + 136
$11$
\( T^{6} - 6 T^{5} - 15 T^{4} + 141 T^{3} + \cdots + 888 \)
T^6 - 6*T^5 - 15*T^4 + 141*T^3 - 90*T^2 - 648*T + 888
$13$
\( T^{6} - 6 T^{5} - 18 T^{4} + 74 T^{3} + \cdots - 89 \)
T^6 - 6*T^5 - 18*T^4 + 74*T^3 + 75*T^2 - 108*T - 89
$17$
\( T^{6} - 9 T^{5} + 135 T^{3} + \cdots + 459 \)
T^6 - 9*T^5 + 135*T^3 - 108*T^2 - 486*T + 459
$19$
\( T^{6} - 12 T^{5} + 15 T^{4} + \cdots - 296 \)
T^6 - 12*T^5 + 15*T^4 + 155*T^3 - 84*T^2 - 588*T - 296
$23$
\( T^{6} - 12 T^{5} + 21 T^{4} + \cdots + 456 \)
T^6 - 12*T^5 + 21*T^4 + 201*T^3 - 756*T^2 + 396*T + 456
$29$
\( T^{6} + 21 T^{5} + 111 T^{4} + \cdots + 9879 \)
T^6 + 21*T^5 + 111*T^4 - 219*T^3 - 2277*T^2 + 585*T + 9879
$31$
\( T^{6} - 15 T^{5} + 45 T^{4} + \cdots - 1016 \)
T^6 - 15*T^5 + 45*T^4 + 227*T^3 - 1338*T^2 + 2088*T - 1016
$37$
\( T^{6} - 3 T^{5} - 129 T^{4} + \cdots - 16109 \)
T^6 - 3*T^5 - 129*T^4 + 461*T^3 + 3165*T^2 - 9237*T - 16109
$41$
\( T^{6} - 12 T^{5} - 60 T^{4} + \cdots + 23109 \)
T^6 - 12*T^5 - 60*T^4 + 849*T^3 - 126*T^2 - 13329*T + 23109
$43$
\( T^{6} - 6 T^{5} - 108 T^{4} + \cdots + 17344 \)
T^6 - 6*T^5 - 108*T^4 + 632*T^3 + 1776*T^2 - 13536*T + 17344
$47$
\( T^{6} - 15 T^{5} - 42 T^{4} + \cdots + 17736 \)
T^6 - 15*T^5 - 42*T^4 + 1059*T^3 - 576*T^2 - 16596*T + 17736
$53$
\( T^{6} - 9 T^{5} - 81 T^{4} + \cdots + 1944 \)
T^6 - 9*T^5 - 81*T^4 + 729*T^3 + 1620*T^2 - 14580*T + 1944
$59$
\( T^{6} + 6 T^{5} - 105 T^{4} + \cdots - 408 \)
T^6 + 6*T^5 - 105*T^4 - 375*T^3 + 684*T^2 + 1404*T - 408
$61$
\( T^{6} - 24 T^{5} + 117 T^{4} + \cdots + 1576 \)
T^6 - 24*T^5 + 117*T^4 + 551*T^3 - 3264*T^2 - 3564*T + 1576
$67$
\( T^{6} - 15 T^{5} - 54 T^{4} + \cdots - 17288 \)
T^6 - 15*T^5 - 54*T^4 + 1019*T^3 + 1794*T^2 - 13644*T - 17288
$71$
\( T^{6} - 180 T^{4} + 864 T^{3} + \cdots + 29376 \)
T^6 - 180*T^4 + 864*T^3 + 2592*T^2 - 20736*T + 29376
$73$
\( T^{6} - 12 T^{5} - 156 T^{4} + \cdots + 57601 \)
T^6 - 12*T^5 - 156*T^4 + 2054*T^3 + 159*T^2 - 37020*T + 57601
$79$
\( T^{6} - 24 T^{5} + 27 T^{4} + \cdots - 93176 \)
T^6 - 24*T^5 + 27*T^4 + 2675*T^3 - 23622*T^2 + 78372*T - 93176
$83$
\( T^{6} - 6 T^{5} - 87 T^{4} + \cdots + 4344 \)
T^6 - 6*T^5 - 87*T^4 + 501*T^3 + 954*T^2 - 5976*T + 4344
$89$
\( T^{6} - 9 T^{5} - 225 T^{4} + \cdots - 123957 \)
T^6 - 9*T^5 - 225*T^4 + 999*T^3 + 10395*T^2 - 17577*T - 123957
$97$
\( T^{6} + 21 T^{5} + 72 T^{4} + \cdots - 18251 \)
T^6 + 21*T^5 + 72*T^4 - 1159*T^3 - 10482*T^2 - 28098*T - 18251
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