[N,k,chi] = [2004,2,Mod(1,2004)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2004, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2004.1");
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
\(2\)
\(-1\)
\(3\)
\(-1\)
\(167\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{9} - 9T_{5}^{8} + 7T_{5}^{7} + 126T_{5}^{6} - 259T_{5}^{5} - 405T_{5}^{4} + 964T_{5}^{3} + 506T_{5}^{2} - 856T_{5} - 466 \)
T5^9 - 9*T5^8 + 7*T5^7 + 126*T5^6 - 259*T5^5 - 405*T5^4 + 964*T5^3 + 506*T5^2 - 856*T5 - 466
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2004))\).
$p$
$F_p(T)$
$2$
\( T^{9} \)
T^9
$3$
\( (T - 1)^{9} \)
(T - 1)^9
$5$
\( T^{9} - 9 T^{8} + 7 T^{7} + 126 T^{6} + \cdots - 466 \)
T^9 - 9*T^8 + 7*T^7 + 126*T^6 - 259*T^5 - 405*T^4 + 964*T^3 + 506*T^2 - 856*T - 466
$7$
\( T^{9} - 2 T^{8} - 39 T^{7} + 109 T^{6} + \cdots - 288 \)
T^9 - 2*T^8 - 39*T^7 + 109*T^6 + 351*T^5 - 1365*T^4 + 480*T^3 + 1640*T^2 - 912*T - 288
$11$
\( T^{9} - 7 T^{8} - 23 T^{7} + 168 T^{6} + \cdots + 36 \)
T^9 - 7*T^8 - 23*T^7 + 168*T^6 + 213*T^5 - 989*T^4 - 520*T^3 + 1108*T^2 - 384*T + 36
$13$
\( T^{9} - 6 T^{8} - 45 T^{7} + \cdots - 5520 \)
T^9 - 6*T^8 - 45*T^7 + 334*T^6 + 90*T^5 - 3243*T^4 + 1820*T^3 + 9600*T^2 - 5712*T - 5520
$17$
\( T^{9} - 7 T^{8} - 58 T^{7} + \cdots - 1562 \)
T^9 - 7*T^8 - 58*T^7 + 540*T^6 - 263*T^5 - 5653*T^4 + 7872*T^3 + 11546*T^2 - 8736*T - 1562
$19$
\( T^{9} - 2 T^{8} - 69 T^{7} + 134 T^{6} + \cdots + 64 \)
T^9 - 2*T^8 - 69*T^7 + 134*T^6 + 684*T^5 + 223*T^4 - 700*T^3 - 448*T^2 + 64*T + 64
$23$
\( T^{9} - 19 T^{8} + 47 T^{7} + \cdots - 288512 \)
T^9 - 19*T^8 + 47*T^7 + 1212*T^6 - 9127*T^5 + 4015*T^4 + 164976*T^3 - 615744*T^2 + 786560*T - 288512
$29$
\( T^{9} - 13 T^{8} - 107 T^{7} + \cdots - 272720 \)
T^9 - 13*T^8 - 107*T^7 + 1686*T^6 + 1713*T^5 - 52505*T^4 - 11912*T^3 + 488440*T^2 + 173296*T - 272720
$31$
\( T^{9} - 12 T^{8} - 42 T^{7} + \cdots - 280000 \)
T^9 - 12*T^8 - 42*T^7 + 895*T^6 - 746*T^5 - 18931*T^4 + 32480*T^3 + 131888*T^2 - 178400*T - 280000
$37$
\( T^{9} - 15 T^{8} - 91 T^{7} + \cdots - 5712 \)
T^9 - 15*T^8 - 91*T^7 + 2302*T^6 - 4701*T^5 - 81337*T^4 + 450848*T^3 - 544344*T^2 - 434496*T - 5712
$41$
\( T^{9} - 18 T^{8} + 22 T^{7} + \cdots + 11178 \)
T^9 - 18*T^8 + 22*T^7 + 1079*T^6 - 5714*T^5 + 6255*T^4 + 11862*T^3 - 18522*T^2 - 6318*T + 11178
$43$
\( T^{9} + 6 T^{8} - 290 T^{7} + \cdots - 8589994 \)
T^9 + 6*T^8 - 290*T^7 - 1389*T^6 + 26210*T^5 + 70579*T^4 - 886908*T^3 - 142622*T^2 + 7610890*T - 8589994
$47$
\( T^{9} - 25 T^{8} + 25 T^{7} + \cdots + 51047340 \)
T^9 - 25*T^8 + 25*T^7 + 3713*T^6 - 23500*T^5 - 131535*T^4 + 1431344*T^3 - 556332*T^2 - 22532892*T + 51047340
$53$
\( T^{9} - 17 T^{8} - 159 T^{7} + \cdots + 12400550 \)
T^9 - 17*T^8 - 159*T^7 + 4129*T^6 - 9120*T^5 - 198307*T^4 + 1221572*T^3 - 987166*T^2 - 7077610*T + 12400550
$59$
\( T^{9} - 3 T^{8} - 283 T^{7} + \cdots + 881152 \)
T^9 - 3*T^8 - 283*T^7 + 829*T^6 + 27296*T^5 - 68223*T^4 - 1035632*T^3 + 1538720*T^2 + 13443712*T + 881152
$61$
\( T^{9} - 14 T^{8} - 263 T^{7} + \cdots - 4149788 \)
T^9 - 14*T^8 - 263*T^7 + 4170*T^6 + 12968*T^5 - 289205*T^4 - 172628*T^3 + 5792052*T^2 + 5249068*T - 4149788
$67$
\( T^{9} + 4 T^{8} - 201 T^{7} + \cdots + 150554 \)
T^9 + 4*T^8 - 201*T^7 - 972*T^6 + 10100*T^5 + 50139*T^4 - 155954*T^3 - 729620*T^2 + 260346*T + 150554
$71$
\( T^{9} - 17 T^{8} + \cdots + 343184800 \)
T^9 - 17*T^8 - 320*T^7 + 6815*T^6 + 17563*T^5 - 858773*T^4 + 2518796*T^3 + 31230720*T^2 - 207451056*T + 343184800
$73$
\( T^{9} + 20 T^{8} - 159 T^{7} + \cdots + 45115056 \)
T^9 + 20*T^8 - 159*T^7 - 4594*T^6 + 1642*T^5 + 312735*T^4 + 357948*T^3 - 6872328*T^2 - 5442336*T + 45115056
$79$
\( T^{9} + 8 T^{8} - 240 T^{7} + \cdots + 2054818 \)
T^9 + 8*T^8 - 240*T^7 - 1819*T^6 + 14886*T^5 + 128245*T^4 - 136474*T^3 - 2543762*T^2 - 3887890*T + 2054818
$83$
\( T^{9} + T^{8} - 259 T^{7} + \cdots - 3972576 \)
T^9 + T^8 - 259*T^7 + 497*T^6 + 16090*T^5 - 40425*T^4 - 334868*T^3 + 894312*T^2 + 1719024*T - 3972576
$89$
\( T^{9} - 36 T^{8} + 425 T^{7} + \cdots - 93648 \)
T^9 - 36*T^8 + 425*T^7 - 510*T^6 - 30616*T^5 + 278945*T^4 - 1004108*T^3 + 1378944*T^2 - 152784*T - 93648
$97$
\( T^{9} - 31 T^{8} + \cdots - 481925792 \)
T^9 - 31*T^8 - 160*T^7 + 10451*T^6 - 4645*T^5 - 1237713*T^4 + 1278656*T^3 + 55502112*T^2 - 18464048*T - 481925792
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