[N,k,chi] = [4004,2,Mod(1,4004)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("4004.1");
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
\(2\)
\(-1\)
\(7\)
\(-1\)
\(11\)
\(-1\)
\(13\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{9} - 3T_{3}^{8} - 19T_{3}^{7} + 51T_{3}^{6} + 116T_{3}^{5} - 247T_{3}^{4} - 249T_{3}^{3} + 288T_{3}^{2} + 189T_{3} - 14 \)
T3^9 - 3*T3^8 - 19*T3^7 + 51*T3^6 + 116*T3^5 - 247*T3^4 - 249*T3^3 + 288*T3^2 + 189*T3 - 14
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).
$p$
$F_p(T)$
$2$
\( T^{9} \)
T^9
$3$
\( T^{9} - 3 T^{8} - 19 T^{7} + 51 T^{6} + \cdots - 14 \)
T^9 - 3*T^8 - 19*T^7 + 51*T^6 + 116*T^5 - 247*T^4 - 249*T^3 + 288*T^2 + 189*T - 14
$5$
\( T^{9} - 24 T^{7} + 4 T^{6} + 171 T^{5} + \cdots + 83 \)
T^9 - 24*T^7 + 4*T^6 + 171*T^5 + 5*T^4 - 452*T^3 - 125*T^2 + 311*T + 83
$7$
\( (T - 1)^{9} \)
(T - 1)^9
$11$
\( (T - 1)^{9} \)
(T - 1)^9
$13$
\( (T + 1)^{9} \)
(T + 1)^9
$17$
\( T^{9} - 5 T^{8} - 95 T^{7} + \cdots + 61022 \)
T^9 - 5*T^8 - 95*T^7 + 411*T^6 + 2752*T^5 - 8325*T^4 - 34129*T^3 + 44498*T^2 + 157069*T + 61022
$19$
\( T^{9} - 10 T^{8} - 42 T^{7} + \cdots - 19801 \)
T^9 - 10*T^8 - 42*T^7 + 662*T^6 - 611*T^5 - 11075*T^4 + 31154*T^3 + 3857*T^2 - 51339*T - 19801
$23$
\( T^{9} - 8 T^{8} - 61 T^{7} + \cdots - 11536 \)
T^9 - 8*T^8 - 61*T^7 + 576*T^6 + 733*T^5 - 12613*T^4 + 10176*T^3 + 76964*T^2 - 122080*T - 11536
$29$
\( T^{9} - 14 T^{8} - 47 T^{7} + \cdots - 147280 \)
T^9 - 14*T^8 - 47*T^7 + 1438*T^6 - 4831*T^5 - 19321*T^4 + 158216*T^3 - 384444*T^2 + 400064*T - 147280
$31$
\( T^{9} - 11 T^{8} - 186 T^{7} + \cdots - 15554608 \)
T^9 - 11*T^8 - 186*T^7 + 2254*T^6 + 9793*T^5 - 148923*T^4 - 71368*T^3 + 3315032*T^2 - 2692592*T - 15554608
$37$
\( T^{9} - 188 T^{7} - 175 T^{6} + \cdots + 118400 \)
T^9 - 188*T^7 - 175*T^6 + 9023*T^5 + 12410*T^4 - 134392*T^3 - 148032*T^2 + 586240*T + 118400
$41$
\( T^{9} - 14 T^{8} - 86 T^{7} + \cdots - 1040000 \)
T^9 - 14*T^8 - 86*T^7 + 1445*T^6 + 2309*T^5 - 42230*T^4 - 36824*T^3 + 432224*T^2 + 320512*T - 1040000
$43$
\( T^{9} - 8 T^{8} - 211 T^{7} + \cdots + 1770695 \)
T^9 - 8*T^8 - 211*T^7 + 1806*T^6 + 10221*T^5 - 107699*T^4 - 15742*T^3 + 1661021*T^2 - 3285048*T + 1770695
$47$
\( T^{9} - 10 T^{8} - 241 T^{7} + \cdots - 11326672 \)
T^9 - 10*T^8 - 241*T^7 + 2584*T^6 + 16373*T^5 - 204293*T^4 - 226384*T^3 + 5301524*T^2 - 6222048*T - 11326672
$53$
\( T^{9} - 21 T^{8} + \cdots - 199204765 \)
T^9 - 21*T^8 - 200*T^7 + 5936*T^6 + 4023*T^5 - 524141*T^4 + 568286*T^3 + 18083584*T^2 - 17592009*T - 199204765
$59$
\( T^{9} - 23 T^{8} - 13 T^{7} + \cdots - 775936 \)
T^9 - 23*T^8 - 13*T^7 + 2693*T^6 - 4421*T^5 - 100504*T^4 + 78804*T^3 + 1094656*T^2 + 125552*T - 775936
$61$
\( T^{9} - 34 T^{8} + 311 T^{7} + \cdots + 262558 \)
T^9 - 34*T^8 + 311*T^7 + 1211*T^6 - 33072*T^5 + 151099*T^4 - 88711*T^3 - 379519*T^2 + 157753*T + 262558
$67$
\( T^{9} - 10 T^{8} - 253 T^{7} + \cdots + 235526 \)
T^9 - 10*T^8 - 253*T^7 + 1936*T^6 + 13904*T^5 - 55626*T^4 - 263163*T^3 + 4606*T^2 + 435639*T + 235526
$71$
\( T^{9} - 4 T^{8} - 253 T^{7} + \cdots - 19670560 \)
T^9 - 4*T^8 - 253*T^7 + 1260*T^6 + 18585*T^5 - 104722*T^4 - 449208*T^3 + 2827664*T^2 + 2442384*T - 19670560
$73$
\( T^{9} - 9 T^{8} - 464 T^{7} + \cdots - 41371456 \)
T^9 - 9*T^8 - 464*T^7 + 4593*T^6 + 57870*T^5 - 634711*T^4 - 1016792*T^3 + 14138360*T^2 + 12753440*T - 41371456
$79$
\( T^{9} + 34 T^{8} + 243 T^{7} + \cdots + 3373045 \)
T^9 + 34*T^8 + 243*T^7 - 2746*T^6 - 35595*T^5 + 7905*T^4 + 1082440*T^3 + 1443135*T^2 - 6722294*T + 3373045
$83$
\( T^{9} - 15 T^{8} - 261 T^{7} + \cdots + 743353 \)
T^9 - 15*T^8 - 261*T^7 + 2506*T^6 + 27639*T^5 - 33263*T^4 - 708834*T^3 - 1291590*T^2 + 470694*T + 743353
$89$
\( T^{9} - 538 T^{7} + \cdots - 318322999 \)
T^9 - 538*T^7 - 80*T^6 + 92901*T^5 - 44233*T^4 - 5905196*T^3 + 10882147*T^2 + 113607021*T - 318322999
$97$
\( T^{9} - 15 T^{8} - 90 T^{7} + \cdots + 336832 \)
T^9 - 15*T^8 - 90*T^7 + 1903*T^6 - 492*T^5 - 51883*T^4 - 11368*T^3 + 529720*T^2 + 930720*T + 336832
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