[N,k,chi] = [157,2,Mod(1,157)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(157, 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("157.1");
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
\(157\)
\(-1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 5T_{2}^{6} + 2T_{2}^{5} + 21T_{2}^{4} - 22T_{2}^{3} - 21T_{2}^{2} + 27T_{2} - 1 \)
T2^7 - 5*T2^6 + 2*T2^5 + 21*T2^4 - 22*T2^3 - 21*T2^2 + 27*T2 - 1
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(157))\).
$p$
$F_p(T)$
$2$
\( T^{7} - 5 T^{6} + 2 T^{5} + 21 T^{4} + \cdots - 1 \)
T^7 - 5*T^6 + 2*T^5 + 21*T^4 - 22*T^3 - 21*T^2 + 27*T - 1
$3$
\( T^{7} - 5 T^{6} - T^{5} + 31 T^{4} + \cdots - 4 \)
T^7 - 5*T^6 - T^5 + 31*T^4 - 20*T^3 - 45*T^2 + 44*T - 4
$5$
\( T^{7} + T^{6} - 16 T^{5} + 3 T^{4} + \cdots + 16 \)
T^7 + T^6 - 16*T^5 + 3*T^4 + 73*T^3 - 87*T^2 + 8*T + 16
$7$
\( T^{7} - T^{6} - 16 T^{5} + 19 T^{4} + \cdots - 1 \)
T^7 - T^6 - 16*T^5 + 19*T^4 + 56*T^3 - 75*T^2 + 19*T - 1
$11$
\( T^{7} - 10 T^{6} + 28 T^{5} - 9 T^{4} + \cdots - 8 \)
T^7 - 10*T^6 + 28*T^5 - 9*T^4 - 44*T^3 + 33*T^2 + 8*T - 8
$13$
\( T^{7} + 5 T^{6} - 16 T^{5} - 63 T^{4} + \cdots + 113 \)
T^7 + 5*T^6 - 16*T^5 - 63*T^4 + 128*T^3 + 187*T^2 - 407*T + 113
$17$
\( T^{7} - 5 T^{6} - 44 T^{5} + 152 T^{4} + \cdots + 413 \)
T^7 - 5*T^6 - 44*T^5 + 152*T^4 + 593*T^3 - 890*T^2 - 2384*T + 413
$19$
\( T^{7} + 3 T^{6} - 95 T^{5} + \cdots + 5296 \)
T^7 + 3*T^6 - 95*T^5 - 368*T^4 + 1717*T^3 + 9185*T^2 + 12552*T + 5296
$23$
\( T^{7} - 15 T^{6} + 54 T^{5} + \cdots - 10073 \)
T^7 - 15*T^6 + 54*T^5 + 190*T^4 - 1529*T^3 + 1726*T^2 + 5352*T - 10073
$29$
\( T^{7} - 8 T^{6} - 76 T^{5} + \cdots - 18992 \)
T^7 - 8*T^6 - 76*T^5 + 883*T^4 - 1230*T^3 - 10329*T^2 + 32680*T - 18992
$31$
\( T^{7} + 13 T^{6} + 33 T^{5} + \cdots - 436 \)
T^7 + 13*T^6 + 33*T^5 - 150*T^4 - 613*T^3 + 29*T^2 + 1152*T - 436
$37$
\( T^{7} + 15 T^{6} - 46 T^{5} + \cdots + 39539 \)
T^7 + 15*T^6 - 46*T^5 - 1020*T^4 + 2073*T^3 + 19900*T^2 - 64910*T + 39539
$41$
\( T^{7} - 3 T^{6} - 175 T^{5} + \cdots - 47404 \)
T^7 - 3*T^6 - 175*T^5 + 77*T^4 + 7678*T^3 + 6763*T^2 - 70468*T - 47404
$43$
\( T^{7} - 11 T^{6} - 100 T^{5} + \cdots + 475171 \)
T^7 - 11*T^6 - 100*T^5 + 1196*T^4 + 3193*T^3 - 41900*T^2 - 32356*T + 475171
$47$
\( T^{7} - 8 T^{6} - 70 T^{5} + \cdots - 13444 \)
T^7 - 8*T^6 - 70*T^5 + 323*T^4 + 1874*T^3 - 989*T^2 - 12804*T - 13444
$53$
\( T^{7} - 9 T^{6} - 23 T^{5} + \cdots + 8612 \)
T^7 - 9*T^6 - 23*T^5 + 350*T^4 - 215*T^3 - 3033*T^2 + 1920*T + 8612
$59$
\( T^{7} - 31 T^{6} + 176 T^{5} + \cdots + 863917 \)
T^7 - 31*T^6 + 176*T^5 + 2659*T^4 - 24666*T^3 - 46015*T^2 + 515641*T + 863917
$61$
\( T^{7} + 6 T^{6} - 166 T^{5} + \cdots - 385772 \)
T^7 + 6*T^6 - 166*T^5 - 1043*T^4 + 7088*T^3 + 46747*T^2 - 47412*T - 385772
$67$
\( T^{7} - 4 T^{6} - 161 T^{5} + \cdots + 317732 \)
T^7 - 4*T^6 - 161*T^5 + 662*T^4 + 7134*T^3 - 32157*T^2 - 66540*T + 317732
$71$
\( T^{7} - 14 T^{6} - 220 T^{5} + \cdots + 1683928 \)
T^7 - 14*T^6 - 220*T^5 + 2455*T^4 + 16584*T^3 - 119771*T^2 - 388488*T + 1683928
$73$
\( T^{7} + 3 T^{6} - 178 T^{5} + \cdots - 11564 \)
T^7 + 3*T^6 - 178*T^5 - 386*T^4 + 3409*T^3 + 3787*T^2 - 14816*T - 11564
$79$
\( T^{7} - 6 T^{6} - 213 T^{5} + \cdots - 169324 \)
T^7 - 6*T^6 - 213*T^5 + 384*T^4 + 11138*T^3 + 2965*T^2 - 141844*T - 169324
$83$
\( T^{7} - 41 T^{6} + 583 T^{5} + \cdots + 1053284 \)
T^7 - 41*T^6 + 583*T^5 - 2425*T^4 - 18530*T^3 + 227645*T^2 - 834084*T + 1053284
$89$
\( T^{7} + 13 T^{6} - 225 T^{5} + \cdots + 4949 \)
T^7 + 13*T^6 - 225*T^5 - 2952*T^4 + 12077*T^3 + 165800*T^2 + 59133*T + 4949
$97$
\( T^{7} + 12 T^{6} - 421 T^{5} + \cdots - 13926728 \)
T^7 + 12*T^6 - 421*T^5 - 5594*T^4 + 42556*T^3 + 688139*T^2 + 33536*T - 13926728
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