[N,k,chi] = [331,2,Mod(1,331)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(331, 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("331.1");
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
\(331\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 2T_{2}^{6} - 6T_{2}^{5} - 8T_{2}^{4} + 11T_{2}^{3} + 3T_{2}^{2} - 5T_{2} + 1 \)
T2^7 + 2*T2^6 - 6*T2^5 - 8*T2^4 + 11*T2^3 + 3*T2^2 - 5*T2 + 1
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(331))\).
$p$
$F_p(T)$
$2$
\( T^{7} + 2 T^{6} - 6 T^{5} - 8 T^{4} + \cdots + 1 \)
T^7 + 2*T^6 - 6*T^5 - 8*T^4 + 11*T^3 + 3*T^2 - 5*T + 1
$3$
\( T^{7} - 10 T^{5} - 3 T^{4} + 12 T^{3} + \cdots + 1 \)
T^7 - 10*T^5 - 3*T^4 + 12*T^3 - 4*T + 1
$5$
\( T^{7} + 13 T^{6} + 58 T^{5} + \cdots + 257 \)
T^7 + 13*T^6 + 58*T^5 + 79*T^4 - 114*T^3 - 335*T^2 - 28*T + 257
$7$
\( T^{7} + 8 T^{6} - 6 T^{5} - 184 T^{4} + \cdots + 2377 \)
T^7 + 8*T^6 - 6*T^5 - 184*T^4 - 329*T^3 + 834*T^2 + 2988*T + 2377
$11$
\( T^{7} + 10 T^{6} + 4 T^{5} + \cdots + 1857 \)
T^7 + 10*T^6 + 4*T^5 - 206*T^4 - 431*T^3 + 865*T^2 + 2851*T + 1857
$13$
\( T^{7} - 5 T^{6} - 30 T^{5} + 189 T^{4} + \cdots - 79 \)
T^7 - 5*T^6 - 30*T^5 + 189*T^4 - 66*T^3 - 815*T^2 + 906*T - 79
$17$
\( T^{7} + 17 T^{6} + 75 T^{5} + \cdots - 13061 \)
T^7 + 17*T^6 + 75*T^5 - 280*T^4 - 3706*T^3 - 13443*T^2 - 21554*T - 13061
$19$
\( T^{7} + 19 T^{6} + 76 T^{5} + \cdots - 29763 \)
T^7 + 19*T^6 + 76*T^5 - 708*T^4 - 7810*T^3 - 28962*T^2 - 47920*T - 29763
$23$
\( T^{7} - 15 T^{6} + 43 T^{5} + \cdots + 1543 \)
T^7 - 15*T^6 + 43*T^5 + 263*T^4 - 1292*T^3 + 83*T^2 + 3043*T + 1543
$29$
\( T^{7} + 24 T^{6} + 163 T^{5} + \cdots + 2861 \)
T^7 + 24*T^6 + 163*T^5 - 25*T^4 - 3053*T^3 - 4088*T^2 + 9314*T + 2861
$31$
\( T^{7} - 4 T^{6} - 90 T^{5} + \cdots - 1399 \)
T^7 - 4*T^6 - 90*T^5 + 183*T^4 + 2233*T^3 - 71*T^2 - 4726*T - 1399
$37$
\( T^{7} - 2 T^{6} - 159 T^{5} + \cdots - 182141 \)
T^7 - 2*T^6 - 159*T^5 + 34*T^4 + 7742*T^3 + 6359*T^2 - 119087*T - 182141
$41$
\( T^{7} + 18 T^{6} + 50 T^{5} + \cdots - 5367 \)
T^7 + 18*T^6 + 50*T^5 - 534*T^4 - 2093*T^3 + 4243*T^2 + 14705*T - 5367
$43$
\( T^{7} + 8 T^{6} - 105 T^{5} + \cdots - 56737 \)
T^7 + 8*T^6 - 105*T^5 - 966*T^4 + 1786*T^3 + 28675*T^2 + 41697*T - 56737
$47$
\( T^{7} - 10 T^{6} - 121 T^{5} + \cdots + 8779 \)
T^7 - 10*T^6 - 121*T^5 + 1178*T^4 + 860*T^3 - 8383*T^2 - 3847*T + 8779
$53$
\( T^{7} + 21 T^{6} + 84 T^{5} + \cdots + 2531 \)
T^7 + 21*T^6 + 84*T^5 - 532*T^4 - 1818*T^3 + 8164*T^2 - 8370*T + 2531
$59$
\( T^{7} + 17 T^{6} - 59 T^{5} + \cdots + 48049 \)
T^7 + 17*T^6 - 59*T^5 - 2008*T^4 - 3712*T^3 + 56103*T^2 + 201232*T + 48049
$61$
\( T^{7} + 12 T^{6} - 59 T^{5} + \cdots + 7193 \)
T^7 + 12*T^6 - 59*T^5 - 628*T^4 + 295*T^3 + 8022*T^2 + 14261*T + 7193
$67$
\( T^{7} + 16 T^{6} - 104 T^{5} + \cdots - 15947 \)
T^7 + 16*T^6 - 104*T^5 - 2353*T^4 - 4203*T^3 + 45561*T^2 + 114882*T - 15947
$71$
\( T^{7} - 4 T^{6} - 82 T^{5} + 296 T^{4} + \cdots + 383 \)
T^7 - 4*T^6 - 82*T^5 + 296*T^4 + 944*T^3 - 472*T^2 - 817*T + 383
$73$
\( T^{7} - 11 T^{6} - 195 T^{5} + \cdots + 13193 \)
T^7 - 11*T^6 - 195*T^5 + 2269*T^4 - 1442*T^3 - 16466*T^2 + 7470*T + 13193
$79$
\( T^{7} + 2 T^{6} - 438 T^{5} + \cdots - 2733417 \)
T^7 + 2*T^6 - 438*T^5 - 1540*T^4 + 50284*T^3 + 242430*T^2 - 524003*T - 2733417
$83$
\( T^{7} + 4 T^{6} - 94 T^{5} - 67 T^{4} + \cdots - 219 \)
T^7 + 4*T^6 - 94*T^5 - 67*T^4 + 1507*T^3 + 479*T^2 - 1648*T - 219
$89$
\( T^{7} + 28 T^{6} + 49 T^{5} + \cdots - 363183 \)
T^7 + 28*T^6 + 49*T^5 - 3681*T^4 - 15311*T^3 + 119204*T^2 + 190442*T - 363183
$97$
\( T^{7} + 32 T^{6} + 160 T^{5} + \cdots + 867563 \)
T^7 + 32*T^6 + 160*T^5 - 3589*T^4 - 34715*T^3 - 4927*T^2 + 484652*T + 867563
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