[N,k,chi] = [501,2,Mod(1,501)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(501, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("501.1");
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
\(3\)
\(-1\)
\(167\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 3T_{2}^{7} - 8T_{2}^{6} + 28T_{2}^{5} + 9T_{2}^{4} - 64T_{2}^{3} + 17T_{2}^{2} + 23T_{2} + 1 \)
T2^8 - 3*T2^7 - 8*T2^6 + 28*T2^5 + 9*T2^4 - 64*T2^3 + 17*T2^2 + 23*T2 + 1
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(501))\).
$p$
$F_p(T)$
$2$
\( T^{8} - 3 T^{7} - 8 T^{6} + 28 T^{5} + \cdots + 1 \)
T^8 - 3*T^7 - 8*T^6 + 28*T^5 + 9*T^4 - 64*T^3 + 17*T^2 + 23*T + 1
$3$
\( (T - 1)^{8} \)
(T - 1)^8
$5$
\( T^{8} - 7 T^{7} + 7 T^{6} + 50 T^{5} + \cdots - 18 \)
T^8 - 7*T^7 + 7*T^6 + 50*T^5 - 125*T^4 + 9*T^3 + 196*T^2 - 124*T - 18
$7$
\( T^{8} + 4 T^{7} - 19 T^{6} - 65 T^{5} + \cdots - 16 \)
T^8 + 4*T^7 - 19*T^6 - 65*T^5 + 63*T^4 + 255*T^3 + 84*T^2 - 48*T - 16
$11$
\( T^{8} - 13 T^{7} + 41 T^{6} + \cdots - 1596 \)
T^8 - 13*T^7 + 41*T^6 + 104*T^5 - 691*T^4 + 435*T^3 + 1804*T^2 - 1316*T - 1596
$13$
\( T^{8} - 37 T^{6} - 52 T^{5} + 320 T^{4} + \cdots + 56 \)
T^8 - 37*T^6 - 52*T^5 + 320*T^4 + 973*T^3 + 994*T^2 + 412*T + 56
$17$
\( T^{8} - 11 T^{7} - 20 T^{6} + \cdots + 7822 \)
T^8 - 11*T^7 - 20*T^6 + 626*T^5 - 1801*T^4 - 4863*T^3 + 30254*T^2 - 41000*T + 7822
$19$
\( T^{8} - 12 T^{7} + 5 T^{6} + \cdots + 4384 \)
T^8 - 12*T^7 + 5*T^6 + 298*T^5 - 288*T^4 - 2645*T^3 + 1000*T^2 + 8448*T + 4384
$23$
\( T^{8} - 7 T^{7} - 19 T^{6} + 114 T^{5} + \cdots + 384 \)
T^8 - 7*T^7 - 19*T^6 + 114*T^5 + 119*T^4 - 585*T^3 - 312*T^2 + 928*T + 384
$29$
\( T^{8} - T^{7} - 103 T^{6} + 8 T^{5} + \cdots + 88 \)
T^8 - T^7 - 103*T^6 + 8*T^5 + 1007*T^4 - 231*T^3 - 1858*T^2 + 652*T + 88
$31$
\( T^{8} + 2 T^{7} - 206 T^{6} + \cdots - 1017696 \)
T^8 + 2*T^7 - 206*T^6 - 581*T^5 + 13680*T^4 + 50249*T^3 - 275064*T^2 - 1322576*T - 1017696
$37$
\( T^{8} + 9 T^{7} - 107 T^{6} + \cdots - 42872 \)
T^8 + 9*T^7 - 107*T^6 - 894*T^5 + 2913*T^4 + 18797*T^3 - 26006*T^2 - 83724*T - 42872
$41$
\( T^{8} - 4 T^{7} - 188 T^{6} + \cdots - 859446 \)
T^8 - 4*T^7 - 188*T^6 + 793*T^5 + 10530*T^4 - 50327*T^3 - 154490*T^2 + 962494*T - 859446
$43$
\( T^{8} - 2 T^{7} - 38 T^{6} - 23 T^{5} + \cdots + 114 \)
T^8 - 2*T^7 - 38*T^6 - 23*T^5 + 400*T^4 + 1137*T^3 + 1258*T^2 + 622*T + 114
$47$
\( T^{8} - 17 T^{7} - 33 T^{6} + \cdots - 68732 \)
T^8 - 17*T^7 - 33*T^6 + 1811*T^5 - 6384*T^4 - 26255*T^3 + 111504*T^2 + 89328*T - 68732
$53$
\( T^{8} - 9 T^{7} - 53 T^{6} + \cdots + 20326 \)
T^8 - 9*T^7 - 53*T^6 + 407*T^5 + 1042*T^4 - 4463*T^3 - 10334*T^2 + 8186*T + 20326
$59$
\( T^{8} - 29 T^{7} + 269 T^{6} + \cdots - 21056 \)
T^8 - 29*T^7 + 269*T^6 - 459*T^5 - 5490*T^4 + 20601*T^3 + 23524*T^2 - 122192*T - 21056
$61$
\( T^{8} + 12 T^{7} - 99 T^{6} + \cdots + 51244 \)
T^8 + 12*T^7 - 99*T^6 - 1028*T^5 + 4848*T^4 + 23067*T^3 - 106134*T^2 + 24192*T + 51244
$67$
\( T^{8} - 177 T^{6} - 320 T^{5} + \cdots - 1226366 \)
T^8 - 177*T^6 - 320*T^5 + 10402*T^4 + 36139*T^3 - 178430*T^2 - 1010722*T - 1226366
$71$
\( T^{8} - 13 T^{7} - 34 T^{6} + \cdots + 32816 \)
T^8 - 13*T^7 - 34*T^6 + 1345*T^5 - 8371*T^4 + 21115*T^3 - 14648*T^2 - 24360*T + 32816
$73$
\( T^{8} + 20 T^{7} - 169 T^{6} + \cdots - 9386184 \)
T^8 + 20*T^7 - 169*T^6 - 5200*T^5 - 8468*T^4 + 283203*T^3 + 816678*T^2 - 4503628*T - 9386184
$79$
\( T^{8} - 8 T^{7} - 464 T^{6} + \cdots + 39988958 \)
T^8 - 8*T^7 - 464*T^6 + 3851*T^5 + 67944*T^4 - 541873*T^3 - 3540828*T^2 + 23441222*T + 39988958
$83$
\( T^{8} - 33 T^{7} + 257 T^{6} + \cdots - 2562224 \)
T^8 - 33*T^7 + 257*T^6 + 1629*T^5 - 25110*T^4 + 21827*T^3 + 485892*T^2 - 656928*T - 2562224
$89$
\( T^{8} - 4 T^{7} - 199 T^{6} + \cdots - 216312 \)
T^8 - 4*T^7 - 199*T^6 + 398*T^5 + 12156*T^4 + 561*T^3 - 188850*T^2 - 393620*T - 216312
$97$
\( T^{8} + 31 T^{7} + 50 T^{6} + \cdots - 24584 \)
T^8 + 31*T^7 + 50*T^6 - 5065*T^5 - 26723*T^4 + 124853*T^3 + 547194*T^2 - 1293780*T - 24584
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