[N,k,chi] = [6039,2,Mod(1,6039)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6039, 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("6039.1");
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
\(3\)
\(-1\)
\(11\)
\(1\)
\(61\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{13} + 2 T_{2}^{12} - 19 T_{2}^{11} - 35 T_{2}^{10} + 136 T_{2}^{9} + 220 T_{2}^{8} - 469 T_{2}^{7} - 610 T_{2}^{6} + 841 T_{2}^{5} + 760 T_{2}^{4} - 742 T_{2}^{3} - 366 T_{2}^{2} + 236 T_{2} + 47 \)
T2^13 + 2*T2^12 - 19*T2^11 - 35*T2^10 + 136*T2^9 + 220*T2^8 - 469*T2^7 - 610*T2^6 + 841*T2^5 + 760*T2^4 - 742*T2^3 - 366*T2^2 + 236*T2 + 47
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).
$p$
$F_p(T)$
$2$
\( T^{13} + 2 T^{12} - 19 T^{11} - 35 T^{10} + \cdots + 47 \)
T^13 + 2*T^12 - 19*T^11 - 35*T^10 + 136*T^9 + 220*T^8 - 469*T^7 - 610*T^6 + 841*T^5 + 760*T^4 - 742*T^3 - 366*T^2 + 236*T + 47
$3$
\( T^{13} \)
T^13
$5$
\( T^{13} + 3 T^{12} - 37 T^{11} - 104 T^{10} + \cdots + 292 \)
T^13 + 3*T^12 - 37*T^11 - 104*T^10 + 476*T^9 + 1285*T^8 - 2404*T^7 - 6799*T^6 + 3331*T^5 + 13959*T^4 + 4233*T^3 - 4711*T^2 - 1676*T + 292
$7$
\( T^{13} - 11 T^{12} + 11 T^{11} + \cdots - 148 \)
T^13 - 11*T^12 + 11*T^11 + 253*T^10 - 833*T^9 - 914*T^8 + 7318*T^7 - 6921*T^6 - 12042*T^5 + 26514*T^4 - 15918*T^3 + 2331*T^2 + 640*T - 148
$11$
\( (T + 1)^{13} \)
(T + 1)^13
$13$
\( T^{13} - 13 T^{12} - 3 T^{11} + 657 T^{10} + \cdots + 388 \)
T^13 - 13*T^12 - 3*T^11 + 657*T^10 - 1911*T^9 - 9259*T^8 + 45598*T^7 + 6341*T^6 - 270596*T^5 + 388940*T^4 - 219377*T^3 + 59385*T^2 - 7724*T + 388
$17$
\( T^{13} + 17 T^{12} + 51 T^{11} + \cdots - 4744 \)
T^13 + 17*T^12 + 51*T^11 - 542*T^10 - 3254*T^9 + 2972*T^8 + 50352*T^7 + 48021*T^6 - 264249*T^5 - 460055*T^4 + 285500*T^3 + 585821*T^2 - 219092*T - 4744
$19$
\( T^{13} - 14 T^{12} - 43 T^{11} + \cdots - 10215742 \)
T^13 - 14*T^12 - 43*T^11 + 1384*T^10 - 3540*T^9 - 34742*T^8 + 183151*T^7 + 103356*T^6 - 2358937*T^5 + 3999180*T^4 + 5891140*T^3 - 25651789*T^2 + 28256994*T - 10215742
$23$
\( T^{13} + 7 T^{12} - 126 T^{11} + \cdots + 1840672 \)
T^13 + 7*T^12 - 126*T^11 - 1010*T^10 + 4119*T^9 + 46069*T^8 - 904*T^7 - 803085*T^6 - 1675438*T^5 + 3384342*T^4 + 17153801*T^3 + 24183107*T^2 + 13371112*T + 1840672
$29$
\( T^{13} - 6 T^{12} - 130 T^{11} + \cdots - 11772848 \)
T^13 - 6*T^12 - 130*T^11 + 748*T^10 + 6194*T^9 - 34829*T^8 - 125575*T^7 + 735603*T^6 + 832462*T^5 - 6571640*T^4 + 1847092*T^3 + 15263265*T^2 - 5620360*T - 11772848
$31$
\( T^{13} - 27 T^{12} + 197 T^{11} + \cdots - 1170458 \)
T^13 - 27*T^12 + 197*T^11 + 918*T^10 - 20540*T^9 + 92148*T^8 + 137117*T^7 - 2929198*T^6 + 12849900*T^5 - 29312543*T^4 + 38175799*T^3 - 27561335*T^2 + 9710684*T - 1170458
$37$
\( T^{13} - 10 T^{12} - 194 T^{11} + \cdots + 76337468 \)
T^13 - 10*T^12 - 194*T^11 + 2055*T^10 + 11795*T^9 - 142157*T^8 - 204317*T^7 + 3792266*T^6 - 1573195*T^5 - 34929945*T^4 + 46446297*T^3 + 76582459*T^2 - 166492240*T + 76337468
$41$
\( T^{13} + 3 T^{12} - 207 T^{11} + \cdots + 98100918 \)
T^13 + 3*T^12 - 207*T^11 - 652*T^10 + 15082*T^9 + 46771*T^8 - 472838*T^7 - 1287999*T^6 + 6642759*T^5 + 14299487*T^4 - 35584767*T^3 - 64026067*T^2 + 59989368*T + 98100918
$43$
\( T^{13} - 29 T^{12} + \cdots - 195974572 \)
T^13 - 29*T^12 + 198*T^11 + 1904*T^10 - 32073*T^9 + 100152*T^8 + 598450*T^7 - 4333069*T^6 + 2674445*T^5 + 39129841*T^4 - 83579412*T^3 - 63363431*T^2 + 295981576*T - 195974572
$47$
\( T^{13} + 8 T^{12} - 229 T^{11} + \cdots + 26421804 \)
T^13 + 8*T^12 - 229*T^11 - 1806*T^10 + 17270*T^9 + 140009*T^8 - 425660*T^7 - 4095562*T^6 - 476247*T^5 + 29338618*T^4 + 17567742*T^3 - 58833941*T^2 - 15164268*T + 26421804
$53$
\( T^{13} - 24 T^{12} + \cdots - 93111825698 \)
T^13 - 24*T^12 - 214*T^11 + 9653*T^10 - 32735*T^9 - 1152358*T^8 + 10341350*T^7 + 26246078*T^6 - 670996989*T^5 + 2243534229*T^4 + 6441993519*T^3 - 58108058743*T^2 + 130419884528*T - 93111825698
$59$
\( T^{13} + 13 T^{12} - 219 T^{11} + \cdots - 7051312 \)
T^13 + 13*T^12 - 219*T^11 - 3625*T^10 + 4474*T^9 + 251338*T^8 + 752403*T^7 - 3423411*T^6 - 17909350*T^5 - 11841571*T^4 + 29674727*T^3 + 26215377*T^2 - 11441496*T - 7051312
$61$
\( (T + 1)^{13} \)
(T + 1)^13
$67$
\( T^{13} - 44 T^{12} + \cdots - 6958847006 \)
T^13 - 44*T^12 + 317*T^11 + 11183*T^10 - 172601*T^9 - 535019*T^8 + 20091793*T^7 - 25944476*T^6 - 967935121*T^5 + 2313009717*T^4 + 20936383426*T^3 - 39708587867*T^2 - 169477916206*T - 6958847006
$71$
\( T^{13} + 3 T^{12} - 438 T^{11} + \cdots + 48814344 \)
T^13 + 3*T^12 - 438*T^11 - 794*T^10 + 66344*T^9 + 58883*T^8 - 4096969*T^7 - 1593405*T^6 + 88612272*T^5 + 66765769*T^4 - 431661224*T^3 - 512125217*T^2 + 26962908*T + 48814344
$73$
\( T^{13} - 48 T^{12} + \cdots + 28332933844 \)
T^13 - 48*T^12 + 573*T^11 + 7168*T^10 - 222709*T^9 + 1385794*T^8 + 7918583*T^7 - 130311429*T^6 + 388567146*T^5 + 1189424914*T^4 - 6331127315*T^3 - 4537752459*T^2 + 28117420384*T + 28332933844
$79$
\( T^{13} + 17 T^{12} + \cdots + 388386798444 \)
T^13 + 17*T^12 - 485*T^11 - 8586*T^10 + 90102*T^9 + 1660310*T^8 - 8092221*T^7 - 152293306*T^6 + 376370987*T^5 + 6617813672*T^4 - 9729487411*T^3 - 116113531441*T^2 + 131453923248*T + 388386798444
$83$
\( T^{13} + 50 T^{12} + \cdots - 58130319032 \)
T^13 + 50*T^12 + 719*T^11 - 3026*T^10 - 160405*T^9 - 849124*T^8 + 9250209*T^7 + 97247168*T^6 - 45758771*T^5 - 3078005106*T^4 - 6625979573*T^3 + 19852316459*T^2 + 42617697212*T - 58130319032
$89$
\( T^{13} - 15 T^{12} + \cdots + 120747458 \)
T^13 - 15*T^12 - 399*T^11 + 7027*T^10 + 36857*T^9 - 1010888*T^8 + 1219256*T^7 + 46122395*T^6 - 176988322*T^5 - 474338600*T^4 + 3540622044*T^3 - 5483550149*T^2 + 1450031882*T + 120747458
$97$
\( T^{13} - 27 T^{12} + \cdots - 87087118744 \)
T^13 - 27*T^12 - 415*T^11 + 15357*T^10 + 25546*T^9 - 2997530*T^8 + 7345679*T^7 + 238685361*T^6 - 996124072*T^5 - 6856977798*T^4 + 35031639796*T^3 + 22218327363*T^2 - 145098261164*T - 87087118744
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