[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
\( T_{2}^{13} + 4 T_{2}^{12} - 11 T_{2}^{11} - 55 T_{2}^{10} + 32 T_{2}^{9} + 266 T_{2}^{8} + 13 T_{2}^{7} - 534 T_{2}^{6} - 141 T_{2}^{5} + 404 T_{2}^{4} + 98 T_{2}^{3} - 118 T_{2}^{2} - 16 T_{2} + 11 \)
T2^13 + 4*T2^12 - 11*T2^11 - 55*T2^10 + 32*T2^9 + 266*T2^8 + 13*T2^7 - 534*T2^6 - 141*T2^5 + 404*T2^4 + 98*T2^3 - 118*T2^2 - 16*T2 + 11
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).
$p$
$F_p(T)$
$2$
\( T^{13} + 4 T^{12} - 11 T^{11} - 55 T^{10} + \cdots + 11 \)
T^13 + 4*T^12 - 11*T^11 - 55*T^10 + 32*T^9 + 266*T^8 + 13*T^7 - 534*T^6 - 141*T^5 + 404*T^4 + 98*T^3 - 118*T^2 - 16*T + 11
$3$
\( T^{13} \)
T^13
$5$
\( T^{13} + 7 T^{12} - 13 T^{11} - 170 T^{10} + \cdots + 236 \)
T^13 + 7*T^12 - 13*T^11 - 170*T^10 - 56*T^9 + 1437*T^8 + 1268*T^7 - 5209*T^6 - 4779*T^5 + 8285*T^4 + 5179*T^3 - 4529*T^2 - 212*T + 236
$7$
\( T^{13} - 5 T^{12} - 37 T^{11} + 209 T^{10} + \cdots + 244 \)
T^13 - 5*T^12 - 37*T^11 + 209*T^10 + 371*T^9 - 2916*T^8 + 108*T^7 + 15173*T^6 - 13240*T^5 - 19654*T^4 + 22048*T^3 + 6141*T^2 - 8384*T + 244
$11$
\( (T - 1)^{13} \)
(T - 1)^13
$13$
\( T^{13} + 9 T^{12} - 43 T^{11} + \cdots - 7972 \)
T^13 + 9*T^12 - 43*T^11 - 507*T^10 + 281*T^9 + 8651*T^8 + 4560*T^7 - 52975*T^6 - 63406*T^5 + 92168*T^4 + 192435*T^3 + 86755*T^2 - 6732*T - 7972
$17$
\( T^{13} + 7 T^{12} - 49 T^{11} + \cdots + 20968 \)
T^13 + 7*T^12 - 49*T^11 - 294*T^10 + 1138*T^9 + 4084*T^8 - 13150*T^7 - 19689*T^6 + 56369*T^5 + 45549*T^4 - 88972*T^3 - 53391*T^2 + 39604*T + 20968
$19$
\( T^{13} - 2 T^{12} - 89 T^{11} + \cdots + 69050 \)
T^13 - 2*T^12 - 89*T^11 + 172*T^10 + 2484*T^9 - 4872*T^8 - 24751*T^7 + 64036*T^6 + 60047*T^5 - 306012*T^4 + 249632*T^3 + 89693*T^2 - 199390*T + 69050
$23$
\( T^{13} + 23 T^{12} + 106 T^{11} + \cdots - 50081152 \)
T^13 + 23*T^12 + 106*T^11 - 1136*T^10 - 10239*T^9 + 8411*T^8 + 269346*T^7 + 309009*T^6 - 2903444*T^5 - 5362050*T^4 + 13589065*T^3 + 28682557*T^2 - 22796208*T - 50081152
$29$
\( T^{13} + 16 T^{12} - 38 T^{11} + \cdots - 704192 \)
T^13 + 16*T^12 - 38*T^11 - 1662*T^10 - 3904*T^9 + 54045*T^8 + 246071*T^7 - 473877*T^6 - 4269542*T^5 - 3644654*T^4 + 20257168*T^3 + 47964633*T^2 + 29919840*T - 704192
$31$
\( T^{13} - 9 T^{12} - 159 T^{11} + \cdots + 1383602 \)
T^13 - 9*T^12 - 159*T^11 + 1442*T^10 + 7140*T^9 - 77340*T^8 - 27939*T^7 + 1408876*T^6 - 2914254*T^5 - 1865731*T^4 + 9910215*T^3 - 6661547*T^2 - 1124356*T + 1383602
$37$
\( T^{13} - 14 T^{12} - 66 T^{11} + \cdots + 288220 \)
T^13 - 14*T^12 - 66*T^11 + 1455*T^10 - 91*T^9 - 50763*T^8 + 79815*T^7 + 642402*T^6 - 1559947*T^5 - 1707459*T^4 + 7036129*T^3 - 4193181*T^2 - 1071200*T + 288220
$41$
\( T^{13} + 19 T^{12} + 37 T^{11} + \cdots - 1984822 \)
T^13 + 19*T^12 + 37*T^11 - 1270*T^10 - 7182*T^9 + 17183*T^8 + 182756*T^7 + 42767*T^6 - 1672031*T^5 - 1804903*T^4 + 5090059*T^3 + 7901717*T^2 - 174432*T - 1984822
$43$
\( T^{13} - 7 T^{12} - 148 T^{11} + \cdots + 92060 \)
T^13 - 7*T^12 - 148*T^11 + 482*T^10 + 7507*T^9 - 4678*T^8 - 142282*T^7 - 141879*T^6 + 755309*T^5 + 1240103*T^4 - 637850*T^3 - 1488077*T^2 - 225880*T + 92060
$47$
\( T^{13} + 26 T^{12} + 39 T^{11} + \cdots - 232556 \)
T^13 + 26*T^12 + 39*T^11 - 3292*T^10 - 12194*T^9 + 171203*T^8 + 464126*T^7 - 4369904*T^6 - 1474107*T^5 + 33292256*T^4 - 33971318*T^3 + 9721045*T^2 + 117340*T - 232556
$53$
\( T^{13} + 18 T^{12} + \cdots + 2168121694 \)
T^13 + 18*T^12 - 292*T^11 - 6289*T^10 + 20913*T^9 + 681460*T^8 - 743882*T^7 - 34832120*T^6 + 32022793*T^5 + 889019769*T^4 - 1288023135*T^3 - 9235216529*T^2 + 19878418100*T + 2168121694
$59$
\( T^{13} + 31 T^{12} + \cdots + 306402637808 \)
T^13 + 31*T^12 - 9*T^11 - 10231*T^10 - 104438*T^9 + 606012*T^8 + 15805589*T^7 + 62521825*T^6 - 455047406*T^5 - 4438409915*T^4 - 6222639173*T^3 + 57988397319*T^2 + 254368316344*T + 306402637808
$61$
\( (T + 1)^{13} \)
(T + 1)^13
$67$
\( T^{13} - 14 T^{12} + \cdots + 22880121982 \)
T^13 - 14*T^12 - 409*T^11 + 5869*T^10 + 56359*T^9 - 851301*T^8 - 3240867*T^7 + 51889732*T^6 + 93797949*T^5 - 1290850357*T^4 - 2429645224*T^3 + 11048448033*T^2 + 32966700610*T + 22880121982
$71$
\( T^{13} + 37 T^{12} + \cdots + 3379107441064 \)
T^13 + 37*T^12 - 22*T^11 - 16778*T^10 - 169726*T^9 + 2108879*T^8 + 42645083*T^7 + 40643485*T^6 - 3460155876*T^5 - 21933364033*T^4 + 46035465608*T^3 + 918729907841*T^2 + 3212513453388*T + 3379107441064
$73$
\( T^{13} + 16 T^{12} - 319 T^{11} + \cdots - 41176692 \)
T^13 + 16*T^12 - 319*T^11 - 5488*T^10 + 26137*T^9 + 564544*T^8 - 152321*T^7 - 18415707*T^6 - 24190822*T^5 + 127738776*T^4 + 143625801*T^3 - 187193761*T^2 - 238820064*T - 41176692
$79$
\( T^{13} + 17 T^{12} + \cdots + 381289224964 \)
T^13 + 17*T^12 - 405*T^11 - 8106*T^10 + 47740*T^9 + 1361598*T^8 - 819855*T^7 - 99731470*T^6 - 121354995*T^5 + 3478297556*T^4 + 5437574287*T^3 - 58082770427*T^2 - 62037455936*T + 381289224964
$83$
\( T^{13} + 30 T^{12} + \cdots - 49887527848 \)
T^13 + 30*T^12 - 105*T^11 - 11422*T^10 - 75183*T^9 + 1238182*T^8 + 16175045*T^7 - 4312708*T^6 - 897057769*T^5 - 4125441394*T^4 + 1033807971*T^3 + 38173364627*T^2 + 40253448852*T - 49887527848
$89$
\( T^{13} + 35 T^{12} + \cdots + 820098746 \)
T^13 + 35*T^12 + 27*T^11 - 11711*T^10 - 127329*T^9 + 495590*T^8 + 15681452*T^7 + 89439437*T^6 + 102765412*T^5 - 576371082*T^4 - 1317518276*T^3 + 840347053*T^2 + 2261080550*T + 820098746
$97$
\( T^{13} + T^{12} - 655 T^{11} + \cdots - 3579851464 \)
T^13 + T^12 - 655*T^11 - 1643*T^10 + 145570*T^9 + 381094*T^8 - 14077117*T^7 - 25635811*T^6 + 596279356*T^5 + 196398526*T^4 - 8467394524*T^3 + 15274069775*T^2 - 4774044276*T - 3579851464
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