[N,k,chi] = [684,2,Mod(73,684)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(684, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("684.73");
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
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).
\(n\)
\(325\)
\(343\)
\(533\)
\(\chi(n)\)
\(-\beta_{4} + \beta_{6}\)
\(1\)
\(1\)
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
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 9 T_{5}^{10} + 19 T_{5}^{9} - 72 T_{5}^{8} - 36 T_{5}^{7} + 64 T_{5}^{6} - 1251 T_{5}^{5} + 7398 T_{5}^{4} - 9747 T_{5}^{3} + 4131 T_{5}^{2} + 2916 T_{5} + 729 \)
T5^12 + 9*T5^10 + 19*T5^9 - 72*T5^8 - 36*T5^7 + 64*T5^6 - 1251*T5^5 + 7398*T5^4 - 9747*T5^3 + 4131*T5^2 + 2916*T5 + 729
acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{12} \)
T^12
$3$
\( T^{12} \)
T^12
$5$
\( T^{12} + 9 T^{10} + 19 T^{9} - 72 T^{8} + \cdots + 729 \)
T^12 + 9*T^10 + 19*T^9 - 72*T^8 - 36*T^7 + 64*T^6 - 1251*T^5 + 7398*T^4 - 9747*T^3 + 4131*T^2 + 2916*T + 729
$7$
\( T^{12} - 3 T^{11} + 21 T^{10} - 22 T^{9} + \cdots + 9 \)
T^12 - 3*T^11 + 21*T^10 - 22*T^9 + 201*T^8 - 186*T^7 + 1141*T^6 + 447*T^5 + 1386*T^4 - 366*T^3 + 414*T^2 + 54*T + 9
$11$
\( T^{12} + 3 T^{11} + 33 T^{10} - 2 T^{9} + \cdots + 729 \)
T^12 + 3*T^11 + 33*T^10 - 2*T^9 + 591*T^8 + 300*T^7 + 3331*T^6 - 3231*T^5 + 7452*T^4 - 1890*T^3 + 2430*T^2 + 729
$13$
\( T^{12} + 9 T^{11} + 45 T^{10} + \cdots + 130321 \)
T^12 + 9*T^11 + 45*T^10 + 268*T^9 + 1377*T^8 + 4077*T^7 + 20109*T^6 + 152361*T^5 + 639711*T^4 + 1178665*T^3 + 948708*T^2 + 308655*T + 130321
$17$
\( T^{12} - 3 T^{11} + 9 T^{10} + \cdots + 8982009 \)
T^12 - 3*T^11 + 9*T^10 + 30*T^9 - 1179*T^8 - 999*T^7 + 39861*T^6 + 136809*T^5 + 653913*T^4 + 2532789*T^3 + 5143824*T^2 - 10924065*T + 8982009
$19$
\( T^{12} + 12 T^{11} + 33 T^{10} + \cdots + 47045881 \)
T^12 + 12*T^11 + 33*T^10 - 201*T^9 - 1365*T^8 + 627*T^7 + 24358*T^6 + 11913*T^5 - 492765*T^4 - 1378659*T^3 + 4300593*T^2 + 29713188*T + 47045881
$23$
\( T^{12} - 12 T^{11} + \cdots + 151585344 \)
T^12 - 12*T^11 + 69*T^10 - 251*T^9 + 714*T^8 + 3729*T^7 - 40985*T^6 + 36522*T^5 + 856980*T^4 - 4777056*T^3 + 22161600*T^2 - 75792672*T + 151585344
$29$
\( T^{12} + 27 T^{11} + 369 T^{10} + \cdots + 263169 \)
T^12 + 27*T^11 + 369*T^10 + 3023*T^9 + 15840*T^8 + 41454*T^7 - 54107*T^6 - 832302*T^5 - 1363392*T^4 + 6261165*T^3 + 19768293*T^2 + 872613*T + 263169
$31$
\( T^{12} - 6 T^{11} + 126 T^{10} + \cdots + 130321 \)
T^12 - 6*T^11 + 126*T^10 - 90*T^9 + 8508*T^8 - 14955*T^7 + 205549*T^6 - 321024*T^5 + 3546369*T^4 - 6277068*T^3 + 19798323*T^2 + 1584429*T + 130321
$37$
\( (T^{6} + 6 T^{5} - 81 T^{4} - 639 T^{3} + \cdots + 11096)^{2} \)
(T^6 + 6*T^5 - 81*T^4 - 639*T^3 + 78*T^2 + 7836*T + 11096)^2
$41$
\( T^{12} + 3 T^{11} - 21 T^{10} + \cdots + 360278361 \)
T^12 + 3*T^11 - 21*T^10 - 910*T^9 + 6492*T^8 - 44274*T^7 + 492832*T^6 - 1806273*T^5 + 3831138*T^4 - 21232152*T^3 + 23782896*T^2 + 150671178*T + 360278361
$43$
\( T^{12} - 27 T^{11} + 309 T^{10} + \cdots + 1203409 \)
T^12 - 27*T^11 + 309*T^10 - 2142*T^9 + 11874*T^8 - 50418*T^7 + 137242*T^6 - 370035*T^5 + 771306*T^4 + 1881450*T^3 + 18911724*T^2 - 7325766*T + 1203409
$47$
\( T^{12} - 15 T^{11} - 51 T^{10} + \cdots + 729 \)
T^12 - 15*T^11 - 51*T^10 + 460*T^9 + 32919*T^8 - 417405*T^7 + 1894231*T^6 - 3504195*T^5 + 3161673*T^4 - 812565*T^3 + 1256796*T^2 - 10935*T + 729
$53$
\( T^{12} - 21 T^{11} + 177 T^{10} + \cdots + 95004009 \)
T^12 - 21*T^11 + 177*T^10 - 890*T^9 + 3522*T^8 - 4470*T^7 + 42652*T^6 - 666045*T^5 + 1156302*T^4 + 1832436*T^3 + 34913754*T^2 + 80003376*T + 95004009
$59$
\( T^{12} - 48 T^{11} + \cdots + 12621848409 \)
T^12 - 48*T^11 + 1107*T^10 - 16278*T^9 + 183636*T^8 - 1690956*T^7 + 11066715*T^6 - 40087386*T^5 + 70948224*T^4 - 282191040*T^3 + 812641086*T^2 + 3266938413*T + 12621848409
$61$
\( T^{12} + 6 T^{11} + \cdots + 1418049649 \)
T^12 + 6*T^11 - 141*T^10 - 560*T^9 + 8838*T^8 - 18414*T^7 + 449985*T^6 + 1422378*T^5 + 9943830*T^4 - 39761552*T^3 + 276693828*T^2 - 727194327*T + 1418049649
$67$
\( T^{12} - 24 T^{11} + \cdots + 6080256576 \)
T^12 - 24*T^11 + 483*T^10 - 6139*T^9 + 59610*T^8 - 406245*T^7 + 2284291*T^6 - 4719942*T^5 - 32415444*T^4 + 205544736*T^3 + 19649952*T^2 - 2720114784*T + 6080256576
$71$
\( T^{12} + 45 T^{10} + 487 T^{9} + \cdots + 16842816 \)
T^12 + 45*T^10 + 487*T^9 + 8442*T^8 + 34011*T^7 + 1016551*T^6 + 5907222*T^5 + 13286916*T^4 + 17404416*T^3 + 19478880*T^2 - 43879968*T + 16842816
$73$
\( T^{12} - 30 T^{11} + 375 T^{10} + \cdots + 36864 \)
T^12 - 30*T^11 + 375*T^10 - 2947*T^9 + 21024*T^8 - 109335*T^7 + 259069*T^6 - 66444*T^5 + 211536*T^4 + 89088*T^3 + 78336*T^2 - 82944*T + 36864
$79$
\( T^{12} - 3 T^{11} + \cdots + 1548343801 \)
T^12 - 3*T^11 + 39*T^10 + 259*T^9 - 7938*T^8 + 10548*T^7 + 330321*T^6 - 5080752*T^5 + 80469864*T^4 - 476164415*T^3 + 1473881775*T^2 - 2218221177*T + 1548343801
$83$
\( T^{12} + 3 T^{11} + \cdots + 11939714361 \)
T^12 + 3*T^11 + 189*T^10 + 114*T^9 + 24957*T^8 + 14472*T^7 + 1386243*T^6 - 866295*T^5 + 53308368*T^4 - 19603782*T^3 + 958378392*T^2 - 672659964*T + 11939714361
$89$
\( T^{12} - 18 T^{11} + \cdots + 7695324729 \)
T^12 - 18*T^11 + 117*T^10 - 2205*T^9 + 26244*T^8 + 24948*T^7 + 1486998*T^6 - 37993293*T^5 + 125240742*T^4 + 321329349*T^3 + 1314529155*T^2 + 2430102546*T + 7695324729
$97$
\( T^{12} - 12 T^{11} + \cdots + 16983563041 \)
T^12 - 12*T^11 - 39*T^10 + 2124*T^9 - 19416*T^8 - 14412*T^7 + 4516399*T^6 - 66305706*T^5 + 595015248*T^4 - 3870585684*T^3 + 24477528468*T^2 + 31607403735*T + 16983563041
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