[N,k,chi] = [500,2,Mod(101,500)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(500, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("500.101");
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 \(\left(\mathbb{Z}/500\mathbb{Z}\right)^\times\).
\(n\)
\(251\)
\(377\)
\(\chi(n)\)
\(1\)
\(-1 - \beta_{6} - \beta_{7} + \beta_{8}\)
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
\( T_{3}^{12} + 2 T_{3}^{11} + 8 T_{3}^{10} - 4 T_{3}^{9} + 23 T_{3}^{8} + 124 T_{3}^{7} + 637 T_{3}^{6} + 281 T_{3}^{5} + 1403 T_{3}^{4} + 1414 T_{3}^{3} + 608 T_{3}^{2} - 32 T_{3} + 16 \)
T3^12 + 2*T3^11 + 8*T3^10 - 4*T3^9 + 23*T3^8 + 124*T3^7 + 637*T3^6 + 281*T3^5 + 1403*T3^4 + 1414*T3^3 + 608*T3^2 - 32*T3 + 16
acting on \(S_{2}^{\mathrm{new}}(500, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{12} \)
T^12
$3$
\( T^{12} + 2 T^{11} + 8 T^{10} - 4 T^{9} + \cdots + 16 \)
T^12 + 2*T^11 + 8*T^10 - 4*T^9 + 23*T^8 + 124*T^7 + 637*T^6 + 281*T^5 + 1403*T^4 + 1414*T^3 + 608*T^2 - 32*T + 16
$5$
\( T^{12} \)
T^12
$7$
\( (T^{6} - T^{5} - 29 T^{4} + 18 T^{3} + \cdots - 576)^{2} \)
(T^6 - T^5 - 29*T^4 + 18*T^3 + 244*T^2 - 96*T - 576)^2
$11$
\( T^{12} + 5 T^{11} + 40 T^{10} + \cdots + 160000 \)
T^12 + 5*T^11 + 40*T^10 + 230*T^9 + 1385*T^8 + 500*T^7 + 8400*T^6 - 7600*T^5 + 104400*T^4 + 192000*T^3 + 456000*T^2 + 400000*T + 160000
$13$
\( T^{12} - 2 T^{11} + 23 T^{10} + \cdots + 32761 \)
T^12 - 2*T^11 + 23*T^10 + 89*T^9 + 103*T^8 - 539*T^7 + 12107*T^6 + 30794*T^5 + 72303*T^4 + 185586*T^3 + 646348*T^2 - 81993*T + 32761
$17$
\( T^{12} + T^{11} + 42 T^{10} + 187 T^{9} + \cdots + 1296 \)
T^12 + T^11 + 42*T^10 + 187*T^9 + 813*T^8 + 2877*T^7 + 10578*T^6 + 19087*T^5 + 28513*T^4 + 17712*T^3 + 1872*T^2 - 4104*T + 1296
$19$
\( T^{12} + 8 T^{11} + 51 T^{10} + \cdots + 10471696 \)
T^12 + 8*T^11 + 51*T^10 + 160*T^9 + 790*T^8 + 1768*T^7 + 17484*T^6 + 22528*T^5 + 192865*T^4 + 610360*T^3 + 2051256*T^2 + 252408*T + 10471696
$23$
\( T^{12} - 6 T^{11} + 82 T^{10} + \cdots + 4096 \)
T^12 - 6*T^11 + 82*T^10 - 52*T^9 + 193*T^8 - 112*T^7 + 4783*T^6 + 17153*T^5 + 43073*T^4 + 62368*T^3 + 61312*T^2 + 24064*T + 4096
$29$
\( T^{12} + 18 T^{11} + 241 T^{10} + \cdots + 1042441 \)
T^12 + 18*T^11 + 241*T^10 + 2575*T^9 + 23225*T^8 + 163173*T^7 + 898439*T^6 + 3656818*T^5 + 10864425*T^4 + 21747400*T^3 + 28549076*T^2 + 8559043*T + 1042441
$31$
\( T^{12} - 12 T^{11} + \cdots + 102495376 \)
T^12 - 12*T^11 + 106*T^10 - 490*T^9 + 2465*T^8 - 7482*T^7 + 81249*T^6 - 268957*T^5 + 1786015*T^4 - 6401690*T^3 + 25404756*T^2 - 61128712*T + 102495376
$37$
\( T^{12} + 13 T^{11} + 78 T^{10} + \cdots + 2128681 \)
T^12 + 13*T^11 + 78*T^10 + 224*T^9 + 3868*T^8 + 4861*T^7 - 13503*T^6 + 66654*T^5 + 1702343*T^4 + 969601*T^3 + 3968503*T^2 - 783483*T + 2128681
$41$
\( T^{12} + 23 T^{11} + 306 T^{10} + \cdots + 41422096 \)
T^12 + 23*T^11 + 306*T^10 + 3065*T^9 + 27345*T^8 + 191323*T^7 + 1047234*T^6 + 4395373*T^5 + 14183585*T^4 + 34225680*T^3 + 60270576*T^2 + 69109768*T + 41422096
$43$
\( (T^{6} + 25 T^{5} + 185 T^{4} + 140 T^{3} + \cdots + 6400)^{2} \)
(T^6 + 25*T^5 + 185*T^4 + 140*T^3 - 2720*T^2 - 4800*T + 6400)^2
$47$
\( T^{12} + T^{11} + 72 T^{10} - 43 T^{9} + \cdots + 256 \)
T^12 + T^11 + 72*T^10 - 43*T^9 + 2153*T^8 - 1753*T^7 + 32428*T^6 + 7327*T^5 + 371573*T^4 + 1713652*T^3 + 5123232*T^2 - 58624*T + 256
$53$
\( T^{12} + 21 T^{11} + 272 T^{10} + \cdots + 1745041 \)
T^12 + 21*T^11 + 272*T^10 + 2912*T^9 + 31798*T^8 + 199217*T^7 + 833033*T^6 + 2408462*T^5 + 5019863*T^4 + 7188377*T^3 + 7262957*T^2 + 4756921*T + 1745041
$59$
\( T^{12} - 9 T^{11} + 184 T^{10} + \cdots + 130051216 \)
T^12 - 9*T^11 + 184*T^10 + 175*T^9 - 1925*T^8 + 9311*T^7 + 382576*T^6 + 2262199*T^5 + 9441775*T^4 + 22963850*T^3 + 45751024*T^2 + 69518784*T + 130051216
$61$
\( T^{12} + 26 T^{11} + \cdots + 1661296081 \)
T^12 + 26*T^11 + 394*T^10 + 2965*T^9 + 21260*T^8 - 13034*T^7 + 902591*T^6 - 978266*T^5 + 5289020*T^4 + 15469105*T^3 + 116576914*T^2 - 201920086*T + 1661296081
$67$
\( T^{12} - 37 T^{11} + \cdots + 58285547776 \)
T^12 - 37*T^11 + 938*T^10 - 15216*T^9 + 181433*T^8 - 1541264*T^7 + 9460552*T^6 - 36747856*T^5 + 85717168*T^4 - 98554304*T^3 + 1809216448*T^2 - 14947042688*T + 58285547776
$71$
\( T^{12} - 21 T^{11} + \cdots + 5547866256 \)
T^12 - 21*T^11 + 214*T^10 - 1065*T^9 + 7505*T^8 - 128571*T^7 + 2037676*T^6 - 19746269*T^5 + 134232295*T^4 - 606724590*T^3 + 2028760704*T^2 - 3627072864*T + 5547866256
$73$
\( T^{12} + 18 T^{11} + \cdots + 4336354201 \)
T^12 + 18*T^11 + 208*T^10 + 2099*T^9 + 23028*T^8 + 119596*T^7 + 320437*T^6 + 1110834*T^5 + 27441748*T^4 + 134189611*T^3 + 837078518*T^2 + 1633895012*T + 4336354201
$79$
\( T^{12} + 24 T^{11} + \cdots + 56261942416 \)
T^12 + 24*T^11 + 489*T^10 + 5850*T^9 + 63530*T^8 + 465694*T^7 + 3783696*T^6 + 13143336*T^5 + 87184225*T^4 - 62618090*T^3 + 2106987284*T^2 + 17416353496*T + 56261942416
$83$
\( T^{12} - 46 T^{11} + 1162 T^{10} + \cdots + 36048016 \)
T^12 - 46*T^11 + 1162*T^10 - 18762*T^9 + 221843*T^8 - 1799712*T^7 + 10950283*T^6 - 42008477*T^5 + 104997143*T^4 - 169821562*T^3 + 192804192*T^2 - 120944576*T + 36048016
$89$
\( T^{12} + 2 T^{11} + 301 T^{10} + \cdots + 63744256 \)
T^12 + 2*T^11 + 301*T^10 + 3480*T^9 + 39270*T^8 + 424382*T^7 + 3872924*T^6 + 20425932*T^5 + 75037105*T^4 + 180475020*T^3 + 272354256*T^2 + 197268672*T + 63744256
$97$
\( T^{12} - 7 T^{11} + \cdots + 121807282081 \)
T^12 - 7*T^11 + 168*T^10 - 626*T^9 + 15983*T^8 - 88914*T^7 + 1886397*T^6 - 4515891*T^5 + 189957143*T^4 - 792944519*T^3 + 4873758603*T^2 - 24986252328*T + 121807282081
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