[N,k,chi] = [471,2,Mod(1,471)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(471, 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("471.1");
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
\(3\)
\(-1\)
\(157\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + T_{2}^{11} - 20 T_{2}^{10} - 17 T_{2}^{9} + 149 T_{2}^{8} + 106 T_{2}^{7} - 500 T_{2}^{6} - 294 T_{2}^{5} + 711 T_{2}^{4} + 349 T_{2}^{3} - 290 T_{2}^{2} - 173 T_{2} - 15 \)
T2^12 + T2^11 - 20*T2^10 - 17*T2^9 + 149*T2^8 + 106*T2^7 - 500*T2^6 - 294*T2^5 + 711*T2^4 + 349*T2^3 - 290*T2^2 - 173*T2 - 15
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(471))\).
$p$
$F_p(T)$
$2$
\( T^{12} + T^{11} - 20 T^{10} - 17 T^{9} + \cdots - 15 \)
T^12 + T^11 - 20*T^10 - 17*T^9 + 149*T^8 + 106*T^7 - 500*T^6 - 294*T^5 + 711*T^4 + 349*T^3 - 290*T^2 - 173*T - 15
$3$
\( (T - 1)^{12} \)
(T - 1)^12
$5$
\( T^{12} - 4 T^{11} - 33 T^{10} + 140 T^{9} + \cdots - 400 \)
T^12 - 4*T^11 - 33*T^10 + 140*T^9 + 334*T^8 - 1590*T^7 - 1164*T^6 + 7376*T^5 + 8*T^4 - 12456*T^3 + 4748*T^2 + 2096*T - 400
$7$
\( T^{12} - 8 T^{11} - 34 T^{10} + \cdots - 37376 \)
T^12 - 8*T^11 - 34*T^10 + 404*T^9 + 9*T^8 - 6916*T^7 + 9824*T^6 + 43904*T^5 - 100224*T^4 - 60608*T^3 + 249728*T^2 - 92928*T - 37376
$11$
\( T^{12} + T^{11} - 87 T^{10} + \cdots + 393216 \)
T^12 + T^11 - 87*T^10 - 116*T^9 + 2752*T^8 + 4304*T^7 - 38464*T^6 - 65664*T^5 + 224512*T^4 + 385024*T^3 - 418816*T^2 - 499712*T + 393216
$13$
\( T^{12} - 15 T^{11} + 12 T^{10} + \cdots + 80384 \)
T^12 - 15*T^11 + 12*T^10 + 753*T^9 - 3031*T^8 - 6440*T^7 + 46816*T^6 - 17968*T^5 - 195232*T^4 + 233984*T^3 + 145152*T^2 - 297216*T + 80384
$17$
\( T^{12} - 10 T^{11} - 68 T^{10} + \cdots + 1844992 \)
T^12 - 10*T^11 - 68*T^10 + 890*T^9 + 891*T^8 - 28808*T^7 + 30944*T^6 + 386592*T^5 - 920480*T^4 - 1395072*T^3 + 6477824*T^2 - 6605824*T + 1844992
$19$
\( T^{12} - 14 T^{11} - 20 T^{10} + \cdots - 34560 \)
T^12 - 14*T^11 - 20*T^10 + 1052*T^9 - 3144*T^8 - 18896*T^7 + 106928*T^6 - 51296*T^5 - 536912*T^4 + 861920*T^3 + 133248*T^2 - 637056*T - 34560
$23$
\( T^{12} + 8 T^{11} - 132 T^{10} + \cdots - 6124 \)
T^12 + 8*T^11 - 132*T^10 - 1094*T^9 + 4813*T^8 + 44544*T^7 - 30726*T^6 - 525970*T^5 - 329132*T^4 + 533336*T^3 + 304764*T^2 + 17056*T - 6124
$29$
\( T^{12} - 5 T^{11} - 195 T^{10} + \cdots - 32768 \)
T^12 - 5*T^11 - 195*T^10 + 680*T^9 + 13704*T^8 - 23322*T^7 - 403268*T^6 - 2796*T^5 + 3718096*T^4 + 3696956*T^3 - 2304812*T^2 - 742144*T - 32768
$31$
\( T^{12} - 5 T^{11} - 215 T^{10} + \cdots - 72516928 \)
T^12 - 5*T^11 - 215*T^10 + 820*T^9 + 17820*T^8 - 40492*T^7 - 718736*T^6 + 418128*T^5 + 13547424*T^4 + 12980496*T^3 - 79315024*T^2 - 162596480*T - 72516928
$37$
\( T^{12} - 4 T^{11} - 259 T^{10} + \cdots + 111040 \)
T^12 - 4*T^11 - 259*T^10 + 554*T^9 + 23544*T^8 - 12596*T^7 - 823992*T^6 - 462624*T^5 + 8468032*T^4 + 10352704*T^3 - 7615024*T^2 - 6481888*T + 111040
$41$
\( T^{12} - 18 T^{11} + \cdots + 357250128 \)
T^12 - 18*T^11 - 151*T^10 + 4508*T^9 - 6854*T^8 - 343234*T^7 + 1837124*T^6 + 6323528*T^5 - 67041296*T^4 + 99357128*T^3 + 289782372*T^2 - 773466416*T + 357250128
$43$
\( T^{12} - 28 T^{11} + 173 T^{10} + \cdots - 11950080 \)
T^12 - 28*T^11 + 173*T^10 + 2226*T^9 - 37796*T^8 + 180320*T^7 + 130144*T^6 - 4815488*T^5 + 21037312*T^4 - 42091264*T^3 + 36248064*T^2 - 843776*T - 11950080
$47$
\( T^{12} + 3 T^{11} - 319 T^{10} + \cdots + 110166016 \)
T^12 + 3*T^11 - 319*T^10 - 764*T^9 + 34664*T^8 + 71216*T^7 - 1566976*T^6 - 3104896*T^5 + 26661120*T^4 + 56809472*T^3 - 81396736*T^2 - 120913920*T + 110166016
$53$
\( T^{12} + 18 T^{11} - 134 T^{10} + \cdots - 3297856 \)
T^12 + 18*T^11 - 134*T^10 - 3590*T^9 - 1080*T^8 + 215720*T^7 + 634620*T^6 - 3371336*T^5 - 16632964*T^4 - 12434976*T^3 + 26375776*T^2 + 29278720*T - 3297856
$59$
\( T^{12} - 337 T^{10} + 456 T^{9} + \cdots - 503760 \)
T^12 - 337*T^10 + 456*T^9 + 35262*T^8 - 83134*T^7 - 1427632*T^6 + 4570520*T^5 + 17777240*T^4 - 73368832*T^3 + 35384276*T^2 + 8992408*T - 503760
$61$
\( T^{12} - 20 T^{11} - 77 T^{10} + \cdots - 131072 \)
T^12 - 20*T^11 - 77*T^10 + 3084*T^9 - 5416*T^8 - 96048*T^7 + 94592*T^6 + 1001536*T^5 - 176256*T^4 - 3531008*T^3 - 1524480*T^2 + 1101824*T - 131072
$67$
\( T^{12} - 27 T^{11} + \cdots - 2092751872 \)
T^12 - 27*T^11 - 97*T^10 + 7840*T^9 - 37444*T^8 - 600752*T^7 + 5246592*T^6 + 3841472*T^5 - 125950720*T^4 + 134049024*T^3 + 911254784*T^2 - 888803328*T - 2092751872
$71$
\( T^{12} - 7 T^{11} - 423 T^{10} + \cdots - 27557888 \)
T^12 - 7*T^11 - 423*T^10 + 2308*T^9 + 61872*T^8 - 234016*T^7 - 3609728*T^6 + 7751296*T^5 + 64831488*T^4 - 71909376*T^3 - 238229504*T^2 + 169148416*T - 27557888
$73$
\( T^{12} - 15 T^{11} + \cdots - 21988393984 \)
T^12 - 15*T^11 - 351*T^10 + 5600*T^9 + 38420*T^8 - 738384*T^7 - 1199744*T^6 + 42036032*T^5 - 26243328*T^4 - 1045435392*T^3 + 1862306560*T^2 + 9030212608*T - 21988393984
$79$
\( T^{12} - 4 T^{11} + \cdots - 5143724032 \)
T^12 - 4*T^11 - 425*T^10 + 984*T^9 + 62872*T^8 - 133712*T^7 - 4220768*T^6 + 10922432*T^5 + 126735232*T^4 - 434096896*T^3 - 1156282880*T^2 + 5800259584*T - 5143724032
$83$
\( T^{12} + 13 T^{11} + \cdots + 4457318592 \)
T^12 + 13*T^11 - 401*T^10 - 5104*T^9 + 55180*T^8 + 679334*T^7 - 3341324*T^6 - 38716244*T^5 + 85236792*T^4 + 897017924*T^3 - 681916684*T^2 - 5004251968*T + 4457318592
$89$
\( T^{12} + 9 T^{11} - 596 T^{10} + \cdots - 743226112 \)
T^12 + 9*T^11 - 596*T^10 - 4707*T^9 + 121661*T^8 + 822868*T^7 - 9025504*T^6 - 52499088*T^5 + 103453856*T^4 + 823215040*T^3 + 587875072*T^2 - 1340143104*T - 743226112
$97$
\( T^{12} + 16 T^{11} + \cdots - 127959040 \)
T^12 + 16*T^11 - 285*T^10 - 3664*T^9 + 36888*T^8 + 236240*T^7 - 2274080*T^6 - 1492736*T^5 + 37789824*T^4 - 47622912*T^3 - 115552512*T^2 + 258746368*T - 127959040
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