[N,k,chi] = [418,2,Mod(115,418)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(418, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("418.115");
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 \(\left(\mathbb{Z}/418\mathbb{Z}\right)^\times\).
\(n\)
\(287\)
\(343\)
\(\chi(n)\)
\(1\)
\(\beta_{9}\)
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
\( T_{3}^{16} + 16 T_{3}^{14} + 6 T_{3}^{13} + 162 T_{3}^{12} + 232 T_{3}^{11} + 1524 T_{3}^{10} + 3660 T_{3}^{9} + 14448 T_{3}^{8} + 29096 T_{3}^{7} + 73576 T_{3}^{6} + 111184 T_{3}^{5} + 184560 T_{3}^{4} + 195936 T_{3}^{3} + \cdots + 1936 \)
T3^16 + 16*T3^14 + 6*T3^13 + 162*T3^12 + 232*T3^11 + 1524*T3^10 + 3660*T3^9 + 14448*T3^8 + 29096*T3^7 + 73576*T3^6 + 111184*T3^5 + 184560*T3^4 + 195936*T3^3 + 120080*T3^2 + 23584*T3 + 1936
acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{4} \)
(T^4 + T^3 + T^2 + T + 1)^4
$3$
\( T^{16} + 16 T^{14} + 6 T^{13} + \cdots + 1936 \)
T^16 + 16*T^14 + 6*T^13 + 162*T^12 + 232*T^11 + 1524*T^10 + 3660*T^9 + 14448*T^8 + 29096*T^7 + 73576*T^6 + 111184*T^5 + 184560*T^4 + 195936*T^3 + 120080*T^2 + 23584*T + 1936
$5$
\( T^{16} - 4 T^{15} + 18 T^{14} + \cdots + 43681 \)
T^16 - 4*T^15 + 18*T^14 - 24*T^13 + 60*T^12 - 4*T^11 + 518*T^10 - 208*T^9 + 2734*T^8 + 1578*T^7 + 8472*T^6 + 1076*T^5 + 21961*T^4 + 34006*T^3 + 112458*T^2 + 51832*T + 43681
$7$
\( T^{16} - 6 T^{15} + 50 T^{14} + \cdots + 1413721 \)
T^16 - 6*T^15 + 50*T^14 - 116*T^13 + 621*T^12 + 278*T^11 + 8388*T^10 + 19574*T^9 + 137690*T^8 + 369384*T^7 + 1406998*T^6 + 2843960*T^5 + 6400804*T^4 + 11100422*T^3 + 12656698*T^2 + 6380174*T + 1413721
$11$
\( T^{16} - 2 T^{15} - 26 T^{14} + \cdots + 214358881 \)
T^16 - 2*T^15 - 26*T^14 + 64*T^13 + 359*T^12 - 930*T^11 - 3588*T^10 + 5218*T^9 + 32453*T^8 + 57398*T^7 - 434148*T^6 - 1237830*T^5 + 5256119*T^4 + 10307264*T^3 - 46060586*T^2 - 38974342*T + 214358881
$13$
\( T^{16} + 14 T^{15} + \cdots + 1111822336 \)
T^16 + 14*T^15 + 120*T^14 + 816*T^13 + 6120*T^12 + 31664*T^11 + 151136*T^10 + 710656*T^9 + 3882368*T^8 + 14637440*T^7 + 74357504*T^6 + 299497472*T^5 + 1159764992*T^4 + 2984815616*T^3 + 5134462976*T^2 + 2016645120*T + 1111822336
$17$
\( T^{16} - 8 T^{15} + 86 T^{14} + \cdots + 18139081 \)
T^16 - 8*T^15 + 86*T^14 - 304*T^13 + 2433*T^12 - 18376*T^11 + 194068*T^10 - 1215332*T^9 + 7074754*T^8 - 28608580*T^7 + 115192438*T^6 - 320423592*T^5 + 715407248*T^4 - 488346888*T^3 + 2200953092*T^2 - 321895220*T + 18139081
$19$
\( (T^{4} - T^{3} + T^{2} - T + 1)^{4} \)
(T^4 - T^3 + T^2 - T + 1)^4
$23$
\( (T^{8} - 12 T^{7} - 34 T^{6} + 740 T^{5} + \cdots - 96331)^{2} \)
(T^8 - 12*T^7 - 34*T^6 + 740*T^5 - 729*T^4 - 11876*T^3 + 23340*T^2 + 43890*T - 96331)^2
$29$
\( T^{16} + 22 T^{15} + 308 T^{14} + \cdots + 3610000 \)
T^16 + 22*T^15 + 308*T^14 + 3438*T^13 + 32516*T^12 + 246224*T^11 + 1470744*T^10 + 6817308*T^9 + 24413528*T^8 + 65534784*T^7 + 129752568*T^6 + 170856104*T^5 + 130214696*T^4 + 31634480*T^3 + 12724400*T^2 - 12198000*T + 3610000
$31$
\( T^{16} + 24 T^{15} + 292 T^{14} + \cdots + 25361296 \)
T^16 + 24*T^15 + 292*T^14 + 1958*T^13 + 8326*T^12 + 16712*T^11 + 118828*T^10 + 1370564*T^9 + 10315624*T^8 + 29177904*T^7 + 92803272*T^6 - 2009072*T^5 + 125104960*T^4 - 241195408*T^3 + 238530592*T^2 - 72880992*T + 25361296
$37$
\( T^{16} - 6 T^{15} + \cdots + 657143694736 \)
T^16 - 6*T^15 + 152*T^14 - 1842*T^13 + 20444*T^12 - 104744*T^11 + 1730288*T^10 - 11757420*T^9 + 112038248*T^8 - 908281264*T^7 + 7369635784*T^6 - 47173433560*T^5 + 219949411048*T^4 - 638002260528*T^3 + 1095954746640*T^2 - 914571803376*T + 657143694736
$41$
\( T^{16} + 14 T^{15} + \cdots + 7265159596816 \)
T^16 + 14*T^15 + 142*T^14 + 850*T^13 + 12488*T^12 + 59180*T^11 + 826856*T^10 + 4232668*T^9 + 65158480*T^8 + 216835216*T^7 + 2936168696*T^6 - 4295873128*T^5 + 68655036872*T^4 - 245691839888*T^3 + 1275975250000*T^2 - 3268900797712*T + 7265159596816
$43$
\( (T^{8} - 28 T^{7} + 222 T^{6} + \cdots + 51869)^{2} \)
(T^8 - 28*T^7 + 222*T^6 + 426*T^5 - 15057*T^4 + 81194*T^3 - 173278*T^2 + 108260*T + 51869)^2
$47$
\( T^{16} + 38 T^{15} + \cdots + 3590526241 \)
T^16 + 38*T^15 + 746*T^14 + 9080*T^13 + 76644*T^12 + 433352*T^11 + 1691718*T^10 + 4640482*T^9 + 27447650*T^8 + 146189480*T^7 + 1080479212*T^6 + 2287191404*T^5 + 7623817261*T^4 + 18948838218*T^3 + 28661553298*T^2 + 8267659896*T + 3590526241
$53$
\( T^{16} - 22 T^{15} + \cdots + 665784193936 \)
T^16 - 22*T^15 + 294*T^14 - 2858*T^13 + 25950*T^12 - 177936*T^11 + 1017708*T^10 - 4238088*T^9 + 20482596*T^8 - 107206896*T^7 + 787775984*T^6 - 2832726120*T^5 + 11130940504*T^4 - 35833569152*T^3 + 145853644960*T^2 + 15258377200*T + 665784193936
$59$
\( T^{16} + 28 T^{15} + \cdots + 12168376422400 \)
T^16 + 28*T^15 + 544*T^14 + 8784*T^13 + 121568*T^12 + 1388064*T^11 + 13719200*T^10 + 119366016*T^9 + 912760576*T^8 + 6036181888*T^7 + 34818962176*T^6 + 174144064000*T^5 + 742878772736*T^4 + 2584322718720*T^3 + 6939240202240*T^2 + 12617113907200*T + 12168376422400
$61$
\( T^{16} - 12 T^{15} + \cdots + 22290664247401 \)
T^16 - 12*T^15 + 210*T^14 - 2850*T^13 + 39065*T^12 - 254376*T^11 + 3554392*T^10 - 27967390*T^9 + 262562650*T^8 - 2208547360*T^7 + 20262667942*T^6 - 152578930124*T^5 + 920542283440*T^4 - 3778658489900*T^3 + 10825897799710*T^2 - 19773801127388*T + 22290664247401
$67$
\( (T^{8} + 22 T^{7} - 62 T^{6} - 3864 T^{5} + \cdots + 10564)^{2} \)
(T^8 + 22*T^7 - 62*T^6 - 3864*T^5 - 12356*T^4 + 161648*T^3 + 875888*T^2 + 726568*T + 10564)^2
$71$
\( T^{16} - 16 T^{15} + \cdots + 2859602753296 \)
T^16 - 16*T^15 + 168*T^14 - 30*T^13 - 788*T^12 - 33712*T^11 + 2306204*T^10 - 7126868*T^9 + 7963888*T^8 + 328370824*T^7 + 7272969384*T^6 + 4023900936*T^5 + 311401459320*T^4 + 510063310016*T^3 + 4173393520288*T^2 + 2287802604400*T + 2859602753296
$73$
\( T^{16} - 12 T^{15} + \cdots + 1841383864576 \)
T^16 - 12*T^15 + 204*T^14 - 748*T^13 + 20980*T^12 + 16064*T^11 + 3374608*T^10 + 5875632*T^9 + 322775296*T^8 + 733728704*T^7 + 16409927424*T^6 + 48045880960*T^5 + 325435478144*T^4 + 1396099234048*T^3 + 3197055577600*T^2 + 3630833543680*T + 1841383864576
$79$
\( T^{16} - 48 T^{15} + \cdots + 52063440250000 \)
T^16 - 48*T^15 + 1260*T^14 - 20966*T^13 + 252982*T^12 - 2111224*T^11 + 13473132*T^10 - 60137896*T^9 + 432484212*T^8 - 2554530728*T^7 + 34740481408*T^6 - 70695840248*T^5 + 1124854506536*T^4 + 2181081482000*T^3 + 24175456786000*T^2 + 54157378350000*T + 52063440250000
$83$
\( T^{16} - 14 T^{15} + \cdots + 2289526801 \)
T^16 - 14*T^15 + 236*T^14 - 1494*T^13 + 9688*T^12 - 39796*T^11 + 240986*T^10 - 1690072*T^9 + 10398494*T^8 - 49468818*T^7 + 225444364*T^6 - 854725630*T^5 + 2375237233*T^4 - 4467908688*T^3 + 5693447790*T^2 - 4682024650*T + 2289526801
$89$
\( (T^{8} + 28 T^{7} + 36 T^{6} + \cdots - 31782080)^{2} \)
(T^8 + 28*T^7 + 36*T^6 - 4632*T^5 - 25328*T^4 + 247360*T^3 + 1705536*T^2 - 4265280*T - 31782080)^2
$97$
\( T^{16} + \cdots + 385924296241936 \)
T^16 + 24*T^15 + 292*T^14 + 798*T^13 + 10574*T^12 + 317776*T^11 + 9070708*T^10 + 83294020*T^9 + 579362888*T^8 + 499852976*T^7 + 36751149384*T^6 + 389243777840*T^5 + 6154682964848*T^4 + 23897170251872*T^3 + 111036185601040*T^2 + 180312093024064*T + 385924296241936
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