[N,k,chi] = [450,2,Mod(19,450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(450, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("450.19");
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 \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).
\(n\)
\(101\)
\(127\)
\(\chi(n)\)
\(1\)
\(-\beta_{10}\)
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
\( T_{7}^{16} + 82 T_{7}^{14} + 2683 T_{7}^{12} + 44874 T_{7}^{10} + 407105 T_{7}^{8} + 1927704 T_{7}^{6} + 3943448 T_{7}^{4} + 1326272 T_{7}^{2} + 99856 \)
T7^16 + 82*T7^14 + 2683*T7^12 + 44874*T7^10 + 407105*T7^8 + 1927704*T7^6 + 3943448*T7^4 + 1326272*T7^2 + 99856
acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\).
$p$
$F_p(T)$
$2$
\( (T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2} \)
(T^8 - T^6 + T^4 - T^2 + 1)^2
$3$
\( T^{16} \)
T^16
$5$
\( T^{16} + 4 T^{15} - 4 T^{14} + \cdots + 390625 \)
T^16 + 4*T^15 - 4*T^14 - 6*T^13 + 161*T^12 + 260*T^11 - 580*T^10 + 520*T^9 + 7205*T^8 + 2600*T^7 - 14500*T^6 + 32500*T^5 + 100625*T^4 - 18750*T^3 - 62500*T^2 + 312500*T + 390625
$7$
\( T^{16} + 82 T^{14} + 2683 T^{12} + \cdots + 99856 \)
T^16 + 82*T^14 + 2683*T^12 + 44874*T^10 + 407105*T^8 + 1927704*T^6 + 3943448*T^4 + 1326272*T^2 + 99856
$11$
\( T^{16} + 2 T^{15} + 55 T^{14} + \cdots + 524176 \)
T^16 + 2*T^15 + 55*T^14 + 50*T^13 + 1480*T^12 + 4076*T^11 + 38552*T^10 + 189020*T^9 + 981825*T^8 + 4331970*T^7 + 13990042*T^6 + 26865184*T^5 + 30332320*T^4 + 18282200*T^3 + 5337520*T^2 + 660288*T + 524176
$13$
\( T^{16} - 20 T^{15} + 188 T^{14} + \cdots + 4096 \)
T^16 - 20*T^15 + 188*T^14 - 1060*T^13 + 4478*T^12 - 21960*T^11 + 117681*T^10 - 411280*T^9 + 751065*T^8 - 271020*T^7 - 1261864*T^6 + 889600*T^5 + 1912128*T^4 + 785920*T^3 + 50688*T^2 - 10240*T + 4096
$17$
\( T^{16} - 30 T^{15} + 432 T^{14} + \cdots + 3748096 \)
T^16 - 30*T^15 + 432*T^14 - 3780*T^13 + 21558*T^12 - 82980*T^11 + 246619*T^10 - 790760*T^9 + 3013765*T^8 - 6822760*T^7 + 5488964*T^6 + 5209920*T^5 - 12660192*T^4 + 5089920*T^3 + 7960832*T^2 - 10222080*T + 3748096
$19$
\( T^{16} + 100 T^{14} - 220 T^{13} + \cdots + 2560000 \)
T^16 + 100*T^14 - 220*T^13 + 4320*T^12 - 8000*T^11 + 84200*T^10 - 98800*T^9 + 2344400*T^8 + 2408000*T^7 + 36992000*T^6 - 128224000*T^5 + 164608000*T^4 + 192640000*T^3 + 87680000*T^2 + 15360000*T + 2560000
$23$
\( T^{16} - 10 T^{15} + \cdots + 20533743616 \)
T^16 - 10*T^15 - 62*T^14 + 480*T^13 + 5148*T^12 - 8800*T^11 - 212024*T^10 - 218480*T^9 + 5589040*T^8 + 2033280*T^7 - 20284864*T^6 - 223645440*T^5 + 2328182528*T^4 - 3091287040*T^3 - 793377792*T^2 - 7978721280*T + 20533743616
$29$
\( T^{16} - 10 T^{15} + 80 T^{14} + \cdots + 40960000 \)
T^16 - 10*T^15 + 80*T^14 - 310*T^13 + 2170*T^12 - 20700*T^11 + 227375*T^10 - 1497150*T^9 + 7936025*T^8 - 29294000*T^7 + 89374000*T^6 - 204992000*T^5 + 407328000*T^4 - 629760000*T^3 + 826880000*T^2 - 286720000*T + 40960000
$31$
\( T^{16} + 18 T^{15} + \cdots + 205176976 \)
T^16 + 18*T^15 + 245*T^14 + 2780*T^13 + 28690*T^12 + 183994*T^11 + 920972*T^10 + 2692320*T^9 + 4864705*T^8 + 4114340*T^7 + 16443812*T^6 + 46838996*T^5 + 174024880*T^4 + 223722720*T^3 + 215927560*T^2 + 167705392*T + 205176976
$37$
\( T^{16} - 20 T^{15} + 117 T^{14} + \cdots + 4096 \)
T^16 - 20*T^15 + 117*T^14 - 80*T^13 + 438*T^12 - 8340*T^11 + 15644*T^10 + 29040*T^9 - 37935*T^8 - 38560*T^7 + 882864*T^6 + 2963360*T^5 + 4014848*T^4 + 2629120*T^3 + 778752*T^2 + 71680*T + 4096
$41$
\( T^{16} + 22 T^{15} + \cdots + 78050949376 \)
T^16 + 22*T^15 + 480*T^14 + 6170*T^13 + 73150*T^12 + 673116*T^11 + 5360007*T^10 + 30973970*T^9 + 147587005*T^8 + 420300600*T^7 + 1014402852*T^6 + 1047324864*T^5 + 7925375520*T^4 + 44589326080*T^3 + 115932473600*T^2 + 143389173248*T + 78050949376
$43$
\( T^{16} + 328 T^{14} + \cdots + 15083769856 \)
T^16 + 328*T^14 + 39548*T^12 + 2205376*T^10 + 59235280*T^8 + 770631296*T^6 + 4906686208*T^4 + 14436538368*T^2 + 15083769856
$47$
\( T^{16} + \cdots + 172199901270016 \)
T^16 - 50*T^15 + 1102*T^14 - 11560*T^13 + 1008*T^12 + 1580680*T^11 - 18529776*T^10 + 26371040*T^9 + 1634548240*T^8 - 21291263360*T^7 + 128575435264*T^6 - 318740764160*T^5 - 738130995712*T^4 + 6907001753600*T^3 - 10064501357568*T^2 - 40068754186240*T + 172199901270016
$53$
\( T^{16} + 30 T^{15} + \cdots + 111534721 \)
T^16 + 30*T^15 + 443*T^14 + 2680*T^13 - 2852*T^12 - 96790*T^11 - 207569*T^10 + 1201780*T^9 + 4916665*T^8 - 1943930*T^7 - 37207539*T^6 - 67381290*T^5 + 85146708*T^4 + 379694600*T^3 + 605305833*T^2 + 332565890*T + 111534721
$59$
\( T^{16} + 20 T^{15} + \cdots + 1600000000 \)
T^16 + 20*T^15 + 125*T^14 - 100*T^13 + 18150*T^12 + 614000*T^11 + 8962500*T^10 + 80447500*T^9 + 499400625*T^8 + 2161637500*T^7 + 6565625000*T^6 + 13070000000*T^5 + 15884000000*T^4 + 2500000000*T^3 + 21200000000*T^2 - 3200000000*T + 1600000000
$61$
\( T^{16} - 12 T^{15} + 140 T^{14} + \cdots + 10137856 \)
T^16 - 12*T^15 + 140*T^14 + 240*T^13 + 11750*T^12 - 181456*T^11 + 5810137*T^10 - 18107960*T^9 + 215705265*T^8 - 357951000*T^7 + 3192436112*T^6 - 2661983904*T^5 + 17593011360*T^4 + 5456405760*T^3 + 391080000*T^2 - 131537408*T + 10137856
$67$
\( T^{16} + \cdots + 201983672713216 \)
T^16 + 50*T^15 + 1122*T^14 + 12680*T^13 + 32288*T^12 - 1052840*T^11 - 13694576*T^10 - 32153440*T^9 + 754081040*T^8 + 7429202560*T^7 + 10106302464*T^6 - 246479728640*T^5 - 1485531308032*T^4 + 493948436480*T^3 + 32731352727552*T^2 + 120791446323200*T + 201983672713216
$71$
\( T^{16} - 28 T^{15} + \cdots + 3398330023936 \)
T^16 - 28*T^15 + 540*T^14 - 7540*T^13 + 86800*T^12 - 821344*T^11 + 6474872*T^10 - 39399440*T^9 + 185863440*T^8 - 725636800*T^7 + 3207030912*T^6 - 17766322176*T^5 + 99450024960*T^4 - 425924648960*T^3 + 1330632499200*T^2 - 2682656161792*T + 3398330023936
$73$
\( T^{16} - 20 T^{15} + \cdots + 4778526048256 \)
T^16 - 20*T^15 + 88*T^14 + 3530*T^13 - 48062*T^12 - 49240*T^11 + 5643141*T^10 - 43529070*T^9 + 10209465*T^8 + 1613600120*T^7 - 4569035264*T^6 - 18177437440*T^5 + 181700284928*T^4 + 483201126400*T^3 - 294239997952*T^2 + 509246832640*T + 4778526048256
$79$
\( T^{16} + \cdots + 110872476160000 \)
T^16 + 20*T^15 + 540*T^14 + 5730*T^13 + 85070*T^12 + 547700*T^11 + 7574125*T^10 - 104550*T^9 + 533087025*T^8 - 23885000*T^7 + 75421422000*T^6 + 170478716000*T^5 - 330659872000*T^4 - 14144237440000*T^3 + 191851535040000*T^2 + 83874581760000*T + 110872476160000
$83$
\( T^{16} - 30 T^{15} + \cdots + 25573127056 \)
T^16 - 30*T^15 + 203*T^14 - 3290*T^13 + 160248*T^12 - 1899040*T^11 + 4855216*T^10 - 51058980*T^9 + 1075044065*T^8 - 3232904770*T^7 + 9475604786*T^6 - 91903706160*T^5 + 458530144868*T^4 - 1096003177360*T^3 + 1110319587208*T^2 + 339562436080*T + 25573127056
$89$
\( T^{16} + 70 T^{15} + \cdots + 36100000000 \)
T^16 + 70*T^15 + 2475*T^14 + 55800*T^13 + 896150*T^12 + 10577250*T^11 + 93500000*T^10 + 621983750*T^9 + 3211500625*T^8 + 12861537500*T^7 + 42047312500*T^6 + 115122750000*T^5 + 269750250000*T^4 + 324645000000*T^3 + 260925000000*T^2 + 127300000000*T + 36100000000
$97$
\( T^{16} + 10 T^{15} + \cdots + 91700905971481 \)
T^16 + 10*T^15 - 183*T^14 - 8290*T^13 - 24112*T^12 + 1567060*T^11 + 23851659*T^10 + 102052160*T^9 - 646205785*T^8 - 7943899040*T^7 - 9940762061*T^6 + 161774443900*T^5 + 1028722650748*T^4 - 411721888950*T^3 - 7674896673923*T^2 - 25807000202050*T + 91700905971481
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