[N,k,chi] = [99,7,Mod(10,99)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(99, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("99.10");
S:= CuspForms(chi, 7);
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 \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).
\(n\)
\(46\)
\(56\)
\(\chi(n)\)
\(-1\)
\(1\)
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
\( T_{2}^{12} + 486T_{2}^{10} + 82401T_{2}^{8} + 6062364T_{2}^{6} + 204706260T_{2}^{4} + 2964086784T_{2}^{2} + 15081209856 \)
T2^12 + 486*T2^10 + 82401*T2^8 + 6062364*T2^6 + 204706260*T2^4 + 2964086784*T2^2 + 15081209856
acting on \(S_{7}^{\mathrm{new}}(99, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{12} + 486 T^{10} + \cdots + 15081209856 \)
T^12 + 486*T^10 + 82401*T^8 + 6062364*T^6 + 204706260*T^4 + 2964086784*T^2 + 15081209856
$3$
\( T^{12} \)
T^12
$5$
\( (T^{6} + 112 T^{5} + \cdots + 31513690000)^{2} \)
(T^6 + 112*T^5 - 64516*T^4 - 5073800*T^3 + 978439700*T^2 + 46495666000*T + 31513690000)^2
$7$
\( T^{12} + 626988 T^{10} + \cdots + 20\!\cdots\!04 \)
T^12 + 626988*T^10 + 148849140300*T^8 + 16119134118460512*T^6 + 721944741713242790448*T^4 + 6516232203569041035502272*T^2 + 2037776259271860784843043904
$11$
\( T^{12} - 3464 T^{11} + \cdots + 30\!\cdots\!61 \)
T^12 - 3464*T^11 + 8137514*T^10 - 10918677000*T^9 + 8619795607951*T^8 + 1445621685494096*T^7 - 9476672140132971092*T^6 + 2561006998775606203856*T^5 + 27052611137528462250908671*T^4 - 60706941292729417550270637000*T^3 + 80152337545641896589772932103274*T^2 - 60444729459422514951487696203166664*T + 30912680532870672635673352936887453361
$13$
\( T^{12} + 46108704 T^{10} + \cdots + 10\!\cdots\!24 \)
T^12 + 46108704*T^10 + 861554772991176*T^8 + 8292463943660710967232*T^6 + 42862371228764126225679639312*T^4 + 110407771850334596029814744677333248*T^2 + 105629668770737358248268930854769768858624
$17$
\( T^{12} + 176680836 T^{10} + \cdots + 11\!\cdots\!36 \)
T^12 + 176680836*T^10 + 11548949157795780*T^8 + 352725198703102156652352*T^6 + 5439034642266177853541388513408*T^4 + 40989357718237512219855046678296735744*T^2 + 119513494185052211524951080338756345932071936
$19$
\( T^{12} + 265613868 T^{10} + \cdots + 11\!\cdots\!36 \)
T^12 + 265613868*T^10 + 26469571933698084*T^8 + 1236759622573825872319680*T^6 + 27984318303584497227543359282304*T^4 + 297682950952588947409833346063092768768*T^2 + 1192619364458875447656919284278011893055988736
$23$
\( (T^{6} - 7652 T^{5} + \cdots + 11\!\cdots\!88)^{2} \)
(T^6 - 7652*T^5 - 573539572*T^4 + 4765697710504*T^3 + 71149265430970388*T^2 - 748056724156368317408*T + 1168739726537203428579088)^2
$29$
\( T^{12} + 3585415380 T^{10} + \cdots + 86\!\cdots\!24 \)
T^12 + 3585415380*T^10 + 5044418834310779172*T^8 + 3501913323067967388228129792*T^6 + 1224874716673626748801186324780326912*T^4 + 191196592764882117181757998916066354402426880*T^2 + 8600501207583529758803914894675794752916295488897024
$31$
\( (T^{6} + 29304 T^{5} + \cdots + 52\!\cdots\!52)^{2} \)
(T^6 + 29304*T^5 - 2415684264*T^4 - 62754051266752*T^3 + 983146456690562448*T^2 + 21889237142199802389888*T + 52190280126339920809860352)^2
$37$
\( (T^{6} + 101256 T^{5} + \cdots + 26\!\cdots\!68)^{2} \)
(T^6 + 101256*T^5 - 6757391976*T^4 - 814344696294976*T^3 + 4563007054878499728*T^2 + 1613754295975711434829440*T + 26559956551513173023853299968)^2
$41$
\( T^{12} + 38558024532 T^{10} + \cdots + 76\!\cdots\!76 \)
T^12 + 38558024532*T^10 + 539434893783225790500*T^8 + 3438890889205678257267077218560*T^6 + 10589526903972500574812110503585048052224*T^4 + 14989482169659747259368511415318710663735892901888*T^2 + 7613265377982097910339170367635256420233820194619354136576
$43$
\( T^{12} + 52055166636 T^{10} + \cdots + 61\!\cdots\!16 \)
T^12 + 52055166636*T^10 + 1048950060194934093732*T^8 + 10309502116783578578422838602176*T^6 + 50363126056370349425378472620010382975104*T^4 + 107237101749259490332542352954070527124162462404608*T^2 + 61617074878506909779132019675596779672547541039526380758016
$47$
\( (T^{6} + 258460 T^{5} + \cdots - 25\!\cdots\!96)^{2} \)
(T^6 + 258460*T^5 + 16322405960*T^4 - 799013262437768*T^3 - 121190627072347825996*T^2 - 3857731888726904528940224*T - 25753039416711963867570504896)^2
$53$
\( (T^{6} - 521096 T^{5} + \cdots - 89\!\cdots\!12)^{2} \)
(T^6 - 521096*T^5 + 97193442044*T^4 - 6763479447765272*T^3 - 69164115323603678380*T^2 + 28364120428744784934584848*T - 895298630802535081124344441712)^2
$59$
\( (T^{6} - 230504 T^{5} + \cdots + 59\!\cdots\!16)^{2} \)
(T^6 - 230504*T^5 - 31462590892*T^4 + 9615644439807136*T^3 - 458745729907957246528*T^2 - 9824887357956917607885056*T + 596607639615374857785840169216)^2
$61$
\( T^{12} + 273287036976 T^{10} + \cdots + 30\!\cdots\!44 \)
T^12 + 273287036976*T^10 + 23935690649494900733640*T^8 + 933310317813172990410874510913472*T^6 + 17201365483803612441137174271660585846003216*T^4 + 136900068235917032169784149141302178214316478435274752*T^2 + 304654622230921678514120137827371671319319615211364890124550144
$67$
\( (T^{6} - 182376 T^{5} + \cdots + 95\!\cdots\!32)^{2} \)
(T^6 - 182376*T^5 - 224874463128*T^4 + 47954587227599264*T^3 + 5527764408482479784592*T^2 - 1772494029206137412565768960*T + 95341054187903612457452177996032)^2
$71$
\( (T^{6} - 377588 T^{5} + \cdots - 79\!\cdots\!76)^{2} \)
(T^6 - 377588*T^5 - 325196291944*T^4 + 73121571634637752*T^3 + 28489895199902292871892*T^2 - 3046701291758329673882701760*T - 790409544295709029508214895530176)^2
$73$
\( T^{12} + 777459022416 T^{10} + \cdots + 25\!\cdots\!36 \)
T^12 + 777459022416*T^10 + 188605870571730979332288*T^8 + 15025565239211663487502633462228992*T^6 + 472536881284515449919971658903146860320927744*T^4 + 6015729600654604906299224412981211210329422369476902912*T^2 + 25354968119743309479643915582537631207433925515658518484685684736
$79$
\( T^{12} + 856808323212 T^{10} + \cdots + 17\!\cdots\!36 \)
T^12 + 856808323212*T^10 + 214262305230895807096716*T^8 + 16475255535051218664673065409094496*T^6 + 406311608938342255661591483749330044634469424*T^4 + 892959687252307574868000932584717102316101220047227072*T^2 + 177358308464554055675752042267323987030559899184059155629033536
$83$
\( T^{12} + 2879312245056 T^{10} + \cdots + 46\!\cdots\!44 \)
T^12 + 2879312245056*T^10 + 3207103329659486079267456*T^8 + 1787296999192055526024652574064033792*T^6 + 528194429883273416402138783326847700861763620864*T^4 + 78739886706467063020899781643732202952680729644678262816768*T^2 + 4640324093053885239054645205638371986961443154797284384493986794962944
$89$
\( (T^{6} - 1756772 T^{5} + \cdots + 48\!\cdots\!64)^{2} \)
(T^6 - 1756772*T^5 - 419088862324*T^4 + 1788004191385098208*T^3 - 398443134116354943917776*T^2 - 274080889718910466708405984832*T + 48958242904040734541752882902912064)^2
$97$
\( (T^{6} - 1185096 T^{5} + \cdots + 71\!\cdots\!28)^{2} \)
(T^6 - 1185096*T^5 - 1452301501440*T^4 + 1565252789677522112*T^3 - 103351888762945753358784*T^2 - 45822920211670681010976642048*T + 712920172135903379449251663536128)^2
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