[N,k,chi] = [121,5,Mod(40,121)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([7]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("121.40");
S:= CuspForms(chi, 5);
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 \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).
\(n\)
\(2\)
\(\chi(n)\)
\(-\beta_{3}\)
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
\( T_{2}^{12} - 15 T_{2}^{11} + 75 T_{2}^{10} + 5 T_{2}^{9} - 1305 T_{2}^{8} + 1950 T_{2}^{7} + 21730 T_{2}^{6} - 10180 T_{2}^{5} - 172260 T_{2}^{4} + 27560 T_{2}^{3} + 6145160 T_{2}^{2} - 18581200 T_{2} + 16272080 \)
T2^12 - 15*T2^11 + 75*T2^10 + 5*T2^9 - 1305*T2^8 + 1950*T2^7 + 21730*T2^6 - 10180*T2^5 - 172260*T2^4 + 27560*T2^3 + 6145160*T2^2 - 18581200*T2 + 16272080
acting on \(S_{5}^{\mathrm{new}}(121, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{12} - 15 T^{11} + 75 T^{10} + \cdots + 16272080 \)
T^12 - 15*T^11 + 75*T^10 + 5*T^9 - 1305*T^8 + 1950*T^7 + 21730*T^6 - 10180*T^5 - 172260*T^4 + 27560*T^3 + 6145160*T^2 - 18581200*T + 16272080
$3$
\( T^{12} - 19 T^{11} + \cdots + 124779204081 \)
T^12 - 19*T^11 + 424*T^10 - 6300*T^9 + 91915*T^8 - 722364*T^7 + 5271201*T^6 - 390501*T^5 + 308782125*T^4 - 5287311855*T^3 + 37609533549*T^2 - 58350467826*T + 124779204081
$5$
\( T^{12} - 32 T^{11} + \cdots + 47826408560896 \)
T^12 - 32*T^11 + 596*T^10 - 5270*T^9 + 353550*T^8 - 498932*T^7 + 88583289*T^6 - 691655042*T^5 + 24200562345*T^4 - 260700757100*T^3 + 2739495574256*T^2 - 15456426052032*T + 47826408560896
$7$
\( T^{12} + 200 T^{11} + \cdots + 37\!\cdots\!00 \)
T^12 + 200*T^11 + 18490*T^10 + 1157470*T^9 + 68646740*T^8 + 4376148300*T^7 + 243913206275*T^6 + 9784541272450*T^5 + 263431106185825*T^4 + 4578430968471500*T^3 + 47072001779313000*T^2 + 218076096289513000*T + 374245457353922000
$11$
\( T^{12} \)
T^12
$13$
\( T^{12} - 250 T^{11} + \cdots + 10\!\cdots\!80 \)
T^12 - 250*T^11 - 29000*T^10 + 7157950*T^9 + 1657240790*T^8 - 210575122500*T^7 - 52119345933475*T^6 + 8800600652459800*T^5 + 1079820658342164565*T^4 - 362925493627568374510*T^3 + 29985066601938160885200*T^2 - 705309428716559708333600*T + 10238650152132709772160080
$17$
\( T^{12} + 405 T^{11} + \cdots + 16\!\cdots\!05 \)
T^12 + 405*T^11 - 36530*T^10 + 13348280*T^9 + 16373739045*T^8 - 1192081263550*T^7 + 213179801350205*T^6 + 40905562875470045*T^5 + 4035289870126689675*T^4 - 10375524810573447105*T^3 - 12929762893378696861545*T^2 - 86221402654627371762400*T + 16708078472967893322670205
$19$
\( T^{12} + 530 T^{11} + \cdots + 31\!\cdots\!05 \)
T^12 + 530*T^11 + 9940*T^10 + 78368935*T^9 + 109470643145*T^8 + 29848155851250*T^7 - 8533886324937205*T^6 - 5622571240960265570*T^5 - 84444496158804510245*T^4 + 390056064814502665597865*T^3 + 45069244237823029355065935*T^2 - 4185482921007805811649996125*T + 315687083711678888831378500205
$23$
\( (T^{6} - 842 T^{5} + \cdots - 10\!\cdots\!96)^{2} \)
(T^6 - 842*T^5 - 441080*T^4 + 261415800*T^3 + 50208513760*T^2 - 6830496249632*T - 1003640478312896)^2
$29$
\( T^{12} + 4230 T^{11} + \cdots + 61\!\cdots\!80 \)
T^12 + 4230*T^11 + 7229120*T^10 + 6401784710*T^9 + 3423996992550*T^8 + 1454039664150700*T^7 + 779097774003984685*T^6 + 438454918498887122360*T^5 + 146693261754522549405405*T^4 + 14373440829824411999595690*T^3 + 595260342952553441371468920*T^2 + 420524093462572097347507862000*T + 61675935781316087013884503785680
$31$
\( T^{12} - 104 T^{11} + \cdots + 41\!\cdots\!36 \)
T^12 - 104*T^11 + 343994*T^10 + 602728890*T^9 + 1519375034600*T^8 - 2368717789506184*T^7 + 2700667498625021821*T^6 + 113008204690210422994*T^5 + 217394428784988653302945*T^4 - 10155632684609624871400000*T^3 + 3332942853016724871976431064*T^2 - 508851859195018879519817757976*T + 41123090627926476128938019809936
$37$
\( T^{12} - 454 T^{11} + \cdots + 49\!\cdots\!16 \)
T^12 - 454*T^11 - 379376*T^10 + 2879985930*T^9 + 11990538336970*T^8 + 16648714720620696*T^7 + 32989865963798946501*T^6 + 39750574450633425697644*T^5 + 40011080660719455539201265*T^4 + 26208268058345004444748583610*T^3 + 11577243634780526597645759597004*T^2 + 2561741912740589399041079801071464*T + 491027429121986352153025328592972816
$41$
\( T^{12} - 5385 T^{11} + \cdots + 72\!\cdots\!05 \)
T^12 - 5385*T^11 + 15861930*T^10 - 25608156220*T^9 + 25319954682585*T^8 - 16361785780741950*T^7 + 7027247374208761385*T^6 - 1964080894550536353265*T^5 + 342142412149607599374615*T^4 - 35623294510932493526738255*T^3 + 2263584685149678198856996315*T^2 - 66402245623587417520703626900*T + 727539032106068904070377679805
$43$
\( T^{12} + 17669305 T^{10} + \cdots + 16\!\cdots\!00 \)
T^12 + 17669305*T^10 + 119490870083165*T^8 + 394015264153180458800*T^6 + 664057087597221875119103200*T^4 + 538275376506392137827008625600000*T^2 + 161926647121393786361690914838540832000
$47$
\( T^{12} + 4516 T^{11} + \cdots + 84\!\cdots\!96 \)
T^12 + 4516*T^11 + 23343134*T^10 + 99566885290*T^9 + 377821056148400*T^8 + 550002443780429976*T^7 + 1019907557231583301971*T^6 + 287829477144876505434734*T^5 + 490108561140482382356630145*T^4 - 503494084775131863841875030500*T^3 + 203588291898255386047677846512544*T^2 + 21888219245305353383563220663964744*T + 848584371310609468635634469697146896
$53$
\( T^{12} + 4726 T^{11} + \cdots + 11\!\cdots\!76 \)
T^12 + 4726*T^11 + 11749804*T^10 + 15381160290*T^9 + 74283007872610*T^8 + 13897909611194376*T^7 + 48834675016888126941*T^6 - 30193872168308752610496*T^5 + 21884168101469887800685845*T^4 + 38546777221271871900689608830*T^3 + 20627003674096079144946572431944*T^2 + 1145420835857106006655852995291264*T + 116415007435914106264411432550850576
$59$
\( T^{12} - 11624 T^{11} + \cdots + 38\!\cdots\!41 \)
T^12 - 11624*T^11 + 90088964*T^10 - 400913051225*T^9 + 1157284153708905*T^8 - 1629280382365965794*T^7 + 4327309939333543270371*T^6 - 5309604139037961684968336*T^5 + 1018530516714446482779314475*T^4 + 7458842456094916196662229981215*T^3 + 14843948668740732034976006167648859*T^2 - 3263585965878803607383734214126094321*T + 3807328371586939208924650540356706445641
$61$
\( T^{12} - 5780 T^{11} + \cdots + 15\!\cdots\!80 \)
T^12 - 5780*T^11 + 11175560*T^10 - 102788659630*T^9 + 899412391669170*T^8 - 3059208152308705400*T^7 + 3229037022805692071885*T^6 + 11510514178812852984181050*T^5 + 11764958941952318605239302665*T^4 + 3906472854682107707983538129640*T^3 - 1130758606419240798732581626013600*T^2 - 632388939765328759009036918643257800*T + 150356408631119513999453648469726153680
$67$
\( (T^{6} - 6077 T^{5} + \cdots + 15\!\cdots\!84)^{2} \)
(T^6 - 6077*T^5 - 85187735*T^4 + 594072211480*T^3 + 898033178062280*T^2 - 8535075149147827392*T + 1562403210407222555584)^2
$71$
\( T^{12} + 21646 T^{11} + \cdots + 56\!\cdots\!56 \)
T^12 + 21646*T^11 + 248111394*T^10 + 1852565721170*T^9 + 9777688247860260*T^8 + 36763705894871238576*T^7 + 99008985328781797493001*T^6 + 189587417244145677792180184*T^5 + 261575845473990476900656334885*T^4 + 243703779213120696034196457076330*T^3 + 150676347257223047085457503538877744*T^2 + 67019905955453566525202998866944994864*T + 56944313299226219807240770779743454359056
$73$
\( T^{12} + 1015 T^{11} + \cdots + 14\!\cdots\!05 \)
T^12 + 1015*T^11 - 23737950*T^10 - 235529610140*T^9 + 217842690508245*T^8 + 5457768590457017550*T^7 + 20686830515653939655765*T^6 + 8195242723852097569305535*T^5 - 118341859408413931700566486585*T^4 - 270564988781064394088627453064455*T^3 + 89766560861360723454849688668151115*T^2 + 1158777108884077738290832885193891716500*T + 1439577773094941615244585768871616844966005
$79$
\( T^{12} + 15130 T^{11} + \cdots + 36\!\cdots\!80 \)
T^12 + 15130*T^11 + 83421050*T^10 + 1253571952670*T^9 + 21598105811166120*T^8 + 169457619011217141100*T^7 + 789621561132789288838565*T^6 + 3494766763363494846524681640*T^5 + 19065338811357523049689877665465*T^4 + 88980865638791391563585420154262470*T^3 + 272326886578453273176271330779584897260*T^2 + 476182310462844715397714931112099393823400*T + 364406399695576999394180560816091830885353680
$83$
\( T^{12} - 46770 T^{11} + \cdots + 18\!\cdots\!05 \)
T^12 - 46770*T^11 + 1064080060*T^10 - 14986607018065*T^9 + 141274224074030865*T^8 - 920470488298714969450*T^7 + 4191124551809598064813685*T^6 - 13189532583601438725806544010*T^5 + 27345585103698038834328672758475*T^4 - 32954057260205247547993139828788005*T^3 + 14839640427459937970660231631084655785*T^2 + 12342440177897104959076063152675837303275*T + 1809932797396815417720327732728408735646005
$89$
\( (T^{6} - 2777 T^{5} + \cdots + 66\!\cdots\!44)^{2} \)
(T^6 - 2777*T^5 - 87051905*T^4 - 113630897160*T^3 + 484849972414540*T^2 + 1152321825674003488*T + 661682422817958733744)^2
$97$
\( T^{12} + 16546 T^{11} + \cdots + 18\!\cdots\!01 \)
T^12 + 16546*T^11 + 117609764*T^10 + 3690981565*T^9 + 64070758444093415*T^8 + 1792214919906611729736*T^7 + 34114574241813560672523711*T^6 + 369929844818637858364362485084*T^5 + 3246123210437073290815599219169095*T^4 + 22010974786501355106510959377042078345*T^3 + 144532211808196099449366108514551431918499*T^2 + 495849857017376162406478901921959586600703669*T + 1882110444722972499331746327526844969191435758601
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