[N,k,chi] = [33,4,Mod(4,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33.4");
S:= CuspForms(chi, 4);
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 \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).
\(n\)
\(13\)
\(23\)
\(\chi(n)\)
\(\beta_{5}\)
\(1\)
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
\( T_{2}^{12} + 20 T_{2}^{10} + 64 T_{2}^{9} + 225 T_{2}^{8} + 70 T_{2}^{7} + 3191 T_{2}^{6} + 11905 T_{2}^{5} + 53975 T_{2}^{4} + 92394 T_{2}^{3} + 119980 T_{2}^{2} + 109000 T_{2} + 190096 \)
T2^12 + 20*T2^10 + 64*T2^9 + 225*T2^8 + 70*T2^7 + 3191*T2^6 + 11905*T2^5 + 53975*T2^4 + 92394*T2^3 + 119980*T2^2 + 109000*T2 + 190096
acting on \(S_{4}^{\mathrm{new}}(33, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{12} + 20 T^{10} + 64 T^{9} + \cdots + 190096 \)
T^12 + 20*T^10 + 64*T^9 + 225*T^8 + 70*T^7 + 3191*T^6 + 11905*T^5 + 53975*T^4 + 92394*T^3 + 119980*T^2 + 109000*T + 190096
$3$
\( (T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81)^{3} \)
(T^4 - 3*T^3 + 9*T^2 - 27*T + 81)^3
$5$
\( T^{12} - 28 T^{11} + \cdots + 1310363273521 \)
T^12 - 28*T^11 + 541*T^10 - 3867*T^9 + 73757*T^8 - 150345*T^7 + 27569695*T^6 - 142320746*T^5 + 4127739743*T^4 - 11360858236*T^3 + 202382330190*T^2 - 790034888471*T + 1310363273521
$7$
\( T^{12} - 12 T^{11} + \cdots + 834112161 \)
T^12 - 12*T^11 + 737*T^10 - 4621*T^9 + 171450*T^8 - 1246063*T^7 + 2534112*T^6 + 150760607*T^5 + 1305224350*T^4 + 218115399*T^3 + 2415312387*T^2 + 2229411033*T + 834112161
$11$
\( T^{12} + 54 T^{11} + \cdots + 55\!\cdots\!81 \)
T^12 + 54*T^11 + 7*T^10 - 126342*T^9 - 3232702*T^8 + 71168328*T^7 + 7468757428*T^6 + 94725044568*T^5 - 5726928787822*T^4 - 297907827176322*T^3 + 21968998637047*T^2 + 225571401148445154*T + 5559917313492231481
$13$
\( T^{12} + 18 T^{11} + \cdots + 49\!\cdots\!36 \)
T^12 + 18*T^11 + 9171*T^10 + 252228*T^9 + 34911093*T^8 + 799301196*T^7 + 20331732571*T^6 + 237254589522*T^5 + 33825618887337*T^4 - 308457861295196*T^3 + 8778438041268432*T^2 + 377209262420398152*T + 4940484891888664336
$17$
\( T^{12} + 80 T^{11} + \cdots + 25\!\cdots\!76 \)
T^12 + 80*T^11 + 20720*T^10 + 2450744*T^9 + 248061450*T^8 + 14366887160*T^7 + 664018097291*T^6 + 9917269956780*T^5 + 234885091461305*T^4 - 8681752377488356*T^3 + 200167149104581360*T^2 + 3952570388335784640*T + 25415163684085739776
$19$
\( T^{12} + 280 T^{11} + \cdots + 47\!\cdots\!00 \)
T^12 + 280*T^11 + 56322*T^10 + 6892820*T^9 + 603466004*T^8 + 34697435460*T^7 + 1263745866873*T^6 + 16466489991510*T^5 + 148104484335401*T^4 + 12469315833990050*T^3 + 541839735467499000*T^2 - 14246778346818600000*T + 471669547664070250000
$23$
\( (T^{6} + 196 T^{5} - 22204 T^{4} + \cdots - 3716146836)^{2} \)
(T^6 + 196*T^5 - 22204*T^4 - 4055478*T^3 + 86433399*T^2 + 3397895946*T - 3716146836)^2
$29$
\( T^{12} - 13 T^{11} + \cdots + 95\!\cdots\!00 \)
T^12 - 13*T^11 + 50806*T^10 + 6218856*T^9 + 879189689*T^8 + 92371385988*T^7 + 15726435626764*T^6 + 340654970537824*T^5 + 34339024410332816*T^4 + 1988395459002419840*T^3 + 150771396959080463040*T^2 - 16077874727682491744000*T + 959002830788065056774400
$31$
\( T^{12} - 413 T^{11} + \cdots + 13\!\cdots\!61 \)
T^12 - 413*T^11 + 113256*T^10 - 11632138*T^9 + 806553683*T^8 - 699626310846*T^7 + 407894131912521*T^6 - 83833740964697637*T^5 + 9547642558369802987*T^4 - 598336270647597499909*T^3 + 56418794581917696710727*T^2 - 1630141738585595001479532*T + 135970680238296093828855361
$37$
\( T^{12} - 654 T^{11} + \cdots + 18\!\cdots\!76 \)
T^12 - 654*T^11 + 226719*T^10 - 35032212*T^9 + 5743272621*T^8 + 321355264788*T^7 + 152054409087199*T^6 - 10490476824434046*T^5 + 5349166537578760329*T^4 - 254498819118571396820*T^3 + 12559126097233759544688*T^2 - 228849404512755837822528*T + 1843254500341002429573376
$41$
\( T^{12} + 1490 T^{11} + \cdots + 40\!\cdots\!96 \)
T^12 + 1490*T^11 + 1087621*T^10 + 504847086*T^9 + 173459928941*T^8 + 48548738526342*T^7 + 11505453974373277*T^6 + 2202930335451326698*T^5 + 322810989209815083809*T^4 + 34460171818959164643290*T^3 + 2677529649625585668867336*T^2 + 112601403019410039552518944*T + 4014700480516461395893860496
$43$
\( (T^{6} - 208 T^{5} + \cdots + 92003355120644)^{2} \)
(T^6 - 208*T^5 - 184296*T^4 + 53806542*T^3 + 1571969675*T^2 - 1411579950054*T + 92003355120644)^2
$47$
\( T^{12} + 150 T^{11} + \cdots + 61\!\cdots\!56 \)
T^12 + 150*T^11 + 81635*T^10 + 77262304*T^9 + 36390887280*T^8 + 841732567270*T^7 + 6022735182388646*T^6 + 4000331752513804690*T^5 + 1156863677994152680385*T^4 + 107692542582412140073644*T^3 + 15424901931548370408476560*T^2 - 62077507652934100503176000*T + 61351285707115801360670054656
$53$
\( T^{12} - 1359 T^{11} + \cdots + 35\!\cdots\!41 \)
T^12 - 1359*T^11 + 1148174*T^10 - 586846064*T^9 + 195204052947*T^8 - 34061374329290*T^7 + 4647192829092155*T^6 - 282760044710533877*T^5 + 8860916853052859153*T^4 - 122150176416931919463*T^3 + 782598868437497275405*T^2 - 1758297199164700774892*T + 35725979273045406072841
$59$
\( T^{12} - 1262 T^{11} + \cdots + 22\!\cdots\!25 \)
T^12 - 1262*T^11 + 891548*T^10 - 451151729*T^9 + 330841374675*T^8 - 89434214824844*T^7 + 16353589999174643*T^6 - 1045956763694637732*T^5 + 34037144358256901351*T^4 + 13485309515447943391495*T^3 + 5653856286282609894208135*T^2 - 182912595174073836627954825*T + 22750277532315485979901644025
$61$
\( T^{12} + 1044 T^{11} + \cdots + 24\!\cdots\!36 \)
T^12 + 1044*T^11 + 1185929*T^10 + 1109396654*T^9 + 846226049262*T^8 + 387284147327870*T^7 + 142533965272414440*T^6 + 29159976955378850072*T^5 + 4312511266558304701873*T^4 + 71838053087090905720578*T^3 - 22410186696536382674615520*T^2 + 2268701744577776255249350992*T + 2451995657037397948860116520336
$67$
\( (T^{6} + 264 T^{5} + \cdots - 66\!\cdots\!84)^{2} \)
(T^6 + 264*T^5 - 1828197*T^4 - 476415190*T^3 + 903702780723*T^2 + 211997048320224*T - 66078091069174784)^2
$71$
\( T^{12} - 558 T^{11} + \cdots + 12\!\cdots\!96 \)
T^12 - 558*T^11 + 858757*T^10 - 374667486*T^9 + 296511288352*T^8 - 121938904582968*T^7 + 50195818362678846*T^6 - 13047043845076510434*T^5 + 5248928080330494695773*T^4 - 235443457643962585241202*T^3 - 39443408351019221845241484*T^2 + 2738645367601001603550307128*T + 1230652599447265192522582820496
$73$
\( T^{12} + 699 T^{11} + \cdots + 16\!\cdots\!56 \)
T^12 + 699*T^11 + 1719816*T^10 + 665268310*T^9 + 910076845569*T^8 + 426349214756528*T^7 + 158838143820088572*T^6 + 38528206974381539712*T^5 + 25229043847887975857696*T^4 + 557181709365289949792896*T^3 - 326612204617855120713869824*T^2 + 249087866700756722640112967680*T + 168862490111927008309877999769856
$79$
\( T^{12} + 1252 T^{11} + \cdots + 32\!\cdots\!25 \)
T^12 + 1252*T^11 + 773001*T^10 + 106228771*T^9 + 1377983176394*T^8 + 2388476183098753*T^7 + 3378424060355848174*T^6 + 2667661552201569208329*T^5 + 1790322902800271093229756*T^4 + 860899767074292322642800425*T^3 + 379579333277277360311836464625*T^2 + 113941297603354495461986812165625*T + 32092220110009561648733013819390625
$83$
\( T^{12} + 4464 T^{11} + \cdots + 18\!\cdots\!81 \)
T^12 + 4464*T^11 + 9491437*T^10 + 12191516625*T^9 + 10555769115034*T^8 + 6507976852113483*T^7 + 2980658397323326656*T^6 + 1031436060895762929309*T^5 + 274297337582969144943706*T^4 + 56113173114289063935421281*T^3 + 10194578832735756542782603971*T^2 + 1203910924557842063132038016823*T + 182237911283597956875619433695881
$89$
\( (T^{6} - 158 T^{5} + \cdots - 28\!\cdots\!20)^{2} \)
(T^6 - 158*T^5 - 1870450*T^4 + 262338144*T^3 + 200458974069*T^2 - 24282284969250*T - 2896435367602020)^2
$97$
\( T^{12} - 1608 T^{11} + \cdots + 11\!\cdots\!21 \)
T^12 - 1608*T^11 + 4845213*T^10 - 5023192415*T^9 + 8247817338729*T^8 - 7585823967274081*T^7 + 7772584220304323469*T^6 - 1652172541727419871442*T^5 + 5093264236170178631427941*T^4 - 940237173265632613626799952*T^3 + 416643533892995562939047091104*T^2 - 740745945164826452002291363685729*T + 1196680313344179587502252969339895321
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