[N,k,chi] = [8005,2,Mod(1,8005)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8005, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8005.1");
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
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion .
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
\( p \)
Sign
\(5\)
\(1\)
\(1601\)
\(1\)
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8005))\):
\( T_{2}^{127} + 6 T_{2}^{126} - 166 T_{2}^{125} - 1054 T_{2}^{124} + 13308 T_{2}^{123} + 90068 T_{2}^{122} + \cdots - 324407615 \)
T2^127 + 6*T2^126 - 166*T2^125 - 1054*T2^124 + 13308*T2^123 + 90068*T2^122 - 685557*T2^121 - 4989493*T2^120 + 25462337*T2^119 + 201474303*T2^118 - 724740436*T2^117 - 6321927819*T2^116 + 16387762789*T2^115 + 160482118750*T2^114 - 300647111330*T2^113 - 3387881079462*T2^112 + 4517097282805*T2^111 + 60679492416864*T2^110 - 55354257010627*T2^109 - 936117339677579*T2^108 + 537663312004541*T2^107 + 12586366894054262*T2^106 - 3736134841783352*T2^105 - 148873661149471080*T2^104 + 9598483184355819*T2^103 + 1560902381625086821*T2^102 + 208057022664020045*T2^101 - 14597441026933099827*T2^100 - 4437747729904972617*T2^99 + 122394294565187027401*T2^98 + 55471704342492186516*T2^97 - 924058252385248436839*T2^96 - 540670991830455918569*T2^95 + 6304630555327565198247*T2^94 + 4431301464862276527335*T2^93 - 38990599716843274895536*T2^92 - 31543680173997008102055*T2^91 + 219132535474922815313506*T2^90 + 198430321265629781142876*T2^89 - 1121570521458618957314844*T2^88 - 1114876518643708432383729*T2^87 + 5237072769180396716032481*T2^86 + 5634253636366375445933270*T2^85 - 22342149430576055139932189*T2^84 - 25739175339643347489201802*T2^83 + 87185724801475022163183528*T2^82 + 106677444315525047991502745*T2^81 - 311493996432129613504010529*T2^80 - 402194462830232883278098064*T2^79 + 1019618189446261028049012321*T2^78 + 1382146535493400194013225854*T2^77 - 3059262944323150700166638613*T2^76 - 4335782017410242402497034794*T2^75 + 8416077596836103729741639998*T2^74 + 12428934636225240480295408470*T2^73 - 21230428195396347689828983245*T2^72 - 32580576409399916804933288746*T2^71 + 49105584044721584217290448773*T2^70 + 78129663420303844928725439411*T2^69 - 104116998552354249822164789494*T2^68 - 171420512061009013410714874459*T2^67 + 202282638912166641483946321091*T2^66 + 344072635236159216070447166939*T2^65 - 359919053044074125864626260642*T2^64 - 631581210830709683651889354050*T2^63 + 586087451865689896464644792237*T2^62 + 1059633570249744860420279527601*T2^61 - 872723470559164130885720905435*T2^60 - 1623647989818286943666354748990*T2^59 + 1187247024283348092632877078411*T2^58 + 2269899181155081581048469575259*T2^57 - 1474022361203498073429285218578*T2^56 - 2891835991455426768690767965986*T2^55 + 1668311917061970452803467122781*T2^54 + 3352499842218414735102469360780*T2^53 - 1719279937366957170561810154012*T2^52 - 3530727444335791662922114371790*T2^51 + 1611336422345466327135109636772*T2^50 + 3371490766508540318876868637138*T2^49 - 1371749760536879987928938235879*T2^48 - 2912626118005754342759498674979*T2^47 + 1059531330672989467748778079696*T2^46 + 2270722101832984438680878156750*T2^45 - 741724199756069322365968536149*T2^44 - 1593034468988377643991135276336*T2^43 + 470160381766256238206773946408*T2^42 + 1002463928501480191048836082364*T2^41 - 269614826453552318659753522264*T2^40 - 563775188653730397930204733251*T2^39 + 139747271033609264498201186135*T2^38 + 282180804234500269764776239393*T2^37 - 65395117173307835621450628630*T2^36 - 125101443656904379789132758910*T2^35 + 27579578998895166845614738427*T2^34 + 48856567862282682860101607284*T2^33 - 10452874394531826820366234801*T2^32 - 16700514740467338241453601366*T2^31 + 3544445845295655951457849493*T2^30 + 4959127547624706870991952149*T2^29 - 1068233603622165557068852538*T2^28 - 1267703546908115107137807121*T2^27 + 283574404357143709800869671*T2^26 + 275894891530865393410778885*T2^25 - 65538259693046763898260129*T2^24 - 50405796448988358681361403*T2^23 + 12999426812551339165698126*T2^22 + 7588651045712095702213055*T2^21 - 2175070789014995796835330*T2^20 - 917129713352249938009680*T2^19 + 300716493985978351819715*T2^18 + 85412049638562201494006*T2^17 - 33490727944370542210076*T2^16 - 5677098711808709750499*T2^15 + 2907414489478720718035*T2^14 + 218051350557279365788*T2^13 - 187936342001003801800*T2^12 + 725743077846065088*T2^11 + 8417379479691338272*T2^10 - 617380115721089758*T2^9 - 227156647027734364*T2^8 + 33540208830819227*T2^7 + 2336160370465379*T2^6 - 757748821640976*T2^5 + 30244821019557*T2^4 + 4710956420995*T2^3 - 558699188594*T2^2 + 22535810615*T2 - 324407615
\( T_{3}^{127} + 18 T_{3}^{126} - 79 T_{3}^{125} - 3305 T_{3}^{124} - 5332 T_{3}^{123} + \cdots + 1698850855988 \)
T3^127 + 18*T3^126 - 79*T3^125 - 3305*T3^124 - 5332*T3^123 + 285340*T3^122 + 1227295*T3^121 - 15199498*T3^120 - 99993799*T3^119 + 544819888*T3^118 + 5191315295*T3^117 - 13001157742*T3^116 - 197434801318*T3^115 + 159753763156*T3^114 + 5861479863645*T3^113 + 2080108429817*T3^112 - 140838193027705*T3^111 - 178306654817505*T3^110 + 2801173384736744*T3^109 + 5786280750405943*T3^108 - 46771206729583966*T3^107 - 133236880710345878*T3^106 + 660568139922675644*T3^105 + 2447594915936099867*T3^104 - 7898715511444585915*T3^103 - 37587405532829985309*T3^102 + 79290113477940174204*T3^101 + 494636640287840008351*T3^100 - 650314113674641149888*T3^99 - 5663696635438333759820*T3^98 + 4017835472126379012503*T3^97 + 57009080501349098102185*T3^96 - 12698308913163635912839*T3^95 - 508125586548384946178778*T3^94 - 95043389878586118535921*T3^93 + 4031379751089289377654864*T3^92 + 2205698489467498005235901*T3^91 - 28576721827875734640716333*T3^90 - 25061893076639176978771985*T3^89 + 181440944706623300236173385*T3^88 + 216340871403116718223648250*T3^87 - 1033304446645686413919555775*T3^86 - 1557875954026402605663870241*T3^85 + 5279666765233982769441182469*T3^84 + 9707129841952144960089081628*T3^83 - 24177877493702392840604884218*T3^82 - 53307328182547847574649447063*T3^81 + 98961350546603539747226707667*T3^80 + 260740612662607462649480364819*T3^79 - 360108629464008634713993742395*T3^78 - 1143444616297544903571946611679*T3^77 + 1153704338200688387679312917985*T3^76 + 4515065545716422943614296322808*T3^75 - 3194653479002947675820982211958*T3^74 - 16098002350283066084150879328160*T3^73 + 7348831416701876404638304582101*T3^72 + 51916751975162669359752904636447*T3^71 - 12581030950364813060405716557409*T3^70 - 151600413141613614469498493297612*T3^69 + 8428320831131229928964317809713*T3^68 + 400962353393970513868895485696151*T3^67 + 44360188738276985748722456723991*T3^66 - 960343262105095923754078182580520*T3^65 - 251359822909469014452322496858291*T3^64 + 2081360772113426572966881344806381*T3^63 + 837057755905673354322060042213919*T3^62 - 4076701563795693129662267182260059*T3^61 - 2184689282965031193878525876914427*T3^60 + 7202712142407209122875650087714087*T3^59 + 4802962307580673908364296893749852*T3^58 - 11449629732819982208027590029139091*T3^57 - 9150900851489446677559482968721151*T3^56 + 16319181039814634297597677970958521*T3^55 + 15312965466130106992298683502328616*T3^54 - 20759138617044577453535910009020313*T3^53 - 22654832426795959496625133058666836*T3^52 + 23419372572940300938162992723364844*T3^51 + 29720385536575554477368612098554376*T3^50 - 23220604437014441535390043378399760*T3^49 - 34597431548313258356667330795411729*T3^48 + 19958132340133031975126618945105699*T3^47 + 35705357843622884611685650058183361*T3^46 - 14527050009644010220470073516612001*T3^45 - 32595605961037443403524224458803085*T3^44 + 8544969904442721557979142721153503*T3^43 + 26232167475056387009455300226875756*T3^42 - 3574074666297967099812650201951649*T3^41 - 18523990449081329232037161694071804*T3^40 + 448537939767744250250705283253277*T3^39 + 11408470252732427107988185727749046*T3^38 + 890302854908910276118431772239351*T3^37 - 6080372233333639376820927776377813*T3^36 - 1060017886793648182976437359704630*T3^35 + 2776251020512921329262804808252672*T3^34 + 746514266657087511739885772352783*T3^33 - 1071446207438412185432811332839997*T3^32 - 393640444951640581351944472218786*T3^31 + 342974950269861782425985860854320*T3^30 + 164367697237474693003077791922363*T3^29 - 88458497983777541548306148641043*T3^28 - 55177668265764123649912864889099*T3^27 + 17449333344337387003949170685417*T3^26 + 14871828569485727932867030695755*T3^25 - 2320351622258743767623096739055*T3^24 - 3181901494628604907591386962637*T3^23 + 104138723617293386929215349200*T3^22 + 529513333534884750700531802957*T3^21 + 36343008934451477523607388294*T3^20 - 66385482633452205851672462927*T3^19 - 10634275143414567423321371432*T3^18 + 5951361362135626832974376425*T3^17 + 1537135042847529686171430346*T3^16 - 343945105431321237798655678*T3^15 - 139767872588174028515272680*T3^14 + 9009802454218118970802750*T3^13 + 8159538835262280492665824*T3^12 + 262687783795813634759502*T3^11 - 290401194008766724467220*T3^10 - 30645146690743810840721*T3^9 + 5370153785453758224398*T3^8 + 1035685569723782061592*T3^7 - 20568879878904904272*T3^6 - 14979937813826013032*T3^5 - 729593560784240771*T3^4 + 62508741305936648*T3^3 + 7000043878359812*T3^2 + 208850311171360*T3 + 1698850855988