
// Make newform 4013.2.a.c in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4013_a();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_4013_2_a_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
// To make the character of type GrpDrchElt, type "MakeCharacter_4013_a();"
function MakeCharacter_4013_a()
    N := 4013;
    order := 1;
    char_gens := [2];
    v := [1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_4013_a_Hecke(Kf)
    return MakeCharacter_4013_a();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_4013_2_a_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_4013_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4013_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![14852737934123, -869537189716850, 11333408273978134, 245241320674172915, -7290189548298062859, 22344390657611949733, 987484205482911155115, -8672895695062044668748, -54096998249626854869251, 891726351833414856831905, 661905454504423045195520, -50458415493706665001354869, 83455615925929039164059922, 1851334941638961619071393790, -5957836417871922163024170948, -47399424435484271787897034935, 217661664804348315661447868129, 879282202486493524635115243200, -5409965216856026059701746969153, -11962389604073562154835039508975, 100373443803048636252368314471702, 116444021406659700754094526826609, -1459267744201633711637360523178559, -705163652183613770785166904199900, 17152480399675082822492381593623969, 247145847781453654281279753096008, -166710547648495069389412429290304033, 54611478730747418261453056390450775, 1363103130505560682674723422292492923, -800498531923275814216166664460555928, -9505897783838739453822563858114165489, 7520572482933351312127696928497506899, 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-8548445614564999662211250681150284667556969, 17622300423716344954956299217047683336888235, 13812245429525369702770769234332442338532618, -30234318449490765093852585888492688709079745, -20386437687066818362333098555189967720008165, 47651312973699810835229553410088816137839239, 27503784189355464504811919848579266424467042, -69153733983738662877673022219016772302414617, -33915341141414559525312766145916926677029595, 92607774816047293252593200638660964624920937, 38188658331316004002278705245233163785370725, -114657961061045822999556350152015098844233520, -39175284447845071775190571493489574523003352, 131473781039374655485458204483090260853514338, 36451970597381829906263823364158229080830682, -139840593385901546126027764474523791949922120, -30518092064666480467067690880023988904172757, 138166327781728787417975850254741978011415608, 22639875820812941690437125161223481160247268, -126970282179341222851754674102744039460917423, -14410997929192199501740158447266777722013072, 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-159461082713126321650953545488177655959205, -220976926638246220832436443537008193853652, 72327547961460691960113995891974346703425, 84616730537163810481243931612351261660046, -30472015560893839207984131487979794349213, -30338543207470243038182764304035794121926, 11950159537323737915342792438431719550736, 10181397037015451203912297030904455429206, -4369103365153615595982414911156273167238, -3196449759914229240468589062239186159012, 1490874857016596344688429922087462563994, 938135886862318446315568465151208139483, -475183302507888164187460323286449408454, -257150987535308177824692491455833868463, 141539655020077889427694396448258352191, 65749770881916424599875450002707447747, -39411381688908465542983564915050432504, -15655845415369677500319625882017131080, 10259764636681377798240132237637882173, 3464201399801353003533562072382981527, -2496864548070340483554308887474536188, -710273075781039223759332071868490204, 567927357856776527021521166325726688, 134407546786667249554512351539896716, -120686920686043466386472289360993360, -23341777616048378610101243781680866, 23947373840871811949321380531442981, 3688244985346315035800182204067265, -4433801646726164444002828306068921, -522804571176331784405663567438706, 765300918662887930325983479712927, 64760711333082955766191251787371, -123016763049697373188759244698215, -6608788882293490003960769978931, 18391856959864374761699703822576, 457095649398706460831558183589, -2553728800191915337927626776393, 5911064495569402353511692690, 328748878836200815982436597433, -9232372332341249594028914332, -39158403060543398108988659567, 2065310382126690713176950199, 4305726707363459807971640418, -329737981222532521726722416, -435864312956681273489581256, 43544455505804154946370123, 40491217700365120806629214, -4982354924099947578852993, -3439131230388344107633388, 503534844474906653448440, 265874477822272016226275, -45332815486943098989305, -18608064210441893540602, 3647168915109994421266, 1171209604107301914636, -262200547185870218684, -65739098868025177464, 16804222721469294034, 3254428339373407232, -955942436306375862, -139944511629970412, 47968168910229360, 5108981378665584, -2105331873435996, -152306381669339, 79930622541931, 3413579648150, -2586852998685, -43251253998, 69976285182, -439456004, -1539470485, 41258551, 26461663, -1197084, -333379, 20639, 2738, -211, -11, 1]>],Snew);
    return Vf;
end function;