
// Make newform 8005.2.a.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8005_a();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8005_2_a_f();".
// 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_8005_a();"
function MakeCharacter_8005_a()
    N := 8005;
    order := 1;
    char_gens := [1602, 4806];
    v := [1, 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_8005_a_Hecke(Kf)
    return MakeCharacter_8005_a();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8005_2_a_f();".
// 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_8005_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8005_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-324407615, 22535810615, -558699188594, 4710956420995, 30244821019557, -757748821640976, 2336160370465379, 33540208830819227, -227156647027734364, -617380115721089758, 8417379479691338272, 725743077846065088, -187936342001003801800, 218051350557279365788, 2907414489478720718035, -5677098711808709750499, -33490727944370542210076, 85412049638562201494006, 300716493985978351819715, -917129713352249938009680, -2175070789014995796835330, 7588651045712095702213055, 12999426812551339165698126, -50405796448988358681361403, -65538259693046763898260129, 275894891530865393410778885, 283574404357143709800869671, -1267703546908115107137807121, -1068233603622165557068852538, 4959127547624706870991952149, 3544445845295655951457849493, -16700514740467338241453601366, -10452874394531826820366234801, 48856567862282682860101607284, 27579578998895166845614738427, -125101443656904379789132758910, -65395117173307835621450628630, 282180804234500269764776239393, 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-3736134841783352, 12586366894054262, 537663312004541, -936117339677579, -55354257010627, 60679492416864, 4517097282805, -3387881079462, -300647111330, 160482118750, 16387762789, -6321927819, -724740436, 201474303, 25462337, -4989493, -685557, 90068, 13308, -1054, -166, 6, 1]>,<3,R![1698850855988, 208850311171360, 7000043878359812, 62508741305936648, -729593560784240771, -14979937813826013032, -20568879878904904272, 1035685569723782061592, 5370153785453758224398, -30645146690743810840721, -290401194008766724467220, 262687783795813634759502, 8159538835262280492665824, 9009802454218118970802750, -139767872588174028515272680, -343945105431321237798655678, 1537135042847529686171430346, 5951361362135626832974376425, -10634275143414567423321371432, -66385482633452205851672462927, 36343008934451477523607388294, 529513333534884750700531802957, 104138723617293386929215349200, -3181901494628604907591386962637, -2320351622258743767623096739055, 14871828569485727932867030695755, 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    return Vf;
end function;