
// Make newform 74.5.g.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_74_g();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_74_5_g_b();".
// 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_74_g();"
function MakeCharacter_74_g()
    N := 74;
    order := 12;
    char_gens := [39];
    v := [5];
    // 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_74_g_Hecke(Kf)
    return MakeCharacter_74_g();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_74_5_g_b();".
// 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_74_5_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_74_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<3,R![343857745141128532244736, -452090785883054378471424, -353394797744559171451392, 725123951118933355339776, 834265618027290459291456, -8674487610210661974912, -73907747534619833240448, -1242040198014629378496, 4270679639211550773168, 55405105585558631712, -140592794995354123296, -1322688600802144224, 3308472423411251832, 14799504195242352, -53474148517446288, -107434944613440, 643706238516840, 324041868432, -5476626413928, -372070212, 34519933798, 2118720, -146796902, -5418, 452259, 0, -845, 0, 1]>],Snew);
    return Vf;
end function;