
// Make newform 74.4.f.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_74_f();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_74_4_f_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_f();"
function MakeCharacter_74_f()
    N := 74;
    order := 9;
    char_gens := [39];
    v := [8];
    // 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_f_Hecke(Kf)
    return MakeCharacter_74_f();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_74_4_f_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_4_f_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_74_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![15230982117594271744, 58720448543562470400, 108222135326425443072, 124194938950692594944, 99237288057642146976, 58015057975412625228, 24718849086342077013, 7333581217020224835, 1540247991442471041, 244686660166776463, 8013195319262097, -9905102555943630, -327254493604963, 445632974024238, 73783071219780, -15160355754285, 2498806601595, -655388169723, 71925648728, 6366337320, -702730560, 35826424, 10497225, -4288557, 526422, -15213, 4107, -487, 42, -6, 1]>],Snew);
    return Vf;
end function;