
// Make newform 74.3.i.b in Magma, downloaded from the LMFDB on 03 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_74_i();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_74_3_i_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_i();"
function MakeCharacter_74_i()
    N := 74;
    order := 36;
    char_gens := [39];
    v := [23];
    // 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_i_Hecke(Kf)
    return MakeCharacter_74_i();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_74_3_i_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_3_i_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_74_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![1206330159616229376, 40338345316181999616, 838944748231083196416, -1866916270330866008064, 4486711023508422795264, -6060114933723177320448, 7311337506565764092928, -6606300560118725502720, 2087152725018190094784, 3480176387176255967904, -6837564655539025675992, 4803283251042878634156, 133208893859008387977, -2433184826831753709636, 1530725572965859219665, -207076377819767215446, -147937803720691631787, 43800367003158538242, 11022717285824241327, -4107166948979535024, -2167620756964349535, 1162783651099015956, 27170016272837301, -100940448075269844, 17975233390713667, 3456015579526608, 80862255920244, -897811190691486, 142175981541600, 8060991192306, -3215663416661, -372792063006, 174716970807, 87095219544, -12653824593, 2682124740, 112274337, -29687562, 6881118, -8757774, 770217, -55614, -28400, 5904, -291, 180, 15, 0, 1]>],Snew);
    return Vf;
end function;