
// Make newform 1368.2.e.g in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1368_e();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1368_2_e_g();".
// 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_1368_e();"
function MakeCharacter_1368_e()
    N := 1368;
    order := 2;
    char_gens := [343, 685, 1217, 1009];
    v := [1, 1, 2, 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_1368_e_Hecke(Kf)
    return MakeCharacter_1368_e();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1368_2_e_g();".
// 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_1368_2_e_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1368_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![86528, 0, 841216, 0, 2183776, 0, 2583248, 0, 1652556, 0, 617504, 0, 138793, 0, 18700, 0, 1458, 0, 60, 0, 1]>,<7,R![32768, 0, 552960, 0, 3033344, 0, 6599936, 0, 5604704, 0, 2213664, 0, 448065, 0, 48372, 0, 2822, 0, 84, 0, 1]>],Snew);
    return Vf;
end function;