
// Make newform 603.2.e.b in Magma, downloaded from the LMFDB on 05 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_603_e();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_603_2_e_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_603_e();"
function MakeCharacter_603_e()
    N := 603;
    order := 3;
    char_gens := [470, 136];
    v := [1, 3];
    // 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_603_e_Hecke(Kf)
    return MakeCharacter_603_e();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_603_2_e_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_603_2_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_603_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![874225, -2344980, 39066489, -128427970, 1198189194, -3565411915, 19189229784, -42098824635, 176353881556, -306646333528, 1136354321411, -1568132744777, 5381417081350, -6009120197378, 19723195164072, -18119786907777, 56434741842008, -44686074487465, 128735890009565, -91779327188558, 236114512496766, -158982058194742, 353862859872293, -231093590729428, 439005744647950, -281689806651284, 456277226244110, -287890964171894, 400737980804629, -247622228191562, 299522877208482, -180049141191737, 191506850332159, -111264586427672, 105222358053272, -58739630093407, 49863640375071, -26615999309011, 20443540071284, -10390733667874, 7267135712713, -3504418188409, 2242669452202, -1022569007004, 600890677013, -258154092472, 139613631553, -56296314634, 28048076817, -10566163011, 4848399604, -1696544042, 715678251, -230847831, 89208866, -26250482, 9236537, -2442106, 774989, -179887, 50694, -9922, 2419, -369, 74, -7, 1]>],Snew);
    return Vf;
end function;