
// Make newform 1120.2.bz.e in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1120_bz();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1120_2_bz_e();".
// 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_1120_bz();"
function MakeCharacter_1120_bz()
    N := 1120;
    order := 6;
    char_gens := [351, 421, 897, 801];
    v := [3, 3, 6, 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_1120_bz_Hecke(Kf)
    return MakeCharacter_1120_bz();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1120_2_bz_e();".
// 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_1120_2_bz_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1120_bz();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4096, -21504, 8704, 151872, 11984, -446496, -99516, 947640, 776561, -536664, -764820, 189744, 637314, 269256, -67428, -73188, 2835, 18120, 5528, -492, -494, 0, 48, 12, 1]>,<13,R![81088, 108288, -135600, -196672, 22412, 69028, -424, -9034, 221, 512, -34, -10, 1]>],Snew);
    return Vf;
end function;