
// Make newform 1344.2.s.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1344_s();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1344_2_s_d();".
// 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_1344_s();"
function MakeCharacter_1344_s()
    N := 1344;
    order := 4;
    char_gens := [127, 1093, 449, 577];
    v := [2, 3, 2, 4];
    // 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_1344_s_Hecke(Kf)
    return MakeCharacter_1344_s();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1344_2_s_d();".
// 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_1344_2_s_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1344_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![40960000, 0, 0, 0, 1024916914176, 0, 0, 0, 216312723169280, 0, 0, 0, 1266871673606144, 0, 0, 0, 492296913989888, 0, 0, 0, 67930945379840, 0, 0, 0, 4177756384768, 0, 0, 0, 136749727616, 0, 0, 0, 2599544544, 0, 0, 0, 29529376, 0, 0, 0, 196224, 0, 0, 0, 696, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;