
// Make newform 588.2.o.e in Magma, downloaded from the LMFDB on 02 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_588_o();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_588_2_o_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_588_o();"
function MakeCharacter_588_o()
    N := 588;
    order := 6;
    char_gens := [295, 197, 493];
    v := [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_588_o_Hecke( : Kf := Kf)
    return MakeCharacter_588_o();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_588_2_o_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_588_2_o_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_588_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![64, 0, -4224, 0, 259360, 0, -1248448, 0, 4784928, 0, -4618400, 0, 3291336, 0, -760528, 0, 121032, 0, -11864, 0, 850, 0, -36, 0, 1]>,<11,R![3873527824384, 0, -3862443327488, 0, 2662798688256, 0, -899134988288, 0, 214149619968, 0, -28235791872, 0, 2641219456, 0, -169942016, 0, 8159792, 0, -274816, 0, 6776, 0, -104, 0, 1]>,<19,R![65536, 327680, 1968128, -1632256, 1681472, -143104, 93248, -3584, 3896, -64, 72, 0, 1]>],Snew);
    return Vf;
end function;