
// Make newform 585.2.bs.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_585_bs();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_585_2_bs_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_585_bs();"
function MakeCharacter_585_bs()
    N := 585;
    order := 6;
    char_gens := [326, 352, 496];
    v := [6, 3, 2];
    // 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_585_bs_Hecke(Kf)
    return MakeCharacter_585_bs();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_585_2_bs_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_585_2_bs_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_585_bs();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1296, 0, -7056, 0, 25744, 0, -49696, 0, 67920, 0, -57280, 0, 34968, 0, -14660, 0, 4576, 0, -1000, 0, 160, 0, -16, 0, 1]>],Snew);
    return Vf;
end function;