
// Make newform 63.6.p.b in Magma, downloaded from the LMFDB on 10 March 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_63_p();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_63_6_p_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_63_p();"
function MakeCharacter_63_p()
    N := 63;
    order := 6;
    char_gens := [29, 10];
    v := [3, 1];
    // 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_63_p_Hecke(Kf)
    return MakeCharacter_63_p();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_63_6_p_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_63_6_p_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_63_p();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![3278535641010671616, 0, -1759475339393753088, 0, 796304645351919360, 0, -69376302085328640, 0, 3906108154885840, 0, -139603771636472, 0, 3663641491264, 0, -68764950302, 0, 975017677, 0, -9883100, 0, 73519, 0, -344, 0, 1]>],Snew);
    return Vf;
end function;