
// Make newform 33.6.d.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_33_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_33_d_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_33_6_d_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_33_6_d_a();".
// 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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [3, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_33_d();"
function MakeCharacter_33_d()
    N := 33;
    order := 2;
    char_gens := [23, 13];
    v := [1, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_33_d_Hecke();"
function MakeCharacter_33_d_Hecke(Kf)
    N := 33;
    order := 2;
    char_gens := [23, 13];
    char_values := [[-1, 0], [-1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 6;
    raw_aps := [[0, 0], [16, -1], [29, -58], [0, 0], [121, -242], [0, 0], [0, 0], [0, 0], [-1501, 3002], [0, 0], [7775, 0], [-1267, 0], [0, 0], [0, 0], [5282, -10564], [6476, -12952], [-14281, 28562], [0, 0], [72917, 0], [16025, -32050], [0, 0], [0, 0], [0, 0], [-35725, 71450], [-163183, 0], [0, 0], [-180244, 0], [0, 0], [0, 0], [-57751, 115502], [0, 0], [0, 0], [-106057, 212114], [0, 0], [0, 0], [0, 0], [-280117, 0], [164144, 0], [0, 0], [0, 0], [-51475, 102950], [279875, 0], [292733, -585466], [0, 0], [0, 0], [-799900, 0], [0, 0], [-247531, 0], [0, 0], [1171025, 0], [0, 0], [0, 0], [0, 0], [227525, -455050], [-630232, 1260464], [0, 0], [-624244, 1248488], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [168950, -337900], [-1048819, 0], [429443, -858886], [736925, 0], [0, 0], [0, 0], [0, 0], [499301, -998602], [0, 0], [-1313167, 0], [0, 0], [1888625, 0], [-712501, 1425002], [-666919, 1333838], [1560242, 0], [-895192, 1790384], [0, 0], [2166950, -4333900], [-6857050, 0], [0, 0], [-5105131, 0], [0, 0], [-2148949, 4297898], [2331275, -4662550], [0, 0], [0, 0], [7109819, 0], [-2672293, 5344586], [0, 0], [-4400233, 0], [0, 0], [7479800, 0], [0, 0], [-1267531, 2535062], [384917, -769834], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [8053967, 0], [3191282, -6382564], [0, 0], [2904950, -5809900], [0, 0], [0, 0], [0, 0], [1817768, -3635536], [-1912711, 0], [-12870277, 0], [-4322017, 8644034], [-20967769, 0], [6044207, -12088414], [-2096251, 4192502], [0, 0], [-21510073, 0], [0, 0], [0, 0], [-5347426, 10694852], [25031327, 0], [0, 0], [-25915639, 0], [8359025, -16718050], [17087783, 0], [0, 0], [0, 0], [0, 0], [25609373, 0], [-19576042, 0], [0, 0], [0, 0], [-9353524, 18707048], [0, 0], [10434107, -20868214], [0, 0], [0, 0], [0, 0], [35704781, 0], [0, 0], [19068689, 0], [-3216169, 6432338], [0, 0], [0, 0], [-15048361, 0], [-13033426, 26066852], [0, 0], [13725275, -27450550], [-6299944, 0], [0, 0], [30616808, 0], [7902242, -15804484], [0, 0], [13364144, -26728288], [0, 0], [0, 0], [-15556807, 31113614], [0, 0], [0, 0], [11119667, -22239334], [-11062393, 22124786], [672551, -1345102], [61768100, 0], [0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_33_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_33_6_d_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_33_6_d_a(:prec:=2)
    chi := MakeCharacter_33_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_33_6_d_a();".
// 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_33_6_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_33_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;