
// Make newform 6039.2.a.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6039_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6039_2_a_f();".
// 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_6039_2_a_f();".
// 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 := [8, 49, 8, -201, -41, 281, 29, -173, 0, 48, -5, -5, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2408, -4356, 11277, 6045, -15242, -1364, 7681, -920, -1448, 341, 63, -18], [-2370, -4124, 11261, 8653, -12420, -5060, 4494, 646, -404, 13, -49, 14], [2201, 9797, 1742, -26034, -10130, 23359, 6871, -8599, -1142, 1281, -13, -41], [65, 5306, 7254, -15851, -19521, 14583, 15396, -5970, -4293, 1284, 385, -110], [3442, 9631, -11317, -26756, 17915, 22798, -13927, -7059, 4997, 404, -656, 98], [-854, -2331, -2721, 4384, 13116, -1925, -13117, 620, 4691, -536, -580, 121], [-1059, -1128, 8831, 10989, -19916, -13319, 16769, 4593, -6015, -35, 782, -134], [-404, -8430, -10719, 26323, 26766, -24047, -20052, 8984, 5572, -1735, -539, 154], [-1767, 167, 21402, -454, -42957, 27, 32229, -1142, -10047, 1167, 1139, -236]];
    Rf_basisdens := [1, 1, 1, 313, 313, 313, 313, 313, 313, 313, 313, 313];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_6039_a();"
function MakeCharacter_6039_a()
    N := 6039;
    order := 1;
    char_gens := [1343, 5491, 5248];
    v := [1, 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;

function MakeCharacter_6039_a_Hecke(Kf)
    return MakeCharacter_6039_a();
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 := 2;
    raw_aps := [[1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [-1, 0, -1, 0, 0, 0, 1, 0, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, -2, 0, 1, 1, 1, 0, 0], [3, 0, 0, 0, 0, -1, 1, 0, 1, 0, -1, 1], [-1, 0, 0, 2, 0, 0, 0, -1, -1, 0, 0, -1], [1, 0, 0, 1, 0, -1, 1, 1, -1, 1, 1, -2], [3, -1, 0, 2, -2, 1, 1, -1, 1, 0, 2, 1], [3, 0, 1, 1, -2, 2, 0, 0, 0, 1, 0, -1], [-1, -1, 1, 1, 1, 2, -2, -1, -1, -1, 2, 1], [1, 1, -1, 2, -1, 1, -1, 1, -2, 0, -1, 0], [-3, -3, 0, 2, 0, 2, -3, 0, -4, 0, 0, 0], [-1, 1, -1, -1, 4, -2, -1, 3, 0, 1, -2, -2], [2, -1, -1, 2, 0, 1, -1, 0, 1, 0, -1, 1], [2, 1, -2, -2, 3, -2, 1, 0, 1, 1, -2, -4], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, 1, 0, 2, -3, 0, 0, 0], [1, -3, 0, -1, 1, 0, 1, 3, 0, 1, 1, 1], [-4, 3, -3, 0, -2, 0, 3, 2, 6, 2, -1, 4], [-2, -1, 1, 1, 4, 0, -1, -1, -4, -2, 0, -2], [1, -3, -1, 0, 3, -1, -3, 2, -3, 0, 1, 0], [5, 0, 1, 0, 0, 0, -1, 0, 2, 1, 0, 2], [-1, 1, -3, -2, 1, -4, 4, -1, 4, -1, -2, -1], [3, 0, -3, 0, -1, 1, -1, -5, -1, 0, 0, 0], [-4, 4, -4, 1, 4, -5, 0, 5, 3, 3, -3, -2], [3, -1, 0, 0, -2, 6, 0, -2, -2, -1, 3, 4], [1, -4, 2, 2, -3, 3, -3, 2, -1, 0, 2, 3], [1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 1], [1, -5, 1, 2, 2, 0, -1, -3, -6, -1, 2, -4], [6, -2, 4, 0, -3, 5, 0, -2, -1, 1, 3, 0], [1, 3, 0, -3, 2, -1, 2, 2, 1, 0, -2, 2], [-3, 2, -2, -2, 2, -4, 4, -4, 0, 0, -1, -3], [5, -4, 2, 0, 0, 1, -1, -3, -4, -1, 2, 0], [0, 8, -2, -1, -1, -5, 3, 2, 5, 1, -6, -1], [0, 0, 1, 2, -1, -1, 0, 2, -1, 1, 4, -3], [-1, -2, 1, 2, 0, 2, -2, 1, 1, 1, 1, 3], [-2, -1, -2, 2, 0, -3, -2, -2, -4, 0, 0, -3], [-3, -1, -1, 0, 2, -5, -2, 1, -3, -3, -1, 0], [-3, 6, 0, 1, -2, 1, 3, -1, 4, -3, 0, 0], [4, 2, 0, 2, -1, 1, -3, -3, -1, 1, 1, 1], [0, -1, 1, 2, -4, 1, -4, -2, -1, -3, 0, -1], [-4, 4, 1, -3, 1, -2, 1, 1, 3, 0, -3, 4], [-10, 3, 0, -4, 8, -6, 0, 2, -1, -2, -2, -4], [0, -1, 4, -2, -1, 1, -2, 0, 0, -4, 1, -1], [-8, 4, 0, -1, 0, 5, -1, 1, 0, -1, 1, 3], [4, 0, -2, 1, 1, 0, 4, -1, 1, 4, 1, -2], [11, -5, 1, 2, -5, 2, 2, 3, 3, 3, 2, 3], [4, -3, 1, 3, 2, -3, -3, -3, -5, 1, 1, -6], [11, -5, 3, 6, -2, 4, -3, -4, -6, 0, 6, -1], [-4, -2, 3, 4, -1, 4, -3, -2, -4, -2, 6, 0], [-2, 3, -3, -1, 1, 0, -3, -3, -1, -5, -1, 1], [2, 7, -6, -3, 2, -2, 1, -1, 9, 1, -5, 0], [-8, -2, -4, 2, 4, -6, -4, 3, -3, 0, -4, 0], [-5, -2, 0, 5, -7, 1, -1, 0, 2, -4, 1, 7], [-4, -3, -1, 3, -2, 3, -1, 2, -4, 3, 4, 2], [-4, 5, -3, -5, -2, 0, 3, 3, 2, 1, -2, 3], [-4, 0, -5, 2, 2, 2, -1, 2, -1, 1, 1, -1], [4, -4, 2, -3, 9, 0, -3, 3, -3, -2, 0, -1], [-11, -3, -2, 3, 1, -2, 0, -4, -7, -1, 2, -2], [-3, -6, -2, -3, 0, -3, 0, -1, 3, -1, 2, 5], [-7, 1, -5, -5, 1, -3, 6, -2, 9, 0, -1, 1], [11, -1, 8, 3, -10, 8, 4, -1, 0, 2, 6, 1], [5, 6, -5, -5, -1, -6, 8, 5, 7, 7, -6, -1], [0, 4, 0, -8, 1, -3, 3, 0, -1, -2, -3, -4], [2, 4, -2, 2, 3, -4, -1, 1, -1, -1, -2, -6], [2, 7, -1, -9, 2, 1, 3, -1, 5, 1, -2, 0], [9, -3, 0, 2, 4, -1, -2, 0, -5, 4, 0, -8], [5, 3, 0, 1, -7, -1, 6, -2, 4, -1, 1, 3], [6, 6, -3, 2, -3, -6, 4, 2, 3, 5, 0, -8], [-10, 2, -1, -1, 5, -6, 1, 5, 8, -2, 0, -2], [7, -6, 3, 4, -1, 6, -4, -1, -7, 4, 4, -2], [-2, 0, 0, -3, 4, 0, 0, -1, -1, -4, 2, 0], [-2, -6, -2, 5, -1, -5, -2, -2, -11, -3, 2, -2], [4, 4, -3, -3, 5, -6, -1, 6, 5, 4, -6, 2], [5, 6, 0, -3, -4, 6, 0, -1, 7, 3, -3, 2], [-6, -3, -3, 3, 4, 0, -6, -2, -8, -7, -1, 0], [10, -2, 1, -4, -2, -3, 4, 6, 2, 8, -4, 0], [4, 0, 6, 3, -6, 8, 3, -6, 2, -7, 3, 8], [-6, -4, 3, 0, 5, 2, -3, -6, -9, -10, 1, 2], [6, 0, 3, 0, -1, 6, -6, -8, -7, -7, 3, -4], [7, 4, 2, 1, 1, 0, 6, 1, 3, 3, -2, 0], [6, 5, -1, 2, 0, 2, 1, 6, -1, 3, -4, -3], [9, -5, 1, 0, 1, -2, 3, -2, -3, 1, 0, -6], [5, -2, 0, 5, 2, 0, -3, -5, -9, -3, -2, -7], [10, 11, -3, -1, -2, -1, 3, -3, 3, 5, -5, -4], [9, -15, 4, 1, 1, 2, -4, 2, -9, 2, 2, -3], [-9, -2, -5, 3, 4, 0, -5, -3, -1, 1, -3, 2], [2, 3, 0, 6, -7, 6, -3, -6, 3, -3, 3, 6], [6, 6, 2, -1, -5, -7, 7, 4, 7, 4, -8, 0], [1, 10, 1, -1, -6, -2, 4, 2, 5, 0, -3, 2], [1, -3, -1, 6, -1, 2, -1, -2, -2, -4, 5, 1], [3, -5, -7, 1, 3, 0, 5, 0, -3, 5, -2, 2], [-3, 0, 3, -2, 1, 2, -6, 7, -2, -2, 1, 4], [7, 4, -1, -3, -6, -1, 8, 5, 6, 5, 0, 4], [-1, 0, 1, 1, 5, -3, -3, -4, -10, -1, 1, -11], [-4, -3, -1, 1, 10, -11, -2, 4, 0, 2, -6, -8], [-12, -3, 3, 2, 3, -3, -5, 0, -2, -7, -2, 2], [0, -1, 4, 2, -2, 1, -3, -3, 1, -8, 6, 8], [0, 5, -5, -6, 3, -7, 1, 11, 7, 8, -7, -3], [0, 1, 5, 4, -3, -7, -2, 0, 4, -1, 0, -3], [1, 1, -2, -3, -3, 3, 0, -5, -3, 1, 2, -1], [6, 8, -4, -7, 6, -16, 7, 4, 8, 4, -9, -9], [4, -4, 3, 3, -3, 6, 2, -5, -2, -6, 0, 5], [-2, 0, 1, -5, 8, -1, -3, 1, -3, -2, -1, -3], [3, 7, -3, 0, 1, 1, 5, -3, 4, 3, 5, 3], [-1, -1, -10, -3, 7, -5, 8, 5, 4, 8, -8, 2], [-12, -6, 8, -2, 3, -1, -5, 6, 2, -3, 1, 3], [1, -4, 2, 8, -4, 2, -7, 2, -10, 6, 6, -2], [2, 6, 2, -2, -2, -4, 0, 2, -6, 2, 3, -4], [-1, 6, -3, -6, 6, -9, 4, 7, 6, 1, -8, -4], [10, -3, 5, 1, -7, 5, 2, -6, 0, -6, 6, 2], [11, 4, 1, 1, 4, -2, 0, 1, 10, 4, 0, 0], [-11, 0, 0, -3, -1, -8, 3, 7, -2, 2, 0, -4], [7, 4, 6, -3, -5, -2, -3, -3, 5, -5, 0, -1], [-9, -3, -1, 2, 8, 0, -7, 4, -6, 2, -3, -3], [-4, 8, -2, 1, 6, 2, 0, 4, 5, 1, -5, 5], [-7, 4, 4, -1, -8, 2, 2, -2, 3, 1, 8, 4], [-5, 9, -4, -3, 1, 2, 6, -1, 5, 4, 1, -9], [-1, -5, 4, 1, -1, -7, -4, 3, -1, 0, -2, -4], [-6, 1, 2, -9, 0, 0, 0, 0, 7, -1, 3, 9], [-8, 13, -4, -3, 3, -9, 1, -2, -1, -3, -4, -7], [2, -14, 4, 1, 7, 6, -13, 2, -9, -7, 3, 0], [-10, 0, -1, 1, 7, -4, -5, -6, -6, -7, -2, 1], [1, 8, -3, 3, 4, -7, 4, -1, 3, 5, 1, -6], [11, -4, 1, -4, 0, -10, 4, 10, 12, 6, 1, -2], [7, -1, 5, 10, -2, 8, -8, 3, -17, 5, 5, -5], [5, -2, -7, 0, 8, -1, -1, -3, -5, 3, -1, -11], [-8, 10, 2, 2, -3, -1, -1, 2, 2, 5, -2, -1], [-1, -2, 2, 2, 2, 1, -2, -1, 5, -2, 1, 2], [8, 0, 8, -1, -6, -5, 2, 9, -4, 3, 0, -2], [-1, 1, -3, 7, -10, 10, -3, 1, -10, -5, 2, 8], [6, -3, 4, 2, -4, -1, 0, -2, -1, 1, 0, 2], [-7, 3, 5, 5, -6, 9, -8, -1, -10, -13, 1, 10], [-1, -7, 7, 7, 6, 7, -13, -1, -22, -5, 4, -7], [0, -2, 7, 2, 2, 1, -4, -3, -9, -2, -2, -4], [-2, -7, -4, 2, -3, 2, 3, 0, 8, 9, 3, 6], [3, -1, -1, 3, 9, -16, -2, 5, -2, 6, -8, -16], [10, -7, 7, 4, -4, 5, -10, -1, -8, -10, 0, 8], [-4, -6, 2, 4, 4, -4, -7, 0, 0, -9, 1, -2], [-11, -18, 4, -1, 7, 3, -6, 1, -5, -6, 7, 8], [-12, 6, 2, -13, 4, -13, 5, -3, 5, -4, -7, -4], [12, -13, 3, -2, 3, 4, -3, 8, -4, 2, 0, 4], [11, 0, 4, 7, -9, 1, 1, -7, -6, -5, 9, -2], [1, 2, -1, 13, -2, 4, -7, -8, -11, -1, 2, -10], [0, -2, 12, -3, -5, 2, -3, 8, 6, 2, 2, 7], [-7, 7, -7, 7, -2, -6, 1, 2, -2, 2, 0, -3], [-13, 3, 0, -9, 9, -3, -2, 6, 9, -6, -3, 0], [-2, -8, -2, -5, -2, 0, 7, -5, 2, 4, -3, 7], [-10, 3, -4, -10, 0, -13, 12, 1, 20, -1, -8, 0], [-4, 5, -7, -9, 0, 6, 5, 3, 9, -3, -5, 9], [-2, -11, 2, 8, 3, 3, -9, 3, -7, 0, 2, 2], [-3, 1, -1, -6, -6, 2, 5, 2, 5, 4, -1, -1], [-3, -8, 1, 8, 3, -5, -8, 10, -7, 4, -6, 1], [7, -8, 11, -9, 3, -8, -7, 4, -3, 3, 3, -6], [-5, -6, 2, 1, -2, 5, -4, -11, -3, -7, 9, 7], [-3, -2, 1, -1, -2, 7, -5, -2, -4, -7, 0, 5], [-17, 4, -5, -11, 6, -8, 2, 4, 11, 0, -9, -4], [0, 2, 2, 8, -1, 1, -1, -8, -9, -5, 6, -3], [12, -2, 0, -3, -3, -9, 1, -5, 2, -6, -5, 6], [-13, -4, 3, -6, 1, 10, -6, -2, -4, -9, 4, 7], [-2, -6, -11, -4, -2, 3, 4, 1, 6, 0, -4, 10], [-19, -4, 5, -10, 11, -5, -10, 11, 2, -2, -5, -1], [9, 7, -4, 2, -2, 9, -4, -11, -5, -1, -2, 1], [24, -2, 1, -4, 0, -4, 3, -4, 0, -1, -4, -3], [9, -1, -5, 2, -6, 11, 5, 0, 4, -4, 5, 12], [-16, -4, 11, -7, 4, 1, -5, 1, 3, 0, 2, 3], [-5, -11, 3, 2, -3, 12, -1, 3, -2, -1, 11, 10], [1, -10, 1, 13, -12, 10, -7, -4, -7, -6, 3, 7], [-4, 18, -18, 3, 5, -6, 9, 3, 8, 10, -6, -7], [4, -6, -4, 10, 0, 0, -1, 9, -7, 10, 9, -7], [-9, 9, -1, 7, -14, 3, -1, -5, 2, -5, 5, -5], [18, -13, 6, -10, 8, 0, -4, 2, -2, -2, 2, -4], [3, 3, 1, 2, 5, -16, 8, 5, 1, 6, -5, -6], [1, -13, -2, -3, 1, 7, -8, -1, -2, -5, -3, 9], [-10, 2, 1, 5, -11, 7, 5, -12, -5, -3, 2, 6], [-7, 5, 2, -2, -7, -12, 3, 8, 14, 0, -3, 4], [-10, 4, 3, -4, 5, 1, 4, -1, -10, 1, 4, -11], [2, -2, -6, 12, 10, -3, -12, 2, -9, 1, -2, -6], [-3, 8, 0, -9, 3, -2, 3, 4, 9, 2, -8, 2], [9, -11, 3, 9, -6, -4, -8, -6, -12, -2, 0, 2], [11, 5, -6, -2, -8, 6, 5, -3, 8, -3, 6, 7], [19, 2, -2, 0, -4, 9, 1, -3, 7, -5, 2, 2], [-7, 2, -15, 0, 2, -3, 8, -7, 5, -5, -2, 6], [-13, 21, -5, 1, 3, -6, 4, 1, 10, 0, 0, -4], [12, -5, -6, -2, -5, 5, 8, -7, 4, 6, -1, 5], [-20, 4, -5, -9, 3, -5, 12, 2, 6, 5, -4, 7], [9, -9, 10, -14, 6, -5, -3, 8, 2, 9, -1, -2], [-9, 3, 12, 1, -9, 10, -3, -1, -4, -9, 5, -1], [12, 12, -3, -5, -3, -11, 19, 6, 17, 13, -7, 0], [10, -6, 9, 1, -5, 10, -5, 1, -5, -1, 5, -1], [5, -8, 1, -4, -1, -2, 1, 1, 0, -5, -3, -4], [-14, -2, 0, 9, -4, -1, -6, 3, -3, 3, 8, 4], [12, -5, 11, 0, 1, -2, -13, 8, 7, -4, 2, 0], [-1, -6, 4, -9, -4, 1, 8, -2, 1, -3, 2, -5], [4, -1, -2, 9, -7, 4, 4, -2, 10, -3, 7, 5], [-4, -8, -1, -10, 8, -8, -1, 8, 11, 4, -2, 6], [-12, 3, -7, 3, 1, -4, -2, 8, 10, 7, -5, 0], [14, 18, -7, -10, 2, -5, 15, -5, 20, 6, -2, -1], [16, -6, -1, 3, -1, -7, 5, 6, 5, 7, -2, 4], [17, -4, -2, -3, -12, 3, 6, 1, 10, 5, -5, 1], [-1, 0, -1, -3, 10, 3, 1, 3, 6, 7, -1, 3], [2, -17, 5, 8, 3, 12, -11, 6, -16, 0, 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-1, 5, 6, -16, 4, 13, 4, 4, -6, -16], [-22, 21, -19, 8, -6, 1, 11, -15, -1, 0, -1, 3], [2, -1, 12, 12, 7, -9, -19, 7, -11, -11, -5, -13], [-6, -12, 3, 1, 13, -13, -13, 0, -8, -15, -3, -14], [15, -11, 7, -1, -10, 24, -9, -7, -9, -8, -2, 9], [-4, -3, -13, 13, 2, 4, -7, 5, -11, 1, -9, 3], [12, 1, -16, 8, -6, 11, 13, -10, 15, 15, -3, 1], [42, 0, -12, -4, 4, -8, 4, -6, 6, -1, -9, -5], [-6, -26, 3, 7, 4, -2, -13, 3, -22, -11, 18, 3], [-27, 6, -3, -3, -6, -1, -2, -19, -11, -15, 9, -12], [7, -4, 2, 2, 4, 6, 0, 13, 6, 13, 5, 6], [10, 13, 2, 2, -4, 2, 6, 2, 20, 9, -12, 1], [4, 9, 3, 6, -5, 4, -10, 2, -4, -16, 6, 1], [15, -8, 16, 6, -9, 19, -3, 4, 1, -1, 13, 14], [30, 9, 0, -14, -6, -10, 20, 6, 30, 11, 4, 3], [-17, 0, 5, -12, -3, 3, -9, -5, -7, -18, -7, 0], [7, -3, 5, 14, -5, -6, -8, -2, -22, 5, 13, -25], [6, 19, 2, -1, -1, -6, -1, -9, 3, -5, -9, -8], [17, -15, 21, -18, 4, -7, 7, 5, 15, -4, 4, -4], [-24, 4, -6, -3, 10, -6, -14, -12, -21, -29, -7, 1], [-6, 15, 2, -4, 8, -28, -2, -3, 6, 1, -15, -15], [-7, 5, 4, -10, -13, -2, 15, -13, 12, -4, 12, -6], [7, -15, 4, 9, -7, 6, -15, -27, -17, -10, 7, -4], [0, -2, 3, 4, -17, 10, 5, -3, 9, 3, 5, 25], [5, -4, 9, -7, 2, 0, 5, -17, 1, -11, 7, 6], [-8, 13, 4, 8, -8, 9, -4, -2, -1, -6, 2, 11], [1, 10, -7, -7, 12, -5, -6, -10, 10, -3, -9, 8], [9, 24, -25, -15, 6, -16, 22, -1, 20, 17, -17, -11], [10, 2, 3, -3, -1, 3, 5, -15, 12, -4, 11, 11], [-8, -6, 12, 12, -6, 4, -5, 0, -21, -10, 16, 3], [7, 21, -8, -12, 6, -9, 4, 15, 15, 7, -13, -13], [-9, 1, -5, -3, -3, 12, -3, -7, -12, -10, 0, 1], [10, -19, -4, 10, -9, 2, -7, 16, 4, 8, 6, 11], [-14, -5, -13, 1, -11, 16, -2, 0, 7, 2, 1, 13], [19, -4, 1, -5, 1, 3, 8, 6, 18, 7, -13, 4], [-12, 21, -11, -1, 4, -9, -6, -6, -6, 7, -12, -25], [23, -10, 10, 2, -17, 1, 1, 7, -2, 10, 4, 3], [4, -34, 3, 21, -12, 35, -11, -6, -22, 3, 17, 4], [-3, -14, 3, 0, 12, 15, -19, -1, -34, -3, 4, -1], [-31, -3, 2, -10, 6, 9, -9, 1, 2, 8, 11, -9], [19, -26, -3, -3, 11, -11, 1, 7, 10, 13, -3, -1], [11, -17, 2, 1, -10, 15, 9, -1, 6, 6, 10, 9], [-8, -11, -1, 6, 8, -7, -5, 14, 2, 8, -12, 13], [-12, 0, 12, -12, 0, -11, 3, -4, 17, -16, 10, 18], [-20, -8, -5, -10, 9, 1, -11, 14, -6, 4, 11, 1], [-13, 20, -9, 6, -7, -8, 5, -10, -7, -1, -2, -10], [13, 11, -10, 14, 0, -4, 8, 5, 16, 18, -2, -7], [21, -4, -4, 7, -8, 0, -1, -1, 7, 4, -1, -2], [-19, 6, 2, -8, -2, 5, -3, 10, -4, -5, -2, 2], [8, 10, 2, -14, 0, -1, 9, 9, 2, 3, 2, 9], [-13, -4, 0, -2, 3, -4, 8, -6, 2, -4, 2, 7], [-3, -9, 17, 6, -7, 2, 3, 0, -9, 1, 13, -14], [-8, -18, 5, 9, 6, 1, -21, 2, -9, -15, -3, 8], [2, -19, 17, 24, -19, 17, -11, -2, -23, -7, 28, 6], [-3, -14, 4, 10, -9, -4, -3, -9, 1, 5, 14, -1], [8, -3, 1, -3, 23, -16, -1, 2, 0, 3, -10, -19], [-6, -4, -3, 3, -11, -2, -9, 3, 9, -8, -10, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6039_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6039_2_a_f();".
// 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_6039_2_a_f(:prec:=12)
    chi := MakeCharacter_6039_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_6039_2_a_f();".
// 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_6039_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6039_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -5, 65, 203, -413, -92, 414, -97, -123, 57, 6, -7, 1]>],Snew);
    return Vf;
end function;