
// Make newform 6039.2.a.e 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_e();".
// 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_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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, 13, -93, -71, 257, 108, -238, -59, 93, 13, -16, -1, 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, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [517, -1304, -2446, 2384, 2567, -1242, -998, 125, 138, 39, -4, -6], [293, 844, 226, -1581, -1510, 85, 985, 518, -194, -173, 10, 15], [78, -3137, 860, 7209, -1359, -5921, 775, 2205, -181, -363, 14, 21], [107, -387, -2150, 3646, 4681, -4752, -2617, 2113, 528, -376, -35, 23], [-168, 252, 5326, -1577, -9513, 2694, 5300, -1404, -1155, 294, 86, -22], [-387, -1458, 356, 6913, 1057, -8123, -982, 3550, 265, -644, -23, 41], [185, -6242, 522, 15989, 0, -13089, -483, 4620, 196, -739, -21, 44], [-301, 1886, 5780, -8381, -9060, 9621, 4538, -4252, -918, 791, 66, -52], [603, -2490, -5931, 8532, 9060, -9621, -4538, 4252, 918, -791, -66, 52]];
    Rf_basisdens := [1, 1, 1, 151, 151, 151, 151, 151, 151, 151, 151, 151];
    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 := [[0, -1, 0, 0, 0, 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, -1, 0, -1, 0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 1], [-1, 0, 2, -1, 1, 1, 1, -1, -2, 1, 0, 0], [1, 1, 3, -1, 1, 0, 2, -1, -1, 1, 1, 0], [-1, 1, -1, -1, 0, 0, -1, 0, -1, -1, -1, -1], [-2, -1, -1, 0, -1, 1, 0, 0, 0, -2, 1, 0], [-2, 0, -1, 0, -1, 0, 0, 0, 2, 0, 1, 1], [-3, 1, -2, -1, -2, -1, 0, 1, 0, 1, -1, 0], [-3, 0, -2, -1, -1, 0, -1, 2, 0, -2, 0, 0], [1, 1, 1, 1, 2, 0, 1, 0, 1, 2, 0, -1], [0, 0, -2, -1, -2, 2, -2, 0, 0, -2, -1, -1], [2, 3, 0, 2, 3, -2, 1, -1, 1, 1, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, -1, 0, -1, -1, 0, 0, 2, 1, 0, 2], [0, -1, -4, -2, -3, -1, -1, 2, 1, -3, -1, 1], [0, -2, 1, 1, 1, -1, 0, 1, 0, 0, -1, -1], [-1, -3, -1, 2, 0, 2, -3, 0, 3, 0, -2, 1], [4, -2, 0, -2, -3, 0, -1, 1, 0, -3, -1, 2], [-3, 0, -1, 0, -1, 0, 1, 0, 4, 0, 2, 2], [0, 1, -1, -2, -1, 3, -1, 0, -1, -2, -1, 2], [1, 3, -2, 0, -1, -2, -2, 1, 1, 0, 1, 0], [-3, 2, 3, 2, 4, -2, 3, 1, -1, 3, 1, -2], [1, -2, -4, -1, -2, 3, -3, 3, 0, -2, -3, -1], [-2, 2, 1, 0, 3, 1, 2, -1, 0, 2, -2, -3], [2, 2, 2, -1, 3, 0, 2, -1, -3, 2, 0, -1], [0, 0, 1, 0, -3, 1, -1, 1, -3, -3, -3, -1], [-2, -1, -1, 2, 2, 0, 0, 1, -2, 1, -1, -4], [2, -3, 1, -1, -2, 2, -2, 1, 2, -2, 1, 2], [-4, 0, 2, 2, 2, -3, 3, -2, 2, 1, 1, -3], [-3, -3, 1, 1, 1, 2, 2, 0, -2, 1, 2, 0], [-3, 3, -5, 0, -4, -2, -6, 4, 6, -3, -1, 1], [-5, 2, 0, -2, 0, 1, -1, 1, -2, -2, -2, -1], [-6, 6, -1, 6, 5, -2, 2, -2, 2, 3, 1, -3], [4, -1, -5, -3, -6, 1, -5, 2, 1, -3, -3, 1], [0, 1, 0, 3, 1, 1, -1, -4, 3, 2, 0, 0], [2, -6, -1, 0, -2, 3, 0, 1, 0, 0, -1, 2], [-2, 2, 1, -3, -3, 1, -3, 1, -1, -1, -2, 3], [0, 6, 4, 1, 7, -3, 6, -3, -2, 2, 2, 0], [-5, 4, -5, 0, -4, 1, -2, -2, 0, -1, -2, 0], [-2, -3, 0, 1, 1, -4, 1, 1, 1, 1, 1, 3], [-4, 1, 1, 2, 1, 0, 1, 1, 1, 3, -2, -1], [-6, 0, -1, -2, -2, -2, 1, 3, 0, 3, 2, 3], [-4, 2, 2, -1, 3, 3, 1, 0, -6, 1, -4, -4], [-1, 3, 5, 5, 6, -1, 4, -1, -6, 3, -2, -3], [3, -1, 6, 1, 3, 0, 6, -3, -3, 3, 3, -2], [2, -2, 3, -3, 1, 2, 3, -1, -2, -1, -2, -2], [-1, -3, -1, -4, -2, 0, -2, 0, 0, 0, 0, 1], [-2, 4, 5, -5, 0, -1, 2, -1, -4, 4, 0, -2], [-3, -1, -4, 2, -3, 1, -5, 2, 6, -3, -4, -1], [-1, 1, 6, -1, 5, 1, 3, 0, -6, 2, 0, -2], [4, -2, 9, 4, 7, -5, 6, -4, -1, 4, 5, 3], [0, -5, -2, 0, -2, 2, -4, 2, 3, -1, -2, -1], [-1, 2, 5, -1, 6, -2, 7, -5, -3, 3, 0, -1], [2, -4, -1, 1, -6, 3, -9, 3, 3, -4, 0, 2], [-10, -1, 4, 0, 2, -4, 1, -4, 0, 4, 1, -1], [-5, -1, -2, -2, -1, 1, -2, -2, 5, -2, 0, 4], [-5, 0, 3, 2, -2, 3, 0, -1, -1, 0, 0, 3], [-13, -3, -10, 0, -7, -1, -4, 5, 8, -3, -4, 0], [4, -4, -4, -3, -5, 0, -6, 1, 3, -7, 2, 5], [-3, 1, -3, 4, 1, -2, -4, -2, 2, 2, -1, -1], [-8, 0, -3, -2, -5, 0, -1, 5, 2, 1, 1, 2], [-3, -7, 4, 2, 5, 1, 5, -4, -4, 2, 0, -1], [-1, -8, -8, -5, -7, 1, -5, 4, 3, -5, 0, 1], [-1, 0, 7, 0, 5, -3, 2, -2, -4, 5, 6, 0], [5, 0, 9, -6, 4, -1, 5, 0, -1, 0, 3, 3], [-2, 0, 1, 5, 2, -4, 0, -1, 1, 3, -3, -6], [0, 3, 1, 7, 3, 0, -2, 0, 3, -2, 0, 0], [0, -3, -3, -4, -4, 1, -4, 0, 2, -4, -2, 3], [4, 0, 6, -4, -2, 2, -1, -3, -4, 3, 1, 2], [1, 3, 3, -1, 1, -2, 5, 1, 0, 0, 3, -3], [1, -4, -7, 2, -2, 3, -5, 2, 7, -4, 2, 0], [0, -1, 6, 1, -2, -1, -2, -2, 0, 1, 4, 5], [-5, 2, 7, 2, 7, -2, 5, -6, -4, 1, 4, -3], [4, 4, 4, 3, 4, 3, -1, -5, -1, 0, -1, -1], [-1, -11, -2, -7, -6, 6, -8, 2, -2, -5, -2, 3], [10, -6, 3, 3, -1, 5, 0, -2, 1, -1, 2, 5], [-4, 7, -5, 5, 0, -1, -1, -2, 7, 5, 0, 1], [0, 9, 2, 0, 5, -4, 4, 1, -2, 3, 1, 1], [1, 8, -3, 5, 4, 0, -2, -4, 3, 2, -1, -3], [-5, -6, -5, 0, -1, 2, -1, 3, -1, 1, 1, 2], [14, 0, 8, 0, 3, 0, 0, -6, -4, 2, 1, 2], [-9, 1, 0, 3, 5, -4, 4, -3, 1, 3, 1, -2], [5, -12, -4, -3, -5, 5, -6, 5, 5, -3, 0, 4], [-5, -5, -3, -4, -3, -1, -5, 4, -2, -3, -4, -3], [-4, -2, -8, 2, -1, 1, -2, 4, 11, -1, -2, 3], [13, 0, -9, 1, -8, 4, -10, 5, 4, -8, -1, 3], [3, -1, 5, -4, 0, 2, -2, -5, -5, 0, -3, 0], [-4, -2, 1, -2, -6, 3, -3, 2, -6, -3, -5, -4], [-12, 9, -9, -1, -6, -3, -3, 2, 5, 0, 0, -2], [7, -4, -3, -1, -7, -2, -2, 2, 7, -7, 6, 5], [9, 6, 0, 1, 1, -3, 1, -1, 4, 0, 3, 0], [1, -7, -4, 1, -3, 1, -6, 4, 1, -2, 1, 3], [-4, -2, -6, -4, -5, 6, 1, 2, 1, -5, -4, 1], [-9, 3, -2, 12, 8, -7, 7, 1, 3, 4, 1, -6], [-1, -8, -5, 1, -2, 4, -1, 1, -1, -4, -1, 3], [-12, 6, 8, 0, 9, -5, 7, 1, -5, 3, 6, -3], [-5, 11, 0, -1, 3, -3, 8, -4, 3, 3, 3, 2], [-2, 2, 5, -3, 7, 1, 11, -1, -9, 6, 4, -2], [-9, 2, -10, 2, -4, 0, -8, 5, 8, -4, -2, -5], [-8, -5, 11, -2, 8, 0, 9, -3, -1, 5, 8, 0], [-12, -1, 0, -2, -7, 6, 0, 0, -5, 0, -1, -2], [3, 0, -12, 2, -10, 3, -10, 4, 4, -2, -7, -4], [-5, -3, 1, 5, 2, 1, 4, -5, 4, 2, 3, 1], [1, 1, -1, -6, -4, -6, 6, 1, -3, -1, 0, -3], [2, -9, -4, 5, -3, -2, -1, 0, 2, 0, 2, -2], [2, -3, 12, 2, 4, 1, -1, -1, 0, 2, 5, 3], [-2, 3, 2, -4, -4, -2, -1, -4, 0, 3, -2, 2], [0, 3, 4, -5, -2, 0, 6, -2, -5, -3, 2, 0], [-4, 3, -1, -3, -3, 0, -4, -2, 7, -6, 2, 5], [-1, 2, 4, -3, -1, 2, 2, -4, -5, 4, -2, 0], [-6, -7, -8, -3, -12, 8, -7, 0, 6, -6, 1, 5], [-12, -5, 1, -5, -1, 9, -4, -2, 0, -1, 3, 1], [0, 7, 8, 3, 14, -8, 14, -2, -3, 7, 9, -2], [8, 5, -2, -5, 6, -1, 3, -2, -5, 4, 0, -3], [-3, -8, 10, -5, -1, 3, 3, -1, -12, 0, 4, -2], [-10, -7, -7, -2, -1, -7, -1, 4, 3, 3, 2, 0], [-5, 2, -2, -1, -2, 0, 2, 6, -7, -1, 1, -2], [0, -6, 3, -6, -6, 6, -8, 4, 4, -5, 3, 7], [18, 0, 0, 5, 2, -2, -6, 1, 1, -2, -1, 1], [0, 1, -7, 0, 1, -5, -4, 2, -2, 0, 1, -3], [-4, 1, -2, 2, 2, -2, 7, 1, -7, 3, -1, -2], [-7, 0, 9, 6, 8, -1, 7, -5, -2, 6, 3, -5], [3, -11, -1, 2, 3, 2, 4, -2, -4, 1, -1, -3], [-5, 0, 6, 6, 10, -3, 4, -3, -1, 7, -2, -2], [3, 6, 13, 10, 18, -7, 15, -4, -3, 13, 3, -6], [2, -7, -5, -6, -15, 9, -9, 2, -3, -10, 2, 6], [-6, 1, -1, 9, 7, -5, -3, -1, 4, 2, -3, 0], [3, 5, 5, 3, 4, -2, 5, -6, -5, 6, 4, -3], [-1, 4, -1, 9, 3, -3, 4, 4, -4, -1, -2, -6], [-2, 10, 0, -1, 11, -4, 4, 2, -4, 2, -4, -5], [-2, -5, -6, 10, 6, -7, -2, 0, 0, 5, -3, -6], [-1, 1, -8, 2, -3, -2, -4, 3, 1, 4, -5, -5], [9, -6, 4, -3, -4, -1, -1, -3, -8, -2, -3, 2], [13, 5, 11, 10, 14, -4, 5, -2, 3, 8, -2, -6], [3, -4, -2, 9, 2, -6, -7, 2, 2, 1, 0, -2], [-4, 4, -6, 0, -5, -5, -4, -3, 0, -2, -5, -5], [-17, -1, 7, 6, 7, -2, 4, -1, -1, 5, -5, -7], [-1, -3, 5, 2, 7, 4, 2, 1, -1, 7, 3, 4], [2, -2, -5, 6, 0, 0, -8, 0, 10, 4, -2, 3], [-2, 8, 5, 5, 17, 0, 4, -4, -6, 6, -2, -10], [-9, 7, 1, 4, 7, 9, 0, 0, -3, 6, 0, -1], [-3, 4, 0, -9, -11, -3, 0, -2, 2, 0, 11, 7], [1, 1, -7, -10, -12, 7, -7, 0, -2, -8, 1, 2], [2, -3, 5, -6, -3, 7, -4, 4, -8, -8, -2, 0], [-1, -11, -12, -2, -16, 11, -16, 6, 3, -9, -11, 4], [-8, 1, -11, 6, -2, -2, -7, 0, 0, -3, 0, -6], [12, 5, 3, 8, 15, -7, 1, 3, 2, 4, -4, -3], [-5, 5, -1, 8, 7, 0, 6, -5, 11, 4, 3, -2], [3, 5, -9, 8, 0, -3, -7, 7, 13, 2, -6, 0], [-15, 10, 8, 7, 18, -7, 13, -5, -10, 10, -2, -9], [-1, -9, 7, 1, -4, 4, -4, 3, 1, 0, 3, 3], [11, 1, 1, 8, 6, -2, 1, 0, 3, 6, 1, 0], [-3, -7, -4, -4, -10, 6, -4, -1, 3, -14, 2, 5], [-5, 3, -9, -4, -14, -4, -4, 5, 3, -6, -1, 2], [-2, -2, 1, 7, -1, 3, -4, 1, 4, 0, -4, -3], [8, 8, 3, 6, 7, 0, 6, 0, 2, 4, -2, -4], [-15, 7, -7, 4, -3, 0, -3, -1, 4, -5, 0, -5], [6, -9, 10, 1, 10, 4, 5, -6, -4, 1, 6, 2], [-2, 0, 2, 0, -3, 1, -3, 0, 0, 2, -1, 4], [-9, 12, 3, 3, 13, -14, 16, -6, -1, 7, 7, 1], [-7, 9, -5, 6, 3, -2, 7, -7, -2, 3, 0, -4], [3, -9, 4, -4, -11, 10, -3, -2, -1, -4, 4, 6], [-2, -4, 4, -8, -9, 7, -1, 1, -5, -2, 4, 3], [-2, 6, 7, -5, 0, 2, 5, -2, -5, 1, 4, 9], [7, 3, 13, -11, 11, 0, 13, -4, -5, 9, 5, 0], [5, -9, -6, 1, -5, -4, 0, -3, 5, -1, -2, 2], [7, 8, -14, 0, -12, 2, -15, 6, 8, -9, -7, 3], [-2, -2, 8, -6, 8, -4, 13, -3, 0, 5, 2, -1], [-5, -11, -6, 4, -10, 12, -12, 0, 2, -4, -4, 2], [-9, 3, 7, 2, 4, -2, 3, -1, 4, 2, 5, 6], [1, -3, -9, 4, 8, -2, -7, 5, 4, -1, -8, -4], [14, 1, -2, -6, -2, 0, -6, 5, 0, 2, -7, 1], [-1, 8, 10, 4, 8, -9, 9, -7, 0, 4, 9, -2], [2, 5, -8, 3, 1, -4, 4, 1, 2, 0, -6, -3], [6, -3, -3, -1, -3, -1, -4, -2, 4, 1, 4, 0], [-8, -7, -16, -6, -18, 7, -9, 6, 7, -6, -2, 2], [1, -1, 11, -3, -5, 3, 4, -1, -4, -1, 5, 2], [3, 2, -12, -8, -3, -3, -5, 6, -3, -2, -13, -1], [0, -3, -13, 4, 0, 2, -3, 3, 3, -2, -3, -1], [-3, -9, 6, 3, 5, 8, 10, -3, -9, 2, -1, -3], [-11, 1, -5, -6, -8, 2, 1, 2, -1, -6, 2, 0], [0, 6, 11, 2, 13, -5, 7, 0, -8, 4, 7, -8], [-3, 5, -2, -2, -4, 4, -14, -3, -1, -2, -6, -6], [-14, 4, -1, 5, 1, -2, -1, 4, -3, 6, 2, 6], [2, 7, -12, 2, -13, 1, -11, 4, 5, -9, -7, -1], [2, 6, 2, 0, 10, -3, 5, -4, -4, 4, 2, -10], [21, 1, 7, -3, -6, 5, 2, -3, 2, -2, -2, 6], [-5, 5, -17, 3, -12, 4, -15, 8, 1, -7, -8, -1], [-5, -3, -7, -4, -6, 1, 0, 10, -1, -5, -4, -5], [9, -14, -5, -3, -17, 8, -9, -1, -5, -11, 1, 2], [1, 1, 15, 2, 16, -2, 16, -4, -20, 5, -7, -12], [2, -5, -6, 3, -8, 16, -14, 1, 2, -4, -6, -4], [18, -4, 1, 8, 10, -2, 0, 7, -3, 1, -5, -3], [-9, -7, 13, 17, 16, 4, 8, -8, 2, 9, 2, -6], [0, -9, -14, -10, -21, 4, -12, 3, 2, -6, 0, 2], [-4, -1, 12, 4, 16, -10, 14, 1, -4, 6, 7, -5], [16, 3, 9, -1, 9, -2, 2, -5, -4, 1, 2, 1], [-7, -9, -12, -12, -22, 6, -11, 9, 0, -15, -6, 9], [8, 0, 7, -8, 2, 2, -2, 2, -10, -2, -8, 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-1, 12, 1, -4, 2, 2, -10, -1, -5, -2], [17, 19, 5, -4, -1, -7, 9, -6, 4, -1, 12, 11], [9, -9, 13, -14, 3, -5, 12, -1, -6, 2, 10, 2], [7, -4, -10, 3, 3, 1, -4, 3, 7, -1, -9, 8], [14, -3, 5, -8, -1, 5, -3, -6, -23, -3, -5, -2], [5, 8, 0, 4, -1, 0, 8, 1, 4, 11, -4, 6], [9, -18, 16, 0, 14, 2, 12, -6, -5, -2, -5, -5], [1, 9, 7, -9, -11, 13, -9, 3, 3, -3, 8, 14], [15, -11, -11, 7, 3, 6, -12, 13, 3, -9, -8, 2], [-23, -14, -24, 6, -24, 6, -8, 5, 16, -1, 2, -1], [20, -16, 23, -1, 5, -6, 9, -4, 1, 3, 17, 7], [-29, 19, -10, -6, -6, 5, -7, 5, -1, -5, -15, -3], [-10, -9, 14, -13, -5, 4, 5, -5, -19, 2, 12, 5], [-9, 5, 1, -11, 0, 2, 3, 7, 0, -12, 4, 3], [-17, 15, 4, 10, 0, -13, 7, 1, 1, -3, 9, -8], [-12, 11, 4, -7, 9, 4, 2, 11, -12, -4, -2, -6], [-10, -8, -5, -10, -1, 5, -7, 2, 0, 2, -3, -1], [-9, -16, -16, -11, -31, 18, -18, 3, -4, -13, 0, 10], [-1, 4, 3, 9, 4, -1, -8, 6, -4, 7, -7, 0], [-9, -9, 2, 6, -1, -1, 2, 4, -5, -3, 13, -3], [-6, 12, 7, -3, 12, -5, 17, -8, -12, -1, 14, 0], [20, -4, 4, -13, 6, 14, 4, -7, -11, -1, -2, -9], [16, -15, -7, -10, -7, -4, -5, -1, -13, -5, 1, 2], [21, -1, -18, -8, -6, 1, -6, 12, 15, -14, -3, 15], [-12, -8, 18, 5, -1, 15, 3, 2, -16, 9, 0, -5], [-3, 6, 5, -7, 3, 1, 6, 7, -9, 2, -13, -3], [5, 6, 17, -17, 9, 2, 20, -13, -20, 16, 8, 2], [-24, 4, -34, -5, -39, 6, -25, 15, 14, -13, -10, -1], [-21, 13, 7, 16, 3, 7, -6, -4, 3, -1, 4, 3], [8, -3, -4, 13, 13, 3, 8, -4, -6, 2, -6, -17], [-4, 19, -12, 3, 7, -13, 6, 0, -12, 7, -12, -19], [5, 6, 16, 0, 11, 5, 3, -21, -12, 3, -2, -11], [13, -16, 13, 10, 5, -3, 4, 2, -2, 6, 3, 7], [-7, -9, 31, 2, 30, -10, 35, -13, 2, 21, 13, 1]];
    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_e();".
// 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_e(: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_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_6039_2_a_e( : 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, -13, -93, 71, 257, -108, -238, 59, 93, -13, -16, 1, 1]>],Snew);
    return Vf;
end function;