
// Make newform 6039.2.a.i in Magma, downloaded from the LMFDB on 24 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_i();".
// 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_i();".
// 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 := [-47, 236, 366, -742, -760, 841, 610, -469, -220, 136, 35, -19, -2, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4830, 9272, 14993, -10328, -18122, 2278, 11430, 1615, -3970, -577, 646, 47, -38], [-1964, 12785, 21328, -24751, -11934, 29789, -6221, -15191, 5684, 2967, -1217, -192, 80], [5338, -60160, -38492, 131903, 60892, -99199, -31352, 33414, 5943, -5029, -352, 275, -3], [-9293, 35335, 52816, -114506, -66105, 111281, 24220, -44875, -695, 7586, -738, -448, 74], [-8246, 32368, 39084, -105439, -47983, 109003, 12790, -46490, 3275, 8163, -1384, -495, 112], [1171, 1840, -46002, -10180, 100321, 11200, -69946, -4257, 20498, 699, -2670, -42, 128], [-11183, 57426, 16859, -147391, 6149, 132371, -22121, -52242, 12632, 8823, -2507, -524, 162], [8136, -53288, 6321, 134188, -45357, -118535, 47579, 45850, -19377, -7625, 3260, 448, -191], [-17754, 97970, 62706, -270378, -65676, 240723, 11676, -92220, 7541, 15172, -2451, -883, 187], [8700, -79409, 6554, 240692, -59488, -233199, 70611, 94362, -31573, -16038, 5640, 953, -344]];
    Rf_basisdens := [1, 1, 1, 1261, 1261, 1261, 1261, 1261, 1261, 1261, 1261, 1261, 1261];
    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, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, 1, 0, 0, 1, -1, 1, 0, 0, 1, 1], [-2, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1, 1], [2, 0, -1, 0, 0, 1, 1, 2, 0, 1, -2, -1, 1], [-1, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0, 1, 1], [0, -1, 1, 1, 0, -1, 1, -1, 1, 0, 1, 1, 1], [3, 2, -2, 0, -1, 0, 1, 1, 0, 0, -2, -1, 1], [0, 0, 2, -2, 1, -1, -1, 0, -1, -1, 0, 0, -1], [-1, -2, 1, 0, 0, 0, -1, -1, 0, 0, 2, 1, 0], [3, 0, -1, 2, 0, 0, 0, 0, 1, 0, 0, 1, 0], [0, 0, 0, 0, -1, -1, 1, -1, 1, -1, 0, 0, -1], [1, -1, 4, -1, 0, -2, -1, -3, -1, 0, 2, 2, -2], [0, -2, 0, 0, 0, 1, 1, 0, 0, 2, -2, -1, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, -3, 0, 0, 2, 0, 0, -1, 1, 2, -1, -1, -2], [0, 2, -1, -2, -1, 0, -1, 3, -1, -2, -1, 0, 0], [2, 0, 2, -1, 0, 1, -2, -1, -1, -2, 1, 0, -1], [-2, 2, -4, 0, -1, -1, -1, -1, 0, -3, 0, 2, 3], [-5, -2, 1, -1, 2, 2, 0, 0, -2, 0, 0, 0, -1], [2, 1, -1, 1, -2, 2, 1, 2, -1, 1, -2, -2, 1], [0, 3, -2, -1, 0, 0, -3, 0, 0, -4, 2, 2, 1], [1, -3, 2, 2, 2, 1, 1, -2, 1, 1, 1, -2, 1], [-2, 0, 0, -3, 3, 2, -1, -1, -3, -1, 0, -1, 0], [1, -2, -2, 1, 0, 1, 0, 1, 1, 0, -3, 1, 3], [0, 0, -2, 0, 0, 0, 1, 2, -1, 0, 1, 2, 3], [3, -1, 0, -1, 0, 1, -2, 1, 0, 1, 1, 0, -1], [3, 2, -3, -2, -2, -2, 1, 1, -1, -1, -2, 0, 1], [-6, -2, 0, 0, 0, 0, 5, -2, 3, -1, 0, 0, 2], [2, 1, 3, -2, -1, -3, 0, -5, -1, 0, 1, 3, 0], [3, -4, 2, 1, 0, 0, 1, -2, 1, 0, 3, 0, -2], [1, 4, -6, 2, -6, 0, 4, 2, 0, -1, -5, 0, 8], [-6, -3, 2, -1, 5, 3, -2, 0, -5, -1, 1, -3, -2], [-1, 0, 2, -2, 0, 0, 1, -2, -1, 0, -1, -5, -3], [6, 1, -1, -2, 2, 0, -2, -1, 0, 0, -1, 1, 1], [0, 2, 2, 0, -2, -1, 2, -4, 0, 2, 1, 3, 2], [0, -1, -3, 2, 2, 1, 2, 1, 0, 2, 0, 0, -1], [4, -2, 5, -2, 5, 1, -2, 0, -2, 3, -1, -4, -6], [0, 3, 0, -2, 0, 1, -1, -2, 2, -2, 0, 1, -2], [2, 4, 0, -5, 0, -1, 0, 0, 0, 1, -3, -3, -2], [4, 0, -1, 1, -1, 1, 2, 0, 2, -1, -1, 3, 1], [-3, 1, 4, -2, 0, -2, -1, -4, 0, -2, 4, 3, -3], [2, -4, 2, -2, 0, 1, 1, 1, -1, -1, 0, -2, 2], [3, 0, -1, 1, -5, -1, 5, 2, 3, 1, 0, -2, 2], [-2, -3, 8, -4, 5, 1, -3, -2, -4, 0, 1, -3, -4], [-8, 3, -2, 1, -2, -2, 0, 0, 0, -2, 2, -1, 0], [-1, 0, 5, 0, 3, -2, 0, -3, 3, 0, 4, -1, -7], [1, 8, -2, -2, -4, -1, 1, 1, -2, -6, 0, 2, -1], [6, 6, -3, 2, 0, -3, 0, 0, 3, 0, 0, 4, 0], [-2, -6, -1, 3, 1, -2, -1, -3, 2, 1, 5, 4, 2], [-1, 4, -3, -3, -5, -2, 4, 4, -1, 1, -3, 0, 1], [1, 3, -1, -2, 0, 2, -4, -1, -5, -3, 0, 0, 0], [1, -5, -5, 3, -1, -2, 2, 3, 1, 2, 0, 2, 3], [-1, 1, -3, 2, -1, 1, 3, 1, 1, 2, -3, -3, 3], [3, 1, 0, -2, -3, -2, 2, -2, -2, -3, -2, -1, 4], [8, -4, -4, 3, -2, -2, 2, 2, 1, 0, -1, 4, 5], [7, 2, -3, 1, -1, 1, 0, 0, 2, 0, -2, -2, -2], [6, -4, 1, 1, 3, 0, 0, 2, 0, 3, -2, -5, -6], [-2, 1, 3, 1, 3, -2, 1, -3, 1, 0, 1, 2, -3], [4, -3, 4, 0, 1, 0, 2, -3, -3, 1, 2, -1, -1], [1, 2, -7, -1, -5, 2, -3, 6, -3, 0, -2, -1, 5], [2, 4, -1, -1, -1, 1, 2, 4, 0, 2, -2, -1, 6], [-5, 0, 2, -1, 0, -4, 0, -2, 0, -6, 4, 2, 3], [8, 2, -3, 2, -2, 1, -2, 5, 1, 3, 2, 2, 0], [4, -2, -2, 4, 1, 1, -3, -4, 2, -1, 1, 5, 3], [-6, -6, 1, 5, 2, 0, -1, -1, 0, 1, -1, 4, 3], [0, 1, -2, -1, 1, -2, -6, 3, 1, 0, 2, 3, -3], [11, -2, -9, 5, -4, 1, 4, 10, 2, 5, -4, -1, 8], [-3, 6, -6, 1, 1, 0, -6, 1, -3, -2, 0, 0, 0], [-4, 5, 4, -7, 2, 1, -7, -1, -4, -4, 4, 0, -7], [1, 6, -3, 1, -5, -5, -1, -1, 0, -4, -2, 3, 3], [-1, 3, 3, -3, -3, 0, 5, 3, 1, 1, -1, 1, 1], [6, -3, 5, 3, -1, 0, -4, -2, 0, 5, 4, 0, -4], [1, 1, 3, 0, 5, -2, -3, 2, -1, -2, 3, -6, -4], [-1, -3, 4, -1, 3, 2, -2, -1, -1, -2, 5, -1, -1], [9, -3, 3, 2, 4, 2, 1, -1, 3, 2, 3, 0, -1], [2, 1, 3, -2, 1, 0, 3, -3, 1, 3, 1, -5, -2], [-7, 0, 4, 0, 0, 2, -2, 1, -6, -1, 5, 1, 1], [-8, 0, 6, -1, 4, -2, -5, 1, -1, 0, 5, -1, -2], [-6, 2, -2, -1, 1, -2, 1, 1, -5, -4, 2, 4, 3], [1, 4, -6, -2, -8, -1, 6, 3, 3, -2, -2, -1, 6], [-7, -3, 3, -2, 8, -1, -7, -2, -8, -6, 4, 1, -5], [-6, -4, 8, -1, 7, -4, -2, -6, 1, -2, 6, -1, -6], [1, 6, -1, -2, 1, -2, 0, 1, -3, 0, -2, 3, -1], [11, 9, -11, 5, -7, 1, 5, 5, 4, 1, -6, -4, 5], [-6, 3, 1, -6, 3, -2, -1, -8, -2, -6, 1, 3, 0], [1, -4, 3, -2, -2, -2, 0, -3, -1, 1, -1, 3, -3], [-4, -5, -3, 2, 1, 0, 2, 5, 6, 4, -3, -1, 4], [-9, 1, 3, -2, -1, -2, 2, -5, -1, -5, 3, 3, 3], [7, 0, -8, 6, -2, -3, 2, -2, 7, 3, -1, 3, 3], [-2, 1, 2, 0, -3, 2, 3, 2, 2, 8, -2, -3, 1], [2, 4, -4, -2, 2, 2, 2, 8, 2, 2, -5, -6, 3], [-7, 2, 1, -4, 5, 1, -8, 2, -5, -7, 3, -1, -2], [8, 0, 6, -1, -3, -2, 2, -1, 2, 0, 6, 2, -1], [-1, -4, -2, 0, -1, 3, 7, -1, 3, 4, -1, 0, 3], [8, 4, -2, -3, -2, 1, -4, 5, 0, -1, 1, -3, -2], [-8, -5, 10, -4, 6, 0, 1, -4, -7, -5, 3, -1, -3], [0, -1, 7, -4, 4, -5, -3, -2, -5, -4, 0, -1, -6], [3, 0, 3, 1, -2, 0, 3, 8, 4, 5, -7, -5, 1], [-2, 2, 4, 3, 1, -3, 1, -3, 1, 2, 1, 3, -5], [19, 6, -7, -4, -5, 1, -4, 4, -5, -3, -3, 2, 1], [-5, 7, 8, -10, 3, 1, -6, -2, -8, -7, 4, -1, -8], [5, -6, -3, 6, 1, 5, 3, 0, 3, 6, -6, -4, 4], [-8, -1, -7, 2, 0, -3, 8, -2, 4, -2, 0, 2, 5], [-2, -2, -7, 0, 1, 2, 2, 0, -2, 1, -5, -2, 6], [-1, -2, 7, 2, 2, 1, -1, 1, 3, 3, 4, -1, -4], [5, 2, -2, 5, -2, 0, 7, 4, 4, 5, -2, -6, -1], [10, -7, 2, 5, -2, -3, 2, -5, 6, 6, 7, 5, -1], [-1, 3, -1, 3, -2, -2, -1, 0, -4, -4, 3, -4, -1], [4, -4, 7, -4, 3, -2, -5, -7, -3, -2, 5, 5, 1], [7, 4, -3, 4, -5, -5, 1, 1, 7, -1, 0, 3, 7], [-4, -1, 13, -7, 3, -2, -2, -9, -3, -4, 2, 2, -10], [-2, -1, -2, 0, -1, 3, 1, -4, -2, -1, 2, 1, -1], [9, -7, 4, 7, 0, 2, -1, -2, -2, 6, 3, 1, 2], [-3, -5, 6, 4, 3, 1, 9, -10, 7, 7, 1, 0, 2], [-5, 4, 5, -6, -1, -1, -3, -4, -7, -2, 2, -3, -7], [9, -4, -3, 3, -1, 4, -1, 9, -6, 3, -3, 1, 4], [4, 14, -5, 4, -5, -4, -1, -4, 6, -1, 1, 5, 5], [15, 3, 5, -7, 3, 5, 2, 8, -2, 2, -3, -7, -5], [6, 9, -5, 2, -3, 1, -1, 10, 8, 4, -3, -4, -7], [7, 1, -4, 2, -1, 3, 0, -1, -1, 5, -5, -1, 1], [6, 3, -4, 10, -1, 1, -1, 7, 7, 2, 2, 7, 5], [14, 2, -4, 2, -3, -2, 0, -1, 2, 0, 4, 2, 1], [-11, 0, 9, -1, 0, -3, 4, -9, 4, -2, 5, 3, 3], [10, 1, -4, 1, -4, 2, -5, -1, -3, 3, -3, 0, -2], [-3, -1, 3, 1, -4, -2, 8, 2, 1, 1, 4, -1, -1], [4, -6, 8, 4, 2, 0, 0, -10, 0, 2, 5, -1, -10], [-8, -9, 6, -2, 8, 1, -6, -3, -3, -2, 10, 2, -3], [-5, 0, -1, 3, -3, 3, 5, 2, -4, 2, -5, -2, 3], [6, 2, -2, 6, -3, 3, 3, 1, 3, 3, -4, 4, 7], [4, -4, -5, 7, -2, 0, 5, 3, -1, 1, -1, -2, 7], [0, -4, 3, -3, 5, 0, 2, 5, -3, -1, -2, 0, 3], [-2, 0, -5, 1, -5, 2, 2, 5, 0, 2, -3, 4, 4], [3, 3, 1, 1, 2, 4, 4, -4, 5, 0, -7, -3, 0], [-1, -2, -1, 2, 4, 4, 5, 3, 5, 1, -1, -10, 4], [-4, -4, 9, -7, 9, 0, -8, -1, -8, -6, 2, -1, -2], [-2, 4, -5, 1, -4, -4, -11, 8, -3, -2, 1, 4, -1], [-4, 2, 9, -1, 2, -5, 0, -13, 2, -2, 11, 6, -1], [11, 2, -5, 1, 1, 1, 1, -1, 6, 5, -4, 2, 0], [8, 5, 3, -3, 7, -2, -7, -5, 1, 1, 2, 6, -4], [-18, 8, 2, -3, 2, 5, -3, 6, -4, -3, 4, -5, -3], [6, -6, 3, 2, 1, 5, -5, 3, -3, 2, 0, -1, -6], [-5, 1, -6, -3, -3, -1, -6, 0, -1, -7, -2, -1, -2], [0, 3, 0, -1, -5, -9, 2, -5, 3, -9, 6, 4, 2], [-18, -6, 7, -2, 3, 2, -9, -5, -8, -8, 6, 0, -7], [2, 5, 1, -8, 1, 3, -11, 6, -8, -5, 2, -6, -12], [-1, 2, -3, -3, 1, 1, 0, 1, 1, 1, -3, -2, 11], [-9, 7, 8, -7, 2, 0, -1, 0, 0, 0, -5, -2, 3], [-3, 6, 4, -3, 5, 1, -7, 1, -6, -5, 5, -1, -2], [6, 2, -8, -5, -4, 3, -4, 8, -6, -8, -7, 1, 6], [-18, 8, 3, -6, 0, 2, -3, -4, -7, -7, -3, 3, 0], [2, 1, 13, -1, 2, -8, -6, -8, -5, -3, 11, 8, -10], [7, 1, -10, -1, -1, 3, -5, 7, -6, -2, -5, 4, 10], [7, -7, 6, -1, 3, -3, -2, -4, -4, 1, -2, -8, -3], [9, -3, 1, 2, 3, 4, -4, 8, -2, 2, -5, -4, 0], [7, -1, -5, 0, -1, -4, 2, 1, 1, -3, -1, 1, 1], [-2, 8, 6, -12, -4, -6, -5, -2, 0, -4, 1, 2, 1], [2, 8, -11, 0, -3, -1, -7, 3, 0, -10, 2, 11, 8], [-7, 0, -2, 4, 4, 4, 9, -5, 10, 4, -2, 6, 7], [-18, 0, -4, 6, 2, -2, 4, 0, 4, -1, 5, 5, 6], [-7, 3, -8, 0, -10, -3, 6, 4, 0, -2, -2, -1, 11], [9, -5, 9, -4, 6, -1, -6, -3, 0, -3, 3, 3, -12], [-1, -4, 7, -3, 6, -3, 0, -3, -2, -5, 8, 1, 2], [0, -6, 2, -3, 0, -5, -1, 2, 0, 0, -2, 0, 2], [-8, 8, -4, -10, 3, 4, -2, -1, -2, -5, -1, -4, 0], [-10, 9, -12, 1, -6, 0, -1, 2, -3, -3, -3, 8, 12], [19, 12, -11, -1, -11, -3, 6, 0, 6, -2, -5, -2, 3], [-7, 8, -7, 1, -5, 5, 1, 10, 0, 2, -10, -5, 9], [-7, 9, -1, -2, 2, 0, -4, -3, -2, -4, 3, 4, -1], [11, -4, -2, 5, 3, 5, 7, 2, -3, 5, -6, 0, 2], [-8, -2, 1, -3, -4, -4, 1, 6, -7, -7, 0, -5, 5], [-8, 18, -9, -3, 0, -2, -8, 6, -5, -11, 1, 8, 7], [7, 6, -1, -3, 3, -2, -5, -2, 0, -7, 2, 2, 2], [11, 8, -2, -8, -7, -2, 0, 1, -1, 0, -3, -5, 0], [14, -5, -2, 3, -6, -2, 2, 1, 7, 4, -3, -1, 1], [-2, -13, -1, 2, 2, 2, 1, 1, -8, -1, 1, 5, 11], [20, 4, 2, 4, -4, 0, -2, 7, 2, 4, 1, -1, 0], [9, 0, 8, -3, 5, 4, -12, 2, -6, -3, 6, 0, -14], [6, -1, -5, -5, -1, -3, 5, -2, -1, -1, -10, -1, -3], [-10, -7, 15, -3, 10, 5, 5, -4, -5, 8, -2, -8, -3], [7, -6, 4, 10, -4, 0, 7, -4, 2, 7, 1, 7, -1], [-9, 1, 7, -5, 0, -4, -2, -2, -6, 2, 1, -1, 2], [5, 9, 0, -8, 1, 4, -11, 7, -7, -7, 4, 0, 1], [3, 3, -9, -4, 4, 7, 6, 8, 0, 4, -10, -11, 3], [-5, -1, 2, -1, -3, -1, 3, -12, -2, -3, 5, 9, 1], [6, 2, -8, 2, -2, -3, -12, 10, -10, -1, 0, -1, -1], [-4, -6, 15, 0, 9, -2, -5, -5, 1, 0, 5, 0, -6], [-9, -11, 10, -5, 5, -9, 2, -6, 3, -2, 4, 0, -3], [2, -6, 4, 2, 1, -4, 8, 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[-3, 13, 9, -7, -1, 7, -11, 5, -11, 0, -3, -1, -7], [5, 3, -5, -2, 4, 3, -9, 9, -20, -8, -4, 6, 8], [-20, 9, -12, -4, -6, 0, -2, 11, 3, -14, -2, 3, 20], [4, 9, -8, 2, 4, -3, -11, -8, -4, -6, 10, 3, -2], [-8, -21, -3, 3, -7, 1, 5, 5, 7, 2, 2, 10, 2], [4, -10, -10, -3, -6, 4, 15, 1, 2, 0, -14, 2, 6], [-14, 14, 4, -14, 4, -4, -9, 7, -1, -5, -6, -4, 2], [-2, -14, 6, -3, 2, 2, 12, -15, 4, 4, 2, 1, 1], [5, -5, 16, 7, -6, -6, 10, -12, 14, 4, 21, -4, -5], [-1, -10, 8, -16, 12, 2, -20, 9, -23, -11, 2, -10, -19], [2, 8, 4, -2, -1, 0, 2, -16, -6, -5, 7, 4, 0], [-7, -6, -6, 0, -2, -2, -1, 2, -1, 5, 2, -5, 4], [-9, 2, -3, -4, 5, -1, -2, 5, -11, -5, 0, -12, -2], [3, 4, -2, -6, 0, -6, -11, 2, 7, -4, 1, 5, 3], [-5, -3, 12, -10, 12, 1, -10, -11, -6, -4, 13, -4, -3], [-15, 14, -10, -3, -9, -5, 2, -5, 1, -9, 3, 10, 3], [17, -6, 6, 9, 3, 4, -9, 2, -7, 12, 0, -3, -11], [19, -4, -20, 10, -10, 2, 11, 13, 8, 5, -3, 5, 27], [-4, -22, 0, 7, 13, 11, -5, -1, 3, 5, -3, -4, 10], [7, 4, 8, 7, 7, -9, -4, -7, 11, 1, 10, 0, -16], [9, -6, -11, 4, -5, 1, 7, 13, 3, 3, -18, -11, 25], [-3, 11, -6, 4, 0, -10, -1, -2, 5, -6, 4, 2, 2], [-20, 11, 3, -1, 10, 1, -4, -4, -1, -5, -6, 12, 1], [23, -15, -14, 15, -7, 1, 3, 10, 17, 16, -4, 3, 10], [34, 14, -5, -7, -11, -8, 2, 8, 6, -6, 0, 3, 2], [6, -3, 13, -12, 15, -3, -20, 5, -19, -4, 3, -6, -15], [26, 4, -15, 5, -18, 2, 13, 5, 4, -2, -6, 4, 12], [2, -1, -9, 0, 8, -2, 0, 8, 1, 7, 1, 1, -5], [-1, -1, 3, 5, 3, 2, 7, -6, -1, 4, -4, 2, -1], [-28, 0, 10, 0, 15, 0, -10, -13, -2, -8, 12, -4, -17], [-11, -6, 1, 3, 0, 9, -9, 10, -16, 3, 5, 2, 2], [11, -5, -1, 15, -4, -1, 8, 5, 13, 13, -6, -8, 8], [9, -14, 13, 1, 0, -2, 6, -9, -6, 9, 0, 7, -8], [-1, 10, -14, 9, -16, -2, 14, 0, 7, 8, 0, -10, 13], [-14, 6, -2, 8, 14, -1, -15, -3, -4, 0, 14, 7, -11], [1, -8, -2, 9, -2, 12, -3, -5, -7, -5, 6, -1, 12], [-14, -2, 20, 1, -2, -5, 11, -10, 16, 4, 13, -1, -4], [20, 10, -8, -3, -5, -1, -4, 4, -3, 1, -3, 1, 9], [-6, 5, -6, -3, 8, -2, -4, -4, -8, 2, 9, 8, -8], [-33, -11, 12, 0, 9, 7, -5, -6, -11, 1, 1, -10, -10], [-1, -18, 1, 7, 8, -6, 8, -9, 11, 3, -3, 1, 1], [-9, -5, 13, -12, 0, 5, -11, -7, -16, -5, 12, 1, -3], [36, 12, -14, 5, -5, 1, -2, 20, -2, -1, -5, -1, 5], [0, 5, 12, 7, 2, -7, 13, -17, 20, 5, 5, 10, -2], [11, -14, 4, 7, 7, 2, 4, -8, 15, 1, 9, 4, 9], [30, 6, 2, -6, 6, 5, 3, 2, -4, 4, -15, -10, -9], [16, 8, -24, -1, -11, 6, -13, 20, -2, 6, -4, -7, 8], [-16, -2, 5, -1, -1, -3, -4, -7, 1, -9, 19, 17, -11], [9, -21, 1, 7, -7, 2, 9, -4, -2, 7, 2, -1, 0], [0, 13, -6, 7, -8, -15, 11, 1, 9, 1, 8, 2, -1], [1, 0, 11, -5, 14, 4, 4, -8, -4, 1, -2, -10, -17], [31, 10, 0, 1, 9, 3, -1, -8, 0, 6, -8, -1, -9], [9, -9, 2, 16, 0, -10, -1, -5, 4, 8, 8, 1, -11], [0, -3, -6, 2, 1, -8, 3, -10, 2, -7, -5, 7, 4], [11, 0, -3, -1, 3, 3, 5, -10, 6, -6, 0, -10, -16], [0, -3, -8, -6, 3, 0, -2, 10, -1, -6, -5, -7, -4], [-5, 1, 7, 10, 8, -8, 1, -11, 16, -1, 7, 20, 5], [-8, -1, -5, 1, 13, 4, -3, -6, 3, 2, -6, 0, -4], [-16, -1, -12, 1, -16, 1, 4, 7, -4, -6, -3, -2, 16], [-9, -5, 2, -1, -4, 6, -13, 8, 2, 11, 4, 8, 4], [15, 9, -1, -11, 3, 2, -9, 4, 5, -3, 4, -3, -8], [-6, -2, 5, 4, 9, 4, 11, -13, 6, -1, 7, -3, 3], [-2, 15, -23, 5, 0, -3, -5, 2, 4, -1, -3, 5, 3], [-7, 28, -20, -13, -16, 0, 1, 17, -5, -7, -11, -3, 15], [-19, -4, 19, -11, 16, 5, -22, 5, -13, 4, 10, -4, -19], [-7, 7, 2, -15, 6, 1, 6, 2, -9, -9, -14, -4, 2], [15, -17, 11, 10, -12, 3, 8, -6, 4, 15, -6, -6, 2], [7, 36, -12, 0, -15, 0, 3, 16, 3, 4, -11, -15, -6], [5, 12, -14, -5, -8, -6, -4, -3, 8, -11, -1, 2, 10], [38, 13, 2, -6, -6, -5, -9, 7, 1, 1, 1, 1, -9], [-13, 18, -24, -9, -2, -2, 0, 1, 12, -7, -10, 7, 6], [-1, 22, -13, -7, -10, -8, 7, -3, 2, -2, -5, 4, 8], [-11, 6, 12, -12, 3, 0, -1, -2, -10, 1, 0, -24, -14], [-4, 10, -20, -3, -14, 5, -9, 19, -12, -1, -9, -10, 1], [6, -10, 10, 5, 2, 0, 4, -10, 1, 6, -6, 4, 4], [5, 8, -8, -12, 11, 3, 5, 11, -11, 0, -12, -17, -5], [-25, 6, -8, 11, -5, -8, 9, 1, 3, -4, 2, 7, 17]];
    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_i();".
// 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_i(:prec:=13)
    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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6039_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![47, 236, -366, -742, 760, 841, -610, -469, 220, 136, -35, -19, 2, 1]>],Snew);
    return Vf;
end function;