
// Make newform 6039.2.a.d in Magma, downloaded from the LMFDB on 14 May 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_d();".
// 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_d();".
// 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, 14, -29, -76, 115, 55, -109, -2, 37, -6, -4, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, -37, 14, 54, -35, -45, 32, 17, -10, -2, 1], [6, -35, -38, 378, -55, -408, 97, 145, -41, -16, 5], [-30, -62, 16, 387, -64, -408, 97, 145, -41, -16, 5], [6, 109, -83, -396, 188, 339, -137, -107, 40, 11, -4], [-33, -109, 92, 396, -188, -339, 137, 107, -40, -11, 4], [21, -19, -136, 432, -53, -489, 152, 179, -61, -20, 7], [15, 28, -94, -114, 125, 119, -73, -44, 20, 5, -2], [-42, -43, 296, 342, -506, -360, 314, 128, -79, -14, 7], [-6, -226, -46, 1224, -305, -1236, 386, 431, -142, -47, 16]];
    Rf_basisdens := [1, 1, 9, 9, 9, 9, 9, 9, 3, 9, 9];
    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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [1, -1, 1, 0, -1, 1, 1, 1, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 1, -1, -1, 0, -1, -1, -1, 1, 0, -1], [2, -2, 0, 0, -1, 0, 1, 1, 0, 0, 0], [-2, 2, -1, 1, 0, -1, -1, -1, 0, 1, 0], [2, 0, 0, 0, -1, 1, 1, 1, 1, 0, 1], [2, -2, 2, -1, -1, 0, 0, 2, -2, -1, -2], [-2, -1, -1, 0, -1, 1, -1, 1, -1, 1, 0], [-1, 2, 1, -1, 0, -1, 1, 0, 1, -2, -2], [0, 0, 0, -3, 2, -3, 0, 0, -1, 0, -2], [0, -1, -1, -2, -1, 0, 2, -1, 1, 0, 1], [5, 0, 1, 0, -1, 2, 1, 1, -3, -1, 2], [2, -2, -2, 0, -3, 3, 0, -3, 0, 4, 2], [2, 1, 0, -1, 1, -1, 0, -3, 1, 2, -1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 2, -1, -1, -1, -2, 0, 0, 1, -2], [0, 2, -1, -1, 1, -3, -2, 1, -1, -5, -1], [1, -3, 2, -1, 4, 0, 2, 5, -1, -2, -2], [4, -2, -1, 2, 1, 2, 1, 1, -3, -1, 1], [-2, 6, -1, 2, 2, -2, -3, -1, 4, 2, 0], [7, -1, 5, 0, 1, -3, 3, 3, -3, -2, -2], [-2, -4, -2, -2, -2, -1, 2, -1, -2, 1, 0], [0, -4, -1, 1, -2, 2, 0, 0, -2, 3, 0], [-3, 0, 1, 0, 0, 3, -1, 4, -4, -2, 0], [0, 4, -3, 3, -1, 3, -3, 0, 4, -2, 0], [2, 1, 2, 3, 1, 2, -1, 1, 1, -3, -1], [1, 1, -1, 1, 3, -3, -2, 7, 0, -3, -4], [7, -2, 4, 4, 2, 2, 1, 6, 1, -3, -3], [0, 0, 0, -2, 1, -5, 0, -2, -3, 2, -2], [0, 4, -3, 1, -2, -3, -4, -5, 2, 3, 3], [-2, 1, -1, -1, 3, -3, -1, 0, 3, 1, 1], [-1, 1, 0, 2, 2, 2, -3, 0, 2, -1, -2], [-1, 3, 3, 1, 4, -4, -3, 4, 1, -3, -5], [1, -2, -1, 1, 0, -3, -2, 0, 0, -1, -3], [2, -6, 2, -2, -3, -3, 5, 1, -2, 1, 4], [7, -2, 0, 2, 2, 1, 5, 6, -2, -7, -1], [-1, 2, 0, 0, -5, 0, 0, -3, -2, 2, 1], [7, -1, 3, 2, -1, 0, 3, -1, 0, 3, 3], [-2, 4, -8, 0, -3, 2, -2, -10, 1, 6, 8], [-2, 5, 2, -3, 5, -4, -3, 4, -1, -4, -3], [2, -2, 3, -6, -2, -2, -2, -1, -1, 1, -1], [10, -5, 0, 1, -3, 8, 5, 2, -1, -1, -1], [1, -3, 0, -1, -6, 1, 1, -2, -5, 4, -2], [9, 0, 4, 1, 0, -5, 4, 3, 1, -6, -4], [8, -11, 2, 0, -1, 1, 5, 2, -4, 1, 4], [7, 0, 2, 1, 3, -3, 1, -3, 3, 0, 3], [-6, 2, 0, 2, 3, -2, -10, 1, 5, 0, -1], [-3, 0, -1, -2, -5, 1, 1, -3, -4, 1, 3], [-6, 9, -8, -2, 0, 0, -6, -13, 4, 7, 3], [-2, -1, 3, -2, 2, 0, 0, 3, -2, -4, -2], [-3, 6, 0, 2, 3, 2, 0, 1, 0, 2, -2], [10, 0, -1, -1, 4, 3, 3, 1, 3, -4, -2], [5, 2, 2, 1, 2, 1, -1, 2, 2, 0, 4], [2, -3, 0, 0, -4, 2, -4, -6, 2, 2, 0], [-1, 4, -1, 2, -5, 2, -3, -5, 4, 8, 4], [10, 0, -6, 0, -9, 10, -1, -11, 1, 4, 10], [1, 2, 0, 1, 3, 2, -4, 1, 4, 0, -4], [-3, -5, 2, -1, -1, 3, 0, 3, -6, 0, -3], [9, -5, 0, 1, -5, 6, 4, 0, 1, 3, -1], [-7, -2, 1, 2, 2, -2, -1, 3, 8, 1, 0], [-2, -5, -4, 0, -4, 3, -2, 0, -1, 4, 2], [-8, 3, -1, 1, 0, -7, -5, -6, 3, 0, 4], [-2, 2, -5, -6, -6, 1, -2, -9, 2, 8, 2], [-8, 2, 1, -3, 2, -6, -4, 6, 7, -1, -6], [2, 1, 1, -5, -2, -3, 3, -1, -3, -2, 2], [4, 2, 8, -1, -1, -3, 4, 2, 2, -7, -2], [2, 5, -3, -1, -7, 5, 3, -5, -3, 4, 0], [3, 1, -1, 3, 4, 0, -5, 4, 2, -4, -4], [5, -5, 4, 0, -4, -1, 4, 3, -4, -2, 7], [3, 5, 4, -1, 7, -5, 1, 9, -2, -11, -2], [10, -8, 2, -1, -3, 7, 5, -4, -3, 2, 5], [-4, -1, 2, 1, -3, 2, 3, 1, -3, 0, 0], [-1, -3, 1, 3, 2, 6, 9, 2, 3, 0, 5], [3, -3, 7, 0, -6, 4, -2, 4, -7, 0, 1], [3, 3, 6, 1, -4, -2, 2, -1, 0, 0, 1], [11, -10, -6, -4, -6, 0, 2, -2, 1, -2, -1], [10, -5, 1, 3, -1, 3, -1, 4, 5, -4, -4], [3, -6, 3, -2, 0, -5, -4, 7, -1, -6, -7], [3, 0, -10, 6, -3, 5, 0, -8, 6, 6, 13], [-9, -2, 3, 1, 6, -4, 0, 6, 6, -7, -8], [-1, 2, 1, 5, -4, 9, 2, 3, 5, 0, 3], [5, -8, 0, -5, -3, 2, 7, -9, -3, 10, 7], [2, -4, 1, -3, -4, 1, 1, -3, -1, 3, 1], [2, 0, -4, 4, 3, 2, 6, -1, 7, 2, 1], [2, 7, -4, 3, -7, 1, -6, -13, 1, 9, 10], [5, 0, 0, 0, 1, -5, -5, -5, -2, 4, 4], [2, -2, -7, 4, 2, 6, 1, 1, 1, -2, 0], [-2, 11, -1, 1, 4, -6, -9, -10, 4, 0, -2], [6, -2, -2, 2, -7, 2, -1, -9, -1, 3, 5], [-1, 11, 7, 3, 5, -11, -3, 3, -1, -4, -4], [-5, 4, -1, 5, 6, -4, -1, -6, 3, 2, 0], [4, -10, 7, -5, -3, -2, 9, 5, -5, -4, -2], [1, 0, 4, 4, 0, -3, 0, 12, -3, -1, 0], [-5, 1, 6, -1, -2, 2, -2, -2, 0, -2, -1], [2, -11, 5, -2, 1, -2, 8, 1, -1, 8, 4], [-4, 3, -4, 2, -2, -11, -9, -7, 5, 1, -3], [6, -6, 4, -3, 1, -4, 2, 12, -7, -4, -2], [-9, 18, 0, 8, 5, 1, -9, 0, 1, 0, -3], [13, -9, 1, -4, -5, 5, 5, -4, -7, 2, -2], [6, -2, 5, 1, 8, 1, 7, 2, 2, -6, -5], [7, -9, 5, 3, 3, -1, -3, 8, -4, 1, -1], [-4, 9, -7, 2, -1, 1, -12, -18, 4, 9, 2], [12, -1, -2, 8, -5, 16, 6, 5, -4, -5, 6], [13, -12, -2, 1, -6, 7, 6, -2, 4, -3, 5], [-18, 10, -13, -4, 5, -3, -12, -10, 8, 0, -1], [6, -2, 1, 5, 3, -1, -7, 7, 0, 0, 0], [17, -4, 1, 6, 1, 6, 14, 7, 1, -4, 5], [-4, -3, -3, -3, -7, -1, -1, -6, -2, 2, -1], [-7, 4, -3, -8, -1, -9, 0, -7, 1, 9, 3], [2, -7, 6, 4, -2, 9, 3, 3, -2, 1, 0], [-16, 3, -4, -2, 4, 4, -1, -8, 4, 3, 2], [26, -6, 6, 2, -4, 2, 11, 3, -2, 0, 3], [2, -4, -4, 3, -3, -10, -4, -1, 4, -4, -5], [-5, 1, 8, -1, 2, 0, 0, 1, -4, -9, -2], [-16, 9, -2, 0, 5, -9, -8, 1, 2, -8, -2], [-4, 10, 5, -1, 6, -2, 5, 11, 5, -9, 1], [15, -10, 0, -6, 1, -1, 11, 2, -9, -6, 0], [15, -4, -5, -1, -6, 13, 7, -10, -5, 15, 10], [-1, -4, 2, -4, 1, -5, 2, 3, -6, 6, -2], [-10, 4, -3, -1, 2, -6, -9, 2, 3, -5, -5], [9, -5, 6, -2, -7, 2, 1, 2, -7, 0, 1], [9, -9, 1, 6, -3, 8, 0, 12, -1, -6, -3], [-18, 17, -1, -4, 6, -2, -2, -6, 10, 10, 2], [-9, -6, -3, -7, 1, 6, 0, -6, 0, 0, 3], [1, -9, -2, 0, -1, 1, 9, -7, -4, 5, 2], [-15, 3, 1, -8, 0, -4, 2, -6, -4, -3, -4], [0, 1, -6, 4, 4, -8, -10, -9, 8, 2, 5], [-4, -4, -10, 4, 7, 1, -3, 1, 6, -2, 0], [-14, 11, -4, -4, 10, -4, -1, -2, 5, 3, 0], [-1, -6, -5, 2, -5, 4, 3, -7, 1, 3, 2], [-1, 12, -8, -1, 1, 1, 2, -10, 4, -1, 7], [-10, 4, -5, 5, -5, 14, -3, -10, 9, 9, 7], [-1, 3, 3, 4, 5, -5, 0, 7, 7, -12, -7], [-5, 6, -5, 3, 4, -1, -1, -1, 4, 12, 1], [-5, 2, 3, 1, 5, -4, -2, 1, -2, -9, 3], [-7, 0, 1, -2, -1, -9, -4, 2, -9, 8, -4], [-13, 1, 1, 0, 7, -7, -6, -3, 1, -3, -8], [2, -5, -2, 5, -2, 3, 9, -3, 3, 6, 3], [14, 0, -6, 2, -3, 4, 0, -6, 9, -3, 7], [1, 8, 4, 6, -3, -1, -5, 5, 5, -8, -8], [-1, -10, -2, 0, -2, 0, -4, -3, 2, 9, 3], [3, -6, 5, -7, 5, 2, 11, 9, -9, 0, -6], [-2, 1, 4, -1, -1, -17, -8, 6, 6, -3, -8], [1, 4, -13, 2, 3, -5, -7, -4, 1, -6, 10], [-5, 2, -4, 2, 10, -5, -1, -2, 6, 1, 4], [-4, -3, -5, 4, 2, -5, -4, -3, 1, -2, -4], [12, 1, 2, 11, 8, 4, -1, 5, 0, -2, 6], [19, 2, -1, 8, -5, 14, 5, 5, -3, -1, 5], [-6, -1, -6, -4, 7, -11, -1, 1, 5, -9, -4], [-16, 10, -3, -3, 2, 1, -5, -10, 6, 1, 4], [16, -5, 2, 4, -7, 10, 7, 1, -5, -8, 3], [-22, 0, 1, 1, 4, -5, 0, 4, 2, 1, 10], [-12, 8, -6, -4, 7, -9, -2, -8, -1, 0, -1], [-3, -4, -5, -6, 6, 3, 5, 0, 1, 7, 4], [13, -8, 6, -6, 0, -4, 8, 4, -1, 1, 0], [18, 2, -6, 2, -1, 6, 3, -7, -1, 5, 1], [-11, -1, -6, -3, 1, -6, -3, -1, 9, -3, -4], [12, -10, 13, -9, 1, -4, 3, 9, -5, -8, 5], [0, 11, -3, 6, 1, 4, -4, -6, -6, -2, 9], [-2, -16, -1, -10, 3, 5, 10, 3, -3, -3, -4], [-4, 5, 1, -7, 5, -9, -12, 11, -7, -9, -11], [4, 6, -9, 9, -2, 10, 4, 1, 5, 3, 12], [3, -5, 3, -4, -5, 5, 5, 13, -6, -13, -2], [-7, -4, 0, 3, -2, 11, -1, 7, 6, 2, 5], [0, 10, 7, -5, 6, -7, -11, 9, -6, -7, -11], [1, 9, -6, -3, 12, -2, 5, -6, 6, -4, 4], [19, -12, 11, -2, -8, 12, 13, -6, -11, 6, 8], [6, 4, -4, -2, -6, 2, -3, -5, 9, -3, 3], [-19, 7, -9, 4, -18, 10, -6, -24, 9, 17, 12], [-27, 18, -4, 3, 6, -20, -13, -3, 9, -4, -8], [-5, -15, 6, -4, -4, 2, -2, 10, -12, 0, -8], [2, -1, 1, 2, 10, 4, 16, 5, 8, -1, -1], [19, -5, 6, -3, -4, 4, 5, 0, 3, -5, 1], [6, 6, 1, 7, 0, -7, 6, 2, 0, -10, 3], [16, -6, 3, 13, -7, 4, 8, 3, -6, -1, 10], [-5, 10, 10, 0, 4, 4, -1, 15, -1, -13, -7], [-6, 1, 2, -1, -1, 1, 0, -2, 2, -5, -7], [2, -4, -1, -5, -6, 15, 10, 0, -9, -2, 3], [4, -8, 9, -2, -3, -3, 3, 7, -12, -8, 7], [-4, -13, -2, 0, 8, 7, 2, 6, 6, -6, -7], [0, -7, -2, -3, -7, 8, 10, 0, -3, -6, 13], [4, -8, 5, -11, -4, 9, 3, 9, -16, 0, -4], [5, 2, -1, 14, -3, 5, -4, -4, -1, 0, 2], [6, -9, -1, -5, 1, -2, 3, 5, -9, -3, -7], [-16, 10, -8, -2, -6, 1, 5, -19, 5, 8, 7], [17, -13, 0, 7, -8, 8, 6, -3, -4, 6, 8], [10, 4, -1, 3, 9, -5, -6, 1, -1, -3, -9], [6, 7, 0, 4, 7, 2, 5, -2, -4, -2, 0], [11, 12, -5, 8, -6, -2, -2, -7, 4, 4, 8], [2, -10, 12, -3, -2, -8, 3, 18, -6, -14, -4], [-27, 21, -8, 9, 5, 2, -14, -14, 10, 8, 1], [28, -12, 4, 1, -7, 4, 9, 3, 5, -4, -4], [6, -11, 13, -5, 4, -3, 2, 17, -1, -10, 2], [8, 8, 5, -1, -1, 4, 2, 3, 2, 2, 7], [9, -13, 5, 7, -4, 15, 6, 15, -6, 7, 4], [7, -9, -7, 1, 3, -7, -7, 1, 1, -1, -5], [-35, 12, -2, -6, 10, -9, -6, -10, 5, 6, 1], [12, -18, 3, -3, 4, 13, 14, 18, -12, -6, 1], [-2, 6, -9, 5, -1, 5, -12, -18, 0, 8, 1], [-4, 9, -12, 5, -4, -12, -15, -13, 9, 6, 10], [11, -8, 2, -2, -16, 12, 2, -5, -2, 9, -3], [13, -3, 0, -2, -5, 2, -1, 2, -8, 6, 0], [-5, 3, -5, 5, 5, -4, -5, -12, 8, 7, 7], [-3, -12, 3, 0, 6, 6, 17, 17, -3, -2, -6], [-10, -6, 4, 0, 0, 9, -1, 12, 3, 3, 7], [-4, -9, -11, 5, 0, 10, -8, -1, -5, 10, 11], [-11, -2, -6, -3, 3, 2, -2, 7, -6, 12, 1], [18, -8, 6, 1, 0, -9, 1, -6, -5, 9, -4], [-13, 14, 0, 10, 8, -7, -7, -5, 7, 4, -5], [-19, 5, 1, 0, -2, -5, -15, -3, 11, 7, -5], [4, -2, -4, -9, -6, 4, -7, -5, -10, 6, 7], [-7, 4, 0, -1, 2, 6, -5, 2, 10, -7, 11], [16, -10, 15, 0, -6, -3, -3, 11, -17, -4, -6], [6, -7, -4, 2, -5, -16, -5, -3, 3, 10, 4], [6, -2, 10, -4, -10, 0, 11, 2, 0, -8, -6], [19, -6, 1, 2, -2, 4, 12, 1, -1, -8, 7], [1, -18, 4, -13, -10, -6, 16, 6, -12, 2, -5], [9, -2, 1, -5, 12, -17, -4, -1, 7, -16, -3], [-20, 21, 1, 6, 18, -13, -11, -2, 13, -3, -6], [-3, -3, 7, -7, -7, 8, 10, -10, -7, 11, 11], [4, -15, 3, 2, -7, 18, 13, 1, -8, -2, 7], [26, 0, 4, 8, 1, 7, 4, 1, -12, 3, 0], [15, 0, -5, 9, 3, 10, 2, 2, -6, -2, 9], [-7, 10, 9, 0, 3, -3, -1, 10, -4, -3, -5], [-1, 6, 0, 6, -2, 17, 9, 1, -4, -7, 8], [-6, -5, -7, -11, -18, 2, 0, -10, 0, 14, -1], [9, -1, -2, 8, -4, 4, 1, -10, -3, 5, 1], [27, 0, -2, -4, -3, 9, 14, -13, -10, 6, 14], [-27, 14, 0, -7, 8, 0, -1, -10, 15, 0, 1], [-27, -2, 3, -6, 2, -8, -8, -3, 3, 6, 0], [10, 0, 10, -5, 14, -4, 7, 17, -9, -12, 2], [-19, 9, 5, 2, 11, -19, -4, 9, 6, 0, -14], [8, -14, 2, 5, 6, -10, 2, 7, -6, -5, -4], [19, -15, 10, 8, -5, 9, 15, 11, 1, -4, 6], [-6, 11, -3, -6, -1, 0, 7, -1, 2, -9, 1], [2, -8, -2, -9, 6, -10, 1, -1, 0, -10, -12], [-44, 29, -14, 2, 11, -15, -23, -20, 19, 14, -3], [7, -4, -8, -5, -3, -4, 5, -1, 4, 5, -2], [-15, 7, -8, -1, 1, -2, -11, 5, 6, -1, 4], [-10, -4, 11, 2, 3, -5, 1, 17, 5, -3, -2], [-9, 0, -3, 9, 12, -10, -5, 9, 1, 2, -2], [23, -2, 25, 1, 8, -7, 10, 17, -14, -15, -5], [27, -18, 3, 3, -14, 18, 15, 2, 3, 5, 7], [-17, 11, 9, 4, -1, 3, 1, 15, -1, -2, 4], [-9, -14, -1, 2, -3, 12, -2, 13, 0, 0, 4], [20, -9, 9, 5, 12, -3, 13, 12, -9, -7, -4], [18, -7, 10, -1, -10, 16, 6, 1, -1, 8, 9], [6, -13, 9, -2, 4, -5, -5, 14, 1, -7, -11], [-2, 4, 3, -5, -1, 5, 8, -1, 4, 8, -3], [-9, 9, -5, 3, -5, 21, 5, -15, 5, 1, 16], [12, -1, 12, -9, 6, -15, 12, 1, -10, 4, -12], [-4, 5, -1, 5, 1, 1, -8, -4, -1, 3, -8], [-12, 2, -10, 7, 2, 1, -1, 12, 1, -4, -9], [17, 4, 6, -9, 7, -6, 8, -6, -11, -5, -7], [-31, 12, -15, 4, -6, -7, -17, -5, 9, 5, 5], [0, -6, 12, 1, 15, -5, 6, 19, -11, -6, -17], [8, 14, -16, 7, 7, -15, -8, 7, 6, -11, -8], [3, 16, -12, 11, 19, -8, -5, -11, 20, -2, -2], [16, 4, 3, -12, -3, -6, -8, 13, -5, -10, -5], [10, 19, -3, -8, -3, -8, -7, -12, 10, 10, 0], [13, -12, 9, 2, -3, 1, 9, 8, -11, -9, -7], [2, -15, -5, -3, -5, -5, -2, -6, 9, -2, 4], [-16, -5, -5, 10, 10, -8, -1, 9, 4, 5, -7], [-2, 15, 16, 4, 16, 3, 2, 14, -6, -16, -8], [18, 18, -8, 1, -3, 2, -3, -15, 4, -2, 13], [-8, 16, -5, -13, 2, 8, 2, -22, 5, -3, 4], [-28, 19, 1, 4, 9, 3, -12, -4, -1, 0, -4], [8, 19, 7, 3, 5, -4, -7, -4, 8, 1, 6], [-29, 15, -9, 6, 2, 4, -21, -16, 11, 10, 3], [9, -17, 9, 8, 5, 15, 19, 17, -6, -8, -2], [19, -7, 15, -7, 2, 0, -2, 13, -1, -14, -5], [26, -12, 7, -7, -6, 4, 2, 3, -8, -7, 16], [-21, 14, -16, 4, 2, 0, -15, -8, 12, -3, 13], [-17, 8, -8, 1, 3, -15, -5, 5, 5, -5, -5], [-2, -2, 3, -6, -10, 11, -8, -12, -2, 20, 13], [4, 4, -1, -5, 3, 15, 1, -4, 2, 0, -6], [-15, 8, -19, -3, -3, -12, -9, -16, 0, 19, -2], [-13, 4, 7, -4, 10, -4, 5, 9, -7, -1, -10], [7, -10, -8, -1, -16, 15, 7, -10, 1, -3, 6], [15, -4, 5, 6, 5, 8, 3, 9, -6, 6, 4], [18, 2, -2, -2, -11, -4, 2, 3, -4, -1, -1], [-1, 3, 11, 9, -9, 13, 7, -1, -8, 1, 13], [-21, 6, -1, 3, 14, -13, -4, 13, 7, -18, -17], [31, -21, 6, -12, -14, 13, 22, 0, 0, 0, 6], [5, 26, -10, 4, 2, -8, -5, -13, 3, -3, -1], [-5, 3, -5, -12, -12, -5, -3, -4, -11, 2, -3], [10, -13, 5, 2, -6, 13, 8, 10, -5, -7, 13], [-27, 20, -7, -7, -6, -3, -5, -27, 16, 7, -11], [21, -5, -7, 2, 3, 13, 3, -1, 11, -13, 6], [-11, 8, -6, 19, 3, -3, -14, -1, -1, -1, 10], [3, 3, 18, 8, 9, -6, 9, 19, -1, -5, -9], [28, -8, 10, -3, -7, 17, 18, 15, 1, -14, -1], [-12, 5, -7, -1, -11, 14, 6, -21, 5, 20, 20], [34, -20, 3, 1, -14, 4, 7, -2, -7, 6, 3], [-6, 2, -7, -2, -6, 1, 12, -11, -9, 9, 5], [3, -6, 5, -1, 3, -7, 3, 4, 15, 4, -18], [13, -3, 9, -4, -8, 7, 8, 2, -12, -2, -5], [7, -7, -7, 5, 2, 4, -9, -13, 2, 14, 8], [7, 8, -3, -6, -5, -1, -4, -29, 2, 16, 11], [-8, -1, -13, 8, 1, 3, -2, -15, 8, 9, 5], [-7, 9, 6, -11, -6, -1, 4, 4, 0, -2, -5], [-22, 32, -8, -7, 4, -5, -8, -25, 8, 14, 2], [10, 0, 1, 0, -2, 11, 10, -16, 2, 14, 11], [-9, 16, -3, 8, -3, 2, -7, -1, 9, 2, 13], [30, -21, -2, -13, 7, -1, 3, 12, -6, -7, -1], [-16, -4, -1, -17, -5, -7, 3, -3, -2, 4, -15], [4, 24, -9, -1, 2, -10, -15, -22, 6, 7, 6], [13, 20, 8, 3, 0, 6, 9, 8, 2, -4, 14], [4, -2, -5, -7, 13, -8, -1, -2, 2, -6, 0], [-15, 1, 8, -4, 1, 19, 2, 11, -5, 1, 1], [-1, -16, -4, 6, 7, 15, 15, -5, -2, 9, 5], [9, -3, 15, -1, 12, -12, 10, 19, 0, -8, -13], [22, -1, -9, 13, -15, 20, -1, -3, -6, -2, 7], [4, 7, 5, 1, 5, -2, 9, -1, 8, 2, -7], [-7, 2, -6, -2, -12, 5, -6, -19, 5, -2, 13], [22, -17, -2, 10, 5, 18, 12, 7, 5, -4, 8], [8, -6, 12, -12, -11, 14, 15, 3, -18, -8, -2], [-6, 5, -9, 0, -9, 14, 1, -4, 12, 10, -3], [-3, 24, -8, 2, 5, -14, -17, -4, -4, -12, -1], [-4, 5, -3, 3, 8, -4, -19, 7, 9, -3, -8], [16, -4, 14, -10, -10, -11, 9, 17, -2, -9, -17], [-1, 17, -3, -1, 14, -20, -6, 12, 5, -14, 1], [1, 3, 6, -10, 20, -15, -4, 14, 0, -10, -8], [-5, -6, 2, -5, 0, -4, 7, 6, -4, -8, 3], [-15, 3, -2, -4, 18, -18, -3, -3, -11, -2, -7], [7, -9, 3, -14, 6, -16, -4, -13, -14, 10, 6], [-34, 22, 0, -6, 11, -12, -5, 0, 18, 5, -7], [1, 25, -9, 6, 15, -1, -11, -11, 14, 10, 3], [17, -2, -1, 1, -17, 19, 12, -1, -11, -1, 5], [6, -13, -4, 3, 1, 8, -6, 2, -9, 0, -4], [6, 16, -2, 0, 3, 2, -16, 3, 3, -15, 2], [11, -4, -4, -5, 9, 6, 7, 5, 5, 7, 1], [8, 17, 9, -5, 16, -12, -9, -2, 8, -6, -3], [22, -5, 16, 4, -10, 19, 1, 10, -6, -1, -4], [16, -13, 9, 8, -6, 13, 0, -7, -2, -4, 10], [-22, -16, -7, 7, 5, -3, 2, 5, 11, 1, -5], [-3, -1, 6, 0, 2, -23, 0, -2, -8, 9, -9], [-10, 18, 0, -1, -12, 3, -18, -25, 5, 25, 5], [6, -16, -1, 13, 4, 1, -4, 23, -8, -2, -6], [18, -3, -11, 4, 1, -7, -7, 3, 10, 8, -3], [-31, 18, -9, -7, 4, 2, -8, -2, 11, 10, -12], [-8, 21, -10, 9, 13, -1, 7, -6, 14, -5, 11], [-11, -4, -14, -10, 14, -3, 0, -4, -11, -7, 0], [-9, 3, -3, -5, -6, -13, -29, -13, 15, -5, -5], [-3, 20, 10, -3, 0, -15, -9, 9, 15, -9, -8], [8, 14, -1, 5, -19, 19, -2, -3, -8, 4, 10], [-21, -5, 0, -8, 12, -6, -4, 6, -9, -19, -18], [-34, 11, -13, -6, 14, -25, -22, -17, -1, 10, -9], [-23, 24, -2, 10, 1, 11, -5, -6, 19, -3, 4], [-1, -17, -1, 7, -6, -2, 8, 8, -12, 13, -1], [-13, 11, -9, 7, 18, -4, -4, 2, 0, -10, -15], [43, 3, 9, 7, -3, -3, 8, 8, 2, 1, 6], [-29, 28, -5, -3, 14, -6, -17, 4, 7, -5, -13], [-12, 6, -8, -4, -5, -8, -12, -14, -11, 4, -10], [30, -14, 5, 10, 8, 21, 24, 10, -6, -6, 5], [-10, -15, -6, 4, -11, 3, 8, -7, -2, 8, 13], [-10, -14, 10, -5, 16, 0, 12, 1, 1, 5, -6], [-11, 1, -10, 5, -4, 23, 3, -7, -17, -1, 12], [28, -17, -6, 2, -11, 12, 11, -15, 11, -4, 17], [12, -9, -15, 6, -12, 9, 4, -16, -3, 14, 13], [20, -24, -3, -4, -15, 16, 3, 14, -12, 6, 5], [-29, 9, -15, 2, -2, 6, -5, -1, 0, 10, 16], [-1, 5, 0, -11, -11, 0, -2, 2, -5, 4, 5], [-3, 2, 3, 7, 16, -9, -6, 8, 2, -5, 9], [-12, 18, -5, -4, 8, -2, -8, -5, 5, -2, -4], [22, 9, -2, 3, 0, 9, -2, -5, -10, 6, 10], [16, 13, 0, 10, 22, -8, 7, 20, -5, -26, -10], [-4, -1, 10, 14, 6, -5, 14, 15, 7, -15, -3], [2, -6, 3, 0, 4, 8, 8, 14, -16, -9, 1], [-32, 13, -11, 2, 13, -5, -7, 0, 1, 4, 6], [-18, 7, -6, 10, 12, 0, -9, -8, 12, 9, -4], [9, 6, -7, -12, -4, 2, -4, -22, 16, 3, -13], [-3, 1, 8, 2, 1, -12, -2, 6, 3, -8, -6], [-9, 21, -12, -11, -10, 12, -6, -28, 5, -5, 5], [1, -4, 3, -7, 0, -8, -6, 5, -1, -3, 8], [16, 7, 14, -2, -7, -1, 11, 17, -1, -29, -16], [-26, 17, -6, 0, 7, -20, -13, -3, -2, 8, -3], [-5, 1, 1, 9, -5, 1, -15, -4, -5, -7, 0], [-28, 13, -15, 2, 19, -18, -15, -5, 20, 4, -12], [-12, 9, 3, -2, 8, -16, 8, 12, 8, -22, -16], [28, -9, 2, -15, -8, 6, 3, -9, 0, -11, 7], [34, -21, 14, 24, 0, 22, 16, 14, 5, -3, 15], [0, -12, -15, -2, -5, 2, 18, -14, 2, 6, 16], [7, -16, -7, -8, 2, -3, -4, -9, -12, 10, 1], [0, 13, 12, -2, -6, 6, 6, 4, -3, -15, -1], [2, -7, 8, 6, -13, 1, 5, 24, 0, -11, -9], [-27, -1, -2, -3, 16, 9, 5, -2, -6, 2, 8], [21, -18, 19, -12, 14, -7, 13, 23, -15, -23, -24], [5, -7, 2, 3, 3, 5, 1, -8, 11, -8, -2], [20, -11, 20, 8, 7, 11, 16, 17, 0, -6, -10], [-41, 8, -26, -5, -14, 4, -15, -17, 9, 10, -1], [13, 1, 17, -4, -5, -6, -5, 6, -10, -10, 4], [4, -16, 13, -16, 8, 11, 13, 31, -15, -19, -11], [-15, 6, -8, -2, -1, 14, -10, 6, -4, -4, 13], [43, -28, 16, 2, -17, 17, 9, 13, -28, 2, 10], [-6, 10, 0, -5, -1, -17, -18, -10, 15, 6, 0], [2, -11, 4, -9, -18, 5, 1, 9, -23, 6, -2], [-11, 0, -3, -1, -5, -19, -14, 0, 1, 2, 4], [-23, 25, 13, 9, 11, -9, 0, 7, 14, -7, -7], [10, -10, 7, 3, 18, -2, 11, 17, 5, 2, -2], [-16, 13, 8, 2, -5, -17, -19, 5, -3, -1, 1], [2, -4, 2, 4, 4, -27, -16, 15, 5, -5, -19], [18, -15, -6, -4, -2, 1, 1, -3, -3, -12, 9], [-1, -5, -3, 16, 14, -3, -4, 3, 1, 11, 1], [40, 11, 9, 5, -3, -6, 0, -3, 4, 9, 4], [-35, 19, 3, 6, 15, -2, 4, 1, 10, 1, -7], [18, -13, -3, -12, -15, 17, 2, -26, -12, 17, 21], [41, -13, -3, 11, -11, 19, 7, 0, 2, 12, 7], [2, 12, 4, 0, 8, -23, -10, -20, 11, 4, 10], [43, -27, -3, -11, -27, 23, 30, -10, 1, 4, 26], [-26, -1, 23, -9, 5, -27, -10, 25, -8, -13, -26], [5, -10, 5, -14, 19, -30, 3, 1, -11, -6, -12], [19, -11, 7, 4, 3, 16, 12, 29, -5, -23, -6], [-30, 33, -11, 17, 12, 7, -8, -17, 13, 11, 7], [-26, 13, -17, -7, 3, -8, 0, -19, -8, 0, 7], [-12, 1, 4, -17, -11, 2, 9, 1, 9, -2, 11], [18, 9, 7, 8, -5, -5, -6, -7, -7, -4, 15], [-7, 25, 8, 8, 25, -16, -10, 5, 16, -12, -5], [16, 9, 7, 8, -25, 13, 7, 5, -8, 2, 23], [21, -21, 0, 14, 6, 20, -3, 6, -3, 16, 6], [-20, 24, -1, -6, 1, -19, -9, -26, 22, 0, 0], [-9, 19, -7, 2, 6, -19, -6, -7, 9, 6, 8], [36, -13, -1, -12, 0, 7, 6, 12, 5, -13, -18], [-13, 11, -2, 11, -3, -3, -14, -15, 0, 28, 6], [12, -20, 7, -17, -16, 10, 12, 6, -14, 6, 2], [-12, -6, 15, -6, 13, -12, -4, 22, -11, -12, -5], [22, -15, -10, -1, -9, -3, -1, -1, -19, -15, 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_d();".
// 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_d(:prec:=11)
    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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6039_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 14, -29, -76, 115, 55, -109, -2, 37, -6, -4, 1]>],Snew);
    return Vf;
end function;