
// Make newform 5239.2.a.j in Magma, downloaded from the LMFDB on 22 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5239_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_5239_2_a_j();".
// 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_5239_2_a_j();".
// 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, -1, -29, -31, 21, 19, -7, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-3, -1, 1, 0, 0, 0, 0, 0], [0, -4, -1, 1, 0, 0, 0, 0], [3, 9, -3, -3, 1, 0, 0, 0], [0, 3, 9, -3, -3, 1, 0, 0], [2, -9, -30, 3, 21, -3, -4, 1], [2, -20, -34, 18, 20, -7, -3, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1];
    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_5239_a();"
function MakeCharacter_5239_a()
    N := 5239;
    order := 1;
    char_gens := [1861, 1522];
    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;

function MakeCharacter_5239_a_Hecke( : Kf := Kf)
    return MakeCharacter_5239_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]];
    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, -1, 0], [2, 0, 1, 0, 0, 1, 0, 0], [0, 0, -1, -1, -1, 0, 0, 0], [2, 0, 1, 1, 1, -1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 1, -1, 1, 1, 1, -1], [1, 1, 2, 1, 2, 2, -1, 1], [0, 0, 2, -1, 1, 1, 0, 1], [1, 0, 2, 2, 0, -1, -2, 1], [1, 0, 0, 0, 0, 0, 0, 0], [3, -3, 0, -2, -1, -2, 0, -2], [2, 1, 0, 2, 1, 2, 1, -1], [0, 0, -1, -2, -2, 0, 2, -3], [3, 0, 0, -3, -3, -2, -1, 2], [-1, -1, 1, -1, 0, -1, 1, -2], [0, 1, 1, 1, 2, 3, 1, 2], [1, -1, -1, 2, 1, 2, 0, -2], [-2, 1, -1, 0, -1, -1, -1, -1], [1, 0, 1, 0, 2, 2, -2, -1], [-3, 2, 3, 0, 0, 1, 1, 0], [4, 0, -2, -2, 0, -4, 0, 1], [1, 0, -1, -2, 2, -1, -1, -1], [3, 3, -1, 4, -1, 0, -2, 2], [3, 0, 2, 2, 0, 0, 1, 2], [-1, -4, -1, 0, 2, 2, -3, -1], [-1, -1, 3, 4, 5, 3, 0, -2], [-5, -4, 1, -1, 3, 3, 2, -2], [-1, 2, -1, 1, 3, 1, 5, -2], [-6, 3, -2, 0, -3, 0, 1, 0], [1, 3, -1, -1, 0, -3, -2, -2], [-1, 1, 0, -1, 0, 5, 1, 2], [0, 0, -2, -2, 0, 2, 5, -2], [-3, 2, 2, 2, 6, 1, -4, 4], [5, -3, -6, -1, 0, -5, 0, 0], [-5, -4, -1, -3, -1, -4, 2, -2], [-2, 3, 2, -2, 1, -2, -3, 1], [4, -2, 4, -2, 0, 1, 2, -3], [-4, -1, -6, -1, 0, -4, 5, -5], [-9, 5, -3, 3, -2, -1, 2, 1], [-2, 1, -2, -6, -5, -1, 3, -3], [-3, 0, -3, 2, 0, -3, 3, -4], [7, -7, -5, -1, 0, -6, 3, -2], [6, -1, 1, 1, 2, 3, -4, 0], [6, -1, 0, -3, -2, -2, 3, -2], [5, -3, 0, -2, -1, -2, 3, -6], [-1, -4, 2, -4, 0, -2, 0, -1], [-2, -1, -1, -2, -1, 7, 2, 2], [0, 10, -6, 1, -1, -1, -2, 1], [-3, 3, 4, 6, 5, 4, -5, 2], [5, 2, 0, -1, -3, -1, -9, 9], [-3, -6, -6, -5, -5, -2, 8, -4], [-5, 4, -3, 5, 3, 2, -3, -5], [7, 3, -1, 4, 3, -2, 3, 0], [-7, 4, -2, 7, 1, 1, 2, 2], [2, -2, -1, 3, -5, -3, -2, 3], [-16, 3, -7, -1, -2, -2, 2, 1], [6, -5, 7, -1, -2, 0, 2, -3], [4, -3, 2, 3, -4, 0, 2, 3], [5, -2, 2, -1, -3, 3, -1, 6], [0, -1, 2, -2, 7, 3, 0, -3], [18, -3, 3, 5, -2, -2, -2, 3], [-3, 5, -1, 2, -1, 1, -4, 2], [3, 0, 3, -2, -2, 2, -3, -3], [-1, -5, -4, -7, 0, -1, 5, -2], [13, 4, 0, -4, -8, 0, 0, 1], [-8, -3, 0, -1, 6, -6, -2, -3], [-7, 6, -5, -6, -6, -3, 9, 2], [19, 3, 1, 4, -1, -6, -5, -5], [-8, -1, -5, -4, -3, -1, 6, -7], [2, 10, -1, 5, 1, -1, -4, -2], [10, -8, 5, -2, -2, -5, -2, -3], [-3, -4, 2, -4, 4, -1, -1, 2], [-7, 7, -3, 2, -5, -7, 0, -3], [-9, -6, 0, 2, 2, 3, 9, -6], [7, -5, 5, -2, 1, 2, -8, 3], [-1, 5, 4, 5, 4, 9, 2, 4], [7, 0, -2, -1, 1, -7, -7, -1], [4, 3, 3, -1, 2, 2, -2, 7], [-4, 0, 8, -2, 0, 6, 5, 4], [8, -2, 4, 0, -2, -3, 4, -8], [6, -1, -3, 4, 5, 2, -6, 1], [-1, 5, -8, -2, -11, -8, 0, 4], [-5, 7, -4, 2, 3, 0, 6, -5], [9, 3, 5, -6, -3, -3, -6, 5], [-14, 6, -3, -6, 0, 3, -2, 2], [-9, 1, -5, -3, -2, -1, 4, -4], [2, 11, 2, 1, -4, 4, -7, 9], [-2, -2, -3, -4, 2, 2, -6, -2], [0, -1, -1, 6, 7, -1, 6, -2], [-6, 3, -1, 3, 4, -9, -5, -4], [0, 5, 6, 6, 5, 6, 2, -6], [4, 3, 1, -2, -3, -3, -6, 3], [5, -1, -1, 1, 0, 5, -2, -2], [2, -3, 6, 3, 6, 8, 0, -3], [-6, 5, 0, 5, -2, 7, 4, -1], [16, 3, 2, 3, 2, 0, 2, 2], [6, 6, 2, 12, 0, -4, -10, 3], [-1, 7, -3, 3, 2, -3, -4, -3], [-5, 2, 9, 4, 8, 4, 2, 2], [9, 0, 3, 15, 5, 3, -8, 3], [1, 3, 0, -4, -3, -8, -5, -6], [-4, 6, 1, 18, 8, 1, -7, 2], [4, 2, 0, 0, 0, 0, -1, -4], [-10, 1, -1, -3, -6, 2, 0, -1], [-2, -2, 2, -1, 1, -1, 3, -5], [-4, 6, -3, -6, 2, -6, 4, -5], [10, -9, 4, -3, 4, -3, -1, -5], [-12, 7, 11, -7, -2, 9, 1, 2], [10, -4, 5, 8, 10, -2, 4, -2], [10, 2, 14, -8, 0, -4, -5, 1], [3, 10, 6, 4, 2, -3, -7, 7], [14, 0, 9, -1, 5, 5, 0, 0], [7, -1, -6, -5, -2, -3, -1, 5], [15, -1, 8, 8, 3, -2, -9, -3], [1, -5, 0, -2, 7, 2, 2, 1], [-4, 0, -7, -3, -1, -11, 3, -15], [2, 13, -3, 7, 2, 6, -4, 5], [4, -1, 1, -5, -4, -4, -5, 6], [13, -6, -12, -3, -5, -15, -7, -8], [8, 4, -6, -4, -4, -7, -8, 0], [-6, -3, -8, -1, 2, 2, 7, -8], [-23, 2, 0, -7, -3, -2, 3, -3], [6, -7, 1, 1, 2, 1, -4, -4], [18, -2, 5, 6, 8, 9, -2, 1], [13, -4, 2, -8, -12, -2, 6, -8], [10, -2, -2, -3, -5, -9, 0, -4], [0, 7, -4, 3, 4, 0, -5, -3], [-4, 5, 1, 2, -3, 4, -8, 13], [20, -8, 3, -13, -5, -6, -2, -1], [16, 2, 8, 9, 9, 7, -8, 1], [-24, 1, -9, 2, -3, -1, 4, 2], [-2, -3, 5, -6, -9, 0, -4, -2], [17, 2, 5, -5, 3, -2, -4, 2], [7, 0, 2, 5, -3, -12, -6, 5], [19, 1, 0, 13, 4, -6, 0, 3], [9, 3, -7, 5, -2, -9, 1, 0], [14, 8, 4, 1, -5, -9, -7, 0], [-4, 1, 3, -6, -9, -3, -5, 0], [23, -7, -7, -10, -7, -6, -2, -3], [4, -7, 4, 6, -3, -9, -7, 7], [5, 9, -11, 4, -7, -6, -3, -4], [0, -16, -3, 4, 6, 4, 1, -8], [1, 0, -2, -6, 4, 5, 2, -1], [3, 6, -1, 6, -4, -3, -6, -1], [-4, -5, -9, -11, -8, -8, 10, -3], [4, -9, -6, -11, -14, -12, 2, -6], [-9, -4, -2, -10, -2, 5, 15, -3], [13, -1, 3, 18, 7, -6, -11, 10], [-9, 10, -6, 1, -1, -10, -3, 1], [-2, -7, 6, 5, 8, 3, 7, -8], [1, -3, 0, 6, 5, -12, -13, -6], [-1, -2, 5, 4, -4, 11, 3, 9], [2, 5, -9, 9, -2, -7, 0, -5], [-21, 6, -5, 3, 1, 3, 4, 4], [-4, 5, -9, 1, -4, -12, 8, -4], [2, -7, -5, -7, 2, -4, 10, -11], [-8, -1, 1, -2, -3, 7, -1, 8], [6, -5, -3, -1, -2, 10, 6, -1], [14, -8, -5, 4, 0, -14, -8, -5], [-21, 3, -4, -16, -3, 0, 8, -5], [-1, -5, -2, -11, -4, -15, -4, -2], [-15, -4, 5, -5, -5, 0, 4, 2], [5, 4, -1, 3, 3, -14, -11, 0], [6, -6, 12, 0, 2, 13, 4, -8], [13, 8, 9, -7, -5, 8, 5, 5], [-24, 5, -24, -3, -12, -8, 3, 4], [-3, 5, 7, 10, 3, 13, -10, 2], [-7, 12, 3, -1, 1, 8, 2, -2], [-20, -1, -5, 4, 7, -9, -5, -5], [-23, -10, 11, -5, 9, 8, 5, -3], [5, 8, -10, 8, -2, -7, 1, 3], [11, -4, 3, 2, -2, -3, -8, 11], [-10, 1, 9, 5, 10, 5, -5, -1], [9, -8, 4, -7, -11, -3, 5, 0], [-9, -8, -9, -2, 6, 1, 11, -3], [21, -13, 2, 1, -6, -10, -1, -8], [16, 2, -14, 4, -2, -3, 11, -4], [-7, -2, -10, -7, -11, -6, 2, 3], [13, -5, 12, 8, 15, 8, 1, -6], [0, -12, 1, -5, 5, -12, 10, -13], [8, -12, 6, 1, -3, 5, -1, 14], [18, 3, 1, 10, -7, -6, -13, 7], [-7, -6, 1, -1, 3, 4, 2, -11], [14, 5, -11, 2, -9, -4, -7, 7], [10, 12, 11, 1, 1, 9, -7, -1], [9, -16, -1, 0, 2, -7, 8, 0], [11, 11, 1, -4, -3, -3, -10, 5], [-6, 0, -3, 7, -9, 5, 6, 5], [-1, 10, -9, 4, -2, -6, 1, 4], [10, -3, 6, -6, -9, -7, -2, -8], [-8, 7, -2, 5, 4, 3, 4, -2], [-23, -10, -4, 1, 7, -6, 2, -7], [6, -13, -13, -6, -3, -13, 10, -2], [-16, 2, 10, 7, -3, 13, 8, -5], [-10, -2, -18, 1, -3, -13, -2, 3], [5, -2, 4, 9, -7, 4, 2, 3], [-3, -3, -13, 8, -3, -6, -7, 1], [6, 16, 3, -5, -7, 0, -10, -4], [-3, 0, 19, 4, 10, 14, -4, -2], [-1, -18, -12, 1, 1, 5, 3, -3], [-10, -10, -4, -6, -10, -2, 14, 5], [-5, 2, 0, 4, 2, 13, -3, 1], [4, 14, -4, 7, -5, -2, 8, -5], [5, -8, 10, -1, 13, 8, -1, -3], [-12, -2, -11, -2, 4, -14, 2, -6], [-15, -9, -15, 9, 4, -8, -3, 3], [11, 1, -3, 0, -1, 8, 7, -2], [22, 8, 25, -4, 4, 3, -7, 2], [-1, 1, -6, -17, -12, 2, 10, 2], [14, -1, 22, -9, 6, 4, 0, 5], [14, -8, 13, 3, -7, 11, 2, 0], [8, 2, -9, -7, -9, -7, -5, 11], [3, -3, -4, 15, 0, -8, -19, 7], [-6, -6, -2, -10, -6, 16, 8, -11], [9, -5, -9, -3, -10, 4, 8, -7], [22, 17, 7, 4, 1, -1, -16, 8], [-3, 10, -1, -4, -10, -2, 3, -12], [-12, 9, 0, -2, -5, -5, 13, -2], [-4, 7, 8, 1, 6, 11, 1, 10], [-21, -10, -18, 5, 3, 0, 3, -4], [0, 10, -3, -3, 1, 4, 3, -5], [4, -4, 11, 19, 11, 4, 1, 3], [-25, 8, -14, -5, -19, -13, 10, 2], [-16, -5, -11, 0, 5, 2, 0, -3], [14, 6, 1, 7, 7, -17, -5, 3], [-13, 3, 4, 11, 14, 6, -4, 9], [7, -15, -7, -6, 5, 10, 11, 2], [5, -2, -18, -14, -6, -13, -1, -12], [12, 0, -3, -1, -7, -13, 5, 6], [6, -5, 3, -7, 8, 0, -7, -7], [9, -4, 7, 18, 12, 10, 13, -4], [-3, 6, 9, 8, 8, 0, -2, 15], [12, -4, 0, 13, 13, -1, 0, -1], [-23, 3, -17, -11, -12, -11, 2, -10], [-23, 15, -1, 4, -3, 16, 6, 17], [-20, 10, -13, -13, -9, -12, 3, 0], [-2, 0, 2, 0, 2, 4, 1, 3], [-12, 10, 4, 5, 1, -1, -2, -2], [-8, -2, -2, -14, -14, -4, 3, 4], [14, -3, 5, 4, -3, 7, -16, 10], [35, -4, 8, 16, 2, -5, -3, 4], [-8, 7, 6, 7, 8, 16, 19, 2], [-1, -14, 5, 10, 2, 1, -7, 7], [13, 10, 2, -10, -8, 3, 3, -3], [-3, -5, 4, 1, -2, -1, -16, 19], [-16, 13, 19, -3, 10, 20, 3, -2], [-7, 5, -17, 0, -5, 4, 3, -7], [-11, 16, 6, 11, 3, 2, -12, 3], [11, -7, -8, 6, 9, -6, 1, 2], [-8, -10, 3, -5, -3, 6, 10, 11], [-3, 3, 4, -2, 7, 1, -10, -2], [-17, 7, -26, -2, -1, -12, 16, -8], [1, 3, 1, 7, 0, 12, 12, 0], [-16, 0, 8, 1, -7, 2, 8, -2], [-4, 3, 8, 2, 1, 17, -18, 4], [-16, -3, -6, -6, -1, -16, -6, -7], [12, -11, 2, 10, 13, 9, -1, 1], [-31, -2, 4, 2, 4, 10, 14, -4], [-13, 1, -2, -2, 3, -11, -6, -1], [6, 4, 3, 10, 10, 18, 9, -6], [15, 1, 8, 5, 14, 3, 0, -4], [16, -6, 4, 4, -2, 4, 0, -7], [3, -9, -1, -1, 4, -12, 11, 0], [11, -11, 12, -1, 14, 7, -12, 0], [-11, -12, -13, -7, 1, 11, 10, -8], [19, 1, -1, -8, 7, -7, -4, 2], [-2, 7, 12, 16, 9, 5, -2, 15], [-11, -18, 3, -2, 18, 3, -6, -6], [6, 4, 11, -10, 6, 1, -1, 3], [-6, 2, 8, 3, 5, 1, -16, 13], [-27, 1, -4, -1, 6, 11, 8, 1], [-5, -7, -8, 2, -5, 6, 21, -3], [-11, 20, 17, 2, 6, 17, -4, 18], [12, 7, -14, 5, 4, -21, -6, 6], [7, -17, 10, -1, 0, -2, 4, 4], [6, 11, -6, 0, -17, -6, 6, -7], [9, -2, 23, 4, 4, -1, 1, -2], [5, -8, -12, -9, -5, -20, 10, -2], [-14, -4, 12, -6, 0, 16, -8, 12], [11, 6, 15, -1, -3, 3, -5, 0], [42, -6, -7, 6, -12, -19, -3, 3], [2, -8, -3, 8, 16, 1, -18, 2], [33, -13, -3, -3, 2, -10, -11, -7], [0, -8, -13, 8, -12, -5, -9, 15], [2, -8, -1, -4, -12, -16, -9, -7], [37, -4, 8, -1, 3, 0, 2, -3], [20, 15, 7, 9, 6, -12, -13, 17], [20, 0, -7, 7, -3, -25, -7, -6], [6, -2, -3, -10, 2, -16, 10, -10], [5, -2, -9, -1, 7, -13, 3, 4], [12, -8, -17, -9, -5, -8, 0, 1], [2, 16, -20, -9, -3, -3, 3, -3], [-7, 18, -5, 8, 2, -21, 4, -1], [-17, -10, 7, -18, -6, -7, 11, 0], [-6, 14, 4, 1, 7, 7, -18, 1], [16, 7, 5, 8, -1, 13, -2, -5], [21, -4, 16, -4, 8, -13, 2, -9], [-6, 4, 17, 1, 11, 23, 8, 11], [2, 8, 18, 6, 8, 20, 11, 7], [-9, 1, -2, -12, -1, -2, 3, 12], [11, -19, 5, 2, 9, 4, -9, -5], [21, -14, 8, 0, 12, 3, 0, 1], [-13, -15, 14, 6, 7, 0, -7, -3], [25, -7, 8, -7, -4, -15, -15, -6], [32, -12, 6, 7, 7, 13, -8, 11], [10, -15, 3, 0, 1, 1, -1, 1], [-26, 8, -4, 8, 10, 19, 1, 6], [-17, -1, 5, -13, 4, 3, -4, 2], [-2, 8, 0, -2, -8, 1, 4, 12], [-7, 6, 9, 4, 4, -13, 4, 0], [-9, 17, -13, -6, -9, -26, -2, 12], [18, -8, 23, -15, -5, -8, -2, -5], [-17, 1, 21, 3, 4, 1, -4, 1], [6, -6, 10, -14, 8, 1, 11, -9], [-44, 8, -24, 7, -5, 5, 0, 14], [-2, -11, 11, -5, 4, -8, 15, -15], [4, -18, -1, -10, -2, -10, 4, -12], [-29, 4, -2, -22, -16, -8, 7, -2], [32, -1, -6, 0, -13, -13, 2, -9], [-42, -10, 3, -14, -10, 0, 8, 0], [-15, -10, -18, -7, 1, 4, 3, -15], [-27, -9, -16, 5, 0, 12, 10, -3], [8, 8, 0, 9, 5, 6, 12, -6], [3, -24, 2, -2, -4, -1, 9, 6], [-8, 3, 19, 1, 2, 20, -2, -5], [7, -17, -7, 2, -7, -1, 13, -6], [-4, 17, -4, 27, 18, 5, -4, 9], [13, -4, 1, -13, -15, 6, 2, -8], [-4, -11, 7, 8, 7, 1, -2, -10], [-20, 0, 8, 3, 1, -4, 15, -8], [28, 12, -11, 9, -13, -8, 8, -5], [-19, 0, -9, -5, -13, 9, 20, -2], [-3, -4, 9, 1, 13, -6, -9, -6], [17, 1, 25, -10, 9, 13, 1, -5], [-5, 10, -14, 5, 7, -17, -6, -11], [-12, 10, 9, 3, -3, 23, 10, 19], [12, -18, 4, 8, 8, 4, -9, 9], [18, 2, 13, -10, -4, -1, -9, 6], [16, -4, 4, -17, -5, -9, -2, -7], [8, 14, -15, 4, 6, -14, -4, 4], [-2, 3, 17, -14, -3, -1, 0, -3], [3, -23, -4, 5, 0, -11, -21, 0], [-21, -8, 9, 9, 11, -6, -11, 9], [1, 11, -8, 1, 10, -5, 9, -5], [-13, 5, -12, -15, 8, 6, 18, -5], [6, -7, -19, -10, -9, -7, 23, -6], [-24, -19, -3, -3, 4, 14, 21, -19], [9, -2, -7, 0, -6, -1, -13, 7], [-41, -6, -12, -2, 4, 0, 1, -16], [-20, 1, 5, 24, 1, 5, -15, 5], [-22, 4, -21, -9, -19, -5, -8, 11], [2, -7, -3, -4, 11, -10, -4, -15], [12, 2, 25, 11, 3, -3, 0, 2], [-2, -15, 23, 10, 17, 8, -2, -10], [-5, 4, -11, -7, 13, -16, 0, -8], [-18, -5, -1, 0, -11, -1, 5, -5], [21, -23, 8, 4, 9, 4, -3, -6], [20, 19, -22, 21, 0, 2, 2, -3], [-11, -16, 6, -17, -11, 13, 5, -8], [29, -4, 18, 9, -3, 15, -9, 12], [6, 0, 12, -1, -3, 11, -7, 1], [-18, 14, -24, -5, -3, -11, -2, -5], [-9, 0, 10, 0, -8, -8, -5, -2], [-2, -7, 13, -8, 1, 8, -5, -4], [3, -7, 3, -10, -19, -4, -5, -5], [-25, -1, -18, -24, -9, 7, 14, -9], [-13, -2, -22, -13, -3, -6, 16, 5], [13, 9, -21, -9, -4, -19, -7, 8], [-9, 10, 1, -8, -6, -15, 2, 5], [11, -1, -15, -8, -5, -10, -8, -7], [-16, -19, -13, 4, -1, -7, -9, 1], [12, -5, 8, -10, -7, 0, 2, -15], [-25, -11, 17, -21, -2, 20, 14, -3], [3, -6, 23, 13, 7, 0, -1, -5], [-5, -3, 11, 4, -5, 24, -6, 19], [33, -5, 1, 11, -8, -4, 6, -5], [-13, 14, 7, 4, 14, -12, -7, -9], [23, -23, 20, -10, 17, 22, -2, -10], [6, 23, 6, 14, 15, 15, -8, -3], [25, 1, -5, -7, 8, -7, 9, -4], [-20, -12, -3, -10, -6, -14, 7, -13], [12, 3, 19, 8, -3, 9, -21, 23], [7, -21, -1, -3, -18, -13, 10, 2], [0, 10, 3, 9, -1, 5, -2, 16], [53, -3, 0, 14, 5, -11, 0, -3], [-8, 5, -14, -9, -8, -11, -12, 0], [12, 3, 16, 7, 2, 24, 6, -1], [-28, 0, -17, 10, 10, -16, -5, -7], [14, 13, 13, -11, 0, 12, 15, 4], [18, 9, 0, 4, 7, -11, 0, -19], [16, 20, -9, 2, -2, -10, 6, 5], [-26, -12, -8, 9, 13, 22, 22, -6], [-4, -1, 16, 8, 15, 23, -6, -4], [-21, 7, -1, 0, 5, 8, 4, 18], [17, -25, 4, 1, 6, 7, 1, -17], [4, 17, 11, -3, -6, 18, 10, -1], [14, -11, -11, -10, -9, -16, -14, 9], [2, -21, -2, 5, 14, 8, -7, 3], [23, -13, 2, 0, -1, 2, -12, -8], [-21, 37, -2, -4, -9, 2, 2, 20], [32, -19, 19, 14, 21, 6, -7, -6], [-7, 10, 20, -19, 3, 18, 9, -8], [8, 8, -2, -10, 6, 11, 8, -8], [-23, -5, -1, -15, -12, 3, 32, -4], [3, 5, -17, 26, 11, -3, -8, 3], [6, -21, 0, -9, -8, 4, 10, -12], [4, 27, -9, -10, -11, 1, -8, -18], [-40, 13, -9, -10, 1, -3, -4, 16], [-2, -4, 14, 3, -5, 9, -9, 3], [-36, 12, -23, 4, -8, -4, 4, 5], [28, -9, -9, -12, -19, -2, 6, 0], [-7, 5, -13, -4, 1, 4, 36, -13], [-24, -12, -11, -2, -8, -26, -1, 6], [-36, -5, -24, 11, -6, -1, 0, 2], [-18, -21, 3, -19, 8, 31, 16, -10], [-11, -22, -4, 6, 8, -3, 0, -1], [-42, 12, 8, 4, -2, -5, -4, 5], [-24, -5, 11, -15, 4, 14, 0, 5], [14, 3, -14, -17, -6, -15, 4, 5], [16, -10, -12, 17, 15, -13, -14, 5], [41, -7, -6, 26, 7, 6, 11, -6], [-25, 0, 12, 2, 18, 37, 16, -6], [21, 0, 7, -10, 10, 14, 0, -7], [25, -14, 27, 31, 25, 5, -11, -2], [43, -18, -23, -15, -11, -24, -11, -4], [3, -3, -12, -9, -14, -21, -3, -5], [11, -21, -22, -15, 2, -1, 14, -6], [5, -7, -12, -21, -24, -3, 19, -11], [14, -19, 0, -20, -3, -5, 3, -21]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5239_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5239_2_a_j();".
// 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_5239_2_a_j(:prec:=8)
    chi := MakeCharacter_5239_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_5239_2_a_j();".
// 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_5239_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5239_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-29, 28, 54, -54, -24, 30, 0, -5, 1]>,<5,R![3, -90, -158, 225, 99, -192, 83, -15, 1]>],Snew);
    return Vf;
end function;