
// Make newform 8016.2.a.x in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8016_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_8016_2_a_x();".
// 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_8016_2_a_x();".
// 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, 23, 17, -64, 9, 28, -8, -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], [15, 168, -11, -147, 2, 38, 0, -3], [19, 98, -20, -70, 3, 15, 0, -1], [25, 21, -135, 21, 64, -16, -7, 2], [-27, -42, 122, -28, -61, 24, 7, -3], [-5, 8, 37, -25, -18, 10, 2, -1], [-13, -56, -95, 98, 58, -39, -7, 4]];
    Rf_basisdens := [1, 1, 14, 7, 7, 7, 2, 7];
    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_8016_a();"
function MakeCharacter_8016_a()
    N := 8016;
    order := 1;
    char_gens := [3007, 2005, 5345, 673];
    v := [1, 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_8016_a_Hecke(Kf)
    return MakeCharacter_8016_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, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0], [1, -1, 0, 1, 0, 1, 0, 0], [-2, 1, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, -1], [1, -1, -2, 1, 0, 0, -1, 1], [-2, 1, 0, -1, 1, -1, 0, 0], [-1, 0, 0, 1, -1, 1, 0, 1], [1, -1, 0, -2, 0, 0, 0, -1], [-1, 2, -1, 0, -2, -1, 1, 1], [-1, 0, 1, -1, 0, 1, -1, 1], [1, 1, 0, -2, 0, 0, 3, -1], [0, 0, 1, -1, -1, 0, 0, 0], [-2, 1, 1, -2, -2, -1, 1, -1], [1, -1, -1, 1, 1, -1, 0, 0], [-4, 1, -1, -2, 0, -1, 1, 0], [-1, 0, 0, 0, 1, 2, 2, 1], [-1, 2, -2, -1, 1, -1, -1, -1], [-2, 0, -1, 1, 1, -1, -1, -1], [-3, 0, -4, 0, -1, 1, 0, 1], [-2, -2, -1, 1, -1, -2, -4, 2], [-5, 1, 2, 0, -2, -1, 0, 2], [0, 1, 0, -1, -3, -1, 0, 0], [-6, 3, -3, 0, 2, -2, 1, 1], [-2, 1, -4, 2, 1, -2, -1, -1], [0, 2, 0, -2, 2, -1, -1, -1], [-1, 0, 2, -4, -1, -3, 2, -4], [-7, 3, 0, 1, 0, -2, -2, 1], [3, -1, 0, 1, 0, 3, -3, 2], [5, 0, -1, -1, 4, 2, 5, 0], [0, -4, -3, 2, 3, 0, -1, -4], [7, -2, 2, 2, 1, 3, 2, -3], [0, -4, -2, 7, 0, 1, -3, 1], [1, 1, 1, 1, -1, 4, 0, -1], [0, 5, 4, -6, -1, -1, 1, -2], [-4, -1, 5, -2, -2, -1, 1, 0], [3, -3, 2, 3, 2, 5, 3, 4], [1, 0, 0, 0, 0, 0, 0, 0], [-2, 5, 3, -6, 0, -3, 1, -2], [3, -6, -3, 1, 2, 2, 1, -2], [-1, 2, 0, -5, -1, 0, 0, 1], [-1, -3, -3, 3, -1, 4, -3, 2], [0, -5, -3, -3, 4, -4, 1, -3], [-3, 2, -2, 7, -5, 3, -3, 7], [-3, -4, 2, 0, -5, -1, -2, 1], [-3, 3, 2, 5, 0, 4, -4, 3], [-6, 4, 0, 1, 2, 2, 0, 2], [0, -4, 2, 6, 0, 5, 2, 5], [1, 1, -1, 2, 3, -3, 3, -3], [1, 0, 5, 1, -2, -1, -5, -3], [-2, -4, -2, -4, -2, 0, 0, -3], [1, -3, -1, -1, -7, -1, -5, 0], [-10, 4, -3, 1, -3, 0, -3, 6], [4, 0, 9, -2, -3, 2, -2, 0], [1, -3, -5, 4, 3, 5, 1, 2], [0, 5, -1, 1, 2, 2, 4, -1], [-3, 1, 3, 0, 1, -3, 2, -4], [-4, -1, 2, -5, 3, 0, 2, -9], [1, -4, -3, 3, 4, 0, 3, 0], [2, -1, 5, -6, 2, 3, 3, -4], [-3, 1, -5, -1, -1, -8, -3, 1], [-8, 2, 5, -2, -1, -5, 6, -1], [5, -2, 3, 1, 4, -1, 1, -7], [-5, 4, 6, -5, 5, 0, 2, -2], [1, -2, 1, 5, 0, 1, 1, 5], [-4, -3, -6, -1, 3, 1, 1, -2], [-13, 6, -6, 1, 1, -1, 0, 3], [3, -5, 3, 2, 5, 4, -1, 5], [-2, 1, 4, 4, -1, 7, 0, 8], [1, 6, 0, 0, 1, 1, 4, -1], [-2, 2, 5, -3, -7, -3, -1, 3], [-3, -4, -6, -3, 3, -7, 2, -5], [-1, -3, -1, -2, 1, 4, -3, -3], [-5, -5, 1, 1, 5, -3, 0, -4], [10, -5, -4, 1, -1, 0, 2, 2], [1, -6, -3, 1, -2, -4, -6, -2], [-12, 4, 7, 0, -3, 2, -5, 3], [-2, -4, 3, 2, -1, 3, -2, 9], [-4, -5, -6, 3, 1, 2, -6, 0], [-1, -1, 1, 7, -1, -1, -7, -1], [-6, -2, 0, 5, 2, -4, 0, -2], [6, 6, 6, 2, 0, -1, -4, -1], [-1, -2, 7, 2, 2, -1, 1, 0], [-6, 1, -1, -8, -2, -5, 4, -6], [-3, -2, -1, -2, -2, 0, -3, 2], [7, 3, -2, 0, 6, -1, 8, -2], [-16, 4, 0, 1, -4, -3, -4, 1], [8, -10, 6, -2, 0, 9, 0, -7], [-1, 13, -2, 2, 2, 0, -5, 3], [0, 3, -2, 1, 1, -4, -2, 7], [-5, -2, -3, 4, 6, 0, 1, -2], [3, 5, 5, -12, -1, -2, 2, -9], [10, 9, 3, 1, -6, 6, -3, 2], [2, -9, 5, 5, -4, 3, -2, -2], [2, 3, -8, -2, 5, -3, 4, -3], [-3, -8, -2, -1, 3, -2, 0, -4], [-2, -1, 0, 5, -1, -7, -3, 0], [2, -1, -1, 1, 2, 0, -5, -1], [0, -5, -3, -1, -2, -1, 5, -6], [-3, 5, 9, -3, -3, -2, -8, -5], [-2, -12, -2, 2, 0, -1, 0, 3], [2, -4, -7, -3, 7, -1, 1, -5], [2, 3, -1, -1, -4, 0, 0, 7], [-9, 3, -1, -3, 1, -4, 8, -1], [-8, 12, 2, 0, -4, -5, 2, -2], [-11, 0, 2, -2, 1, -2, 0, -8], [-2, 3, 16, -2, -3, 3, 1, 2], [-9, 6, 8, -4, -1, -2, 2, 4], [5, -7, 1, 5, -5, 3, 3, 6], [14, -1, 0, -6, -3, -10, -5, -5], [-15, -4, -10, 9, -1, 5, -2, 6], [0, 0, 8, -1, 2, 6, 2, 4], [-30, 2, -3, -5, -3, -4, -2, 2], [-1, -2, 1, 15, -12, 5, -5, 11], [3, 5, 8, -10, -4, -6, -1, -9], [-3, -1, -7, 0, -1, 2, 4, -1], [-8, -3, -4, -1, -3, -6, -2, -1], [2, -2, -1, -4, 7, -6, -1, -4], [-8, -2, -3, 2, -1, -3, 1, -3], [-1, 7, -10, -7, 6, -11, 2, -2], [-4, 7, 8, -2, 1, 4, 6, -3], [-4, 8, 1, 0, 5, -2, 1, -10], [-7, 3, 0, -12, 2, -6, 2, -11], [-4, -8, 0, 5, 2, 10, 3, 4], [-7, 1, -4, 6, 6, 6, -2, 5], [-7, 10, 1, 1, 0, 1, 5, -3], [1, -4, 3, 2, 10, 5, -1, -1], [1, -4, 4, -5, 5, 5, 5, -5], [-14, 5, -4, 0, -1, -8, -2, -2], [-5, 0, -10, 1, 7, 3, 7, -5], [9, -7, 4, 8, 6, 7, 0, 4], [11, -8, -1, -1, 0, 5, 6, -3], [6, -9, -4, 2, 5, 5, -6, 0], [-9, 5, 1, 6, -1, -2, 1, 5], [11, 1, 11, -11, -1, -1, 5, -2], [8, 11, 0, -8, -5, 3, 5, 0], [-5, -3, 1, -4, -11, -2, -2, -3], [-6, -8, -6, -3, 2, -5, -7, -1], [-4, 11, 5, -5, -2, -3, -9, 0], [-15, -7, 1, 6, -7, 5, 0, 4], [6, -7, 6, -11, 7, 7, 7, -6], [0, -4, -10, 8, 0, 10, -1, 8], [-18, 9, -3, 11, -6, 0, -7, 7], [6, 6, -5, 5, -5, 4, 1, 8], [-15, 4, 4, 4, -1, 9, 0, 8], [-11, 9, 4, -3, 2, -7, -2, 1], [10, -6, -9, 5, -3, -1, -11, 1], [-5, 3, 7, -3, 5, 6, 7, 1], [-3, -8, -7, 5, -2, -8, -13, 2], [-6, 7, 2, 6, -1, 5, -6, 10], [10, -10, -1, -2, 7, -8, 6, -10], [4, -1, 0, 4, 3, 6, -6, 1], [9, 18, 12, -3, -1, -5, -2, -1], [-1, -11, -1, 5, 1, 1, 3, -8], [8, 2, -6, -12, 10, -5, 6, -9], [-5, 0, 1, 4, 0, -4, 5, 4], [23, -6, 2, -3, -1, 4, -6, 0], [-14, 5, 10, -8, 7, -6, 0, -1], [7, -1, 8, 10, -6, 8, 5, 11], [4, 5, -4, 6, -1, 0, 0, -6], [-5, 10, 3, -11, -4, -9, 0, 3], [-5, 4, -3, -3, 0, 10, -1, 4], [10, -11, 0, -1, 1, 7, 2, -2], [19, -11, -7, 4, -5, -7, -6, -6], [2, 2, -6, 5, -10, 6, -2, 12], [5, -4, -6, -2, -3, -7, -9, -1], [14, -7, 0, 4, -3, -5, 2, -5], [-9, -8, -3, -4, 6, -1, 3, -4], [4, -1, -8, 1, -1, -1, 14, 0], [-10, -9, -1, 1, -6, -2, -9, -5], [12, -3, -8, -1, -7, -3, -2, -2], [13, 8, -11, -3, 2, -2, -3, 4], [-19, 1, -10, -2, 2, 4, 10, -2], [-7, -18, -4, 1, -1, 7, -1, 5], [10, 3, 0, 9, -7, 7, 0, 12], [-3, -2, -4, 6, 3, 1, 4, -3], [6, 3, 3, -2, 8, 1, 10, -10], [6, 2, 9, -11, -9, -4, -3, -2], [2, -9, 4, 11, 7, 6, -6, 3], [10, 1, 9, 6, -6, 8, 3, 11], [-14, -2, 4, 1, -4, 0, 0, 8], [6, -12, 5, 12, 1, -1, -3, 1], [0, 6, -2, -4, 4, 3, 9, 3], [0, 2, -8, 10, -6, 0, -2, 14], [5, 8, 1, -5, 2, 5, -1, 5], [23, -4, -2, -6, -7, -1, 0, 0], [-7, -11, -5, 12, 3, 11, -1, 10], [-2, -2, -6, -1, 2, -2, 4, -3], [3, -1, -2, 6, 2, 0, 8, 3], [3, -18, 2, -3, -11, 5, 0, -5], [-1, -3, 1, 2, -5, 5, 3, 10], [-3, 2, -6, -10, 3, -8, 4, -6], [0, -1, 5, 9, 4, 14, -5, 11], [-3, 2, 11, -2, 0, 6, -5, -5], [-1, 7, -9, -4, -3, -17, -3, -6], [-5, 14, 11, -6, -8, 0, 1, 5], [10, -5, -8, -3, 13, 0, 8, -8], [26, -23, 2, 1, 3, -4, 2, -7], [17, 5, 9, -4, 13, -4, 11, -6], [4, 10, 7, -1, 5, -1, -2, -3], [-8, -4, 7, 5, 1, 4, 5, 4], [2, -5, 3, 1, 4, -2, 7, 1], [3, 17, -1, 0, 3, -1, 1, 10], [-14, -5, 5, 1, 0, 2, 11, 7], [23, -5, 5, -6, 3, 1, -3, -4], [-2, 9, 20, 1, -3, 2, -11, -5], [14, 12, -9, -8, 3, -1, 5, -4], [1, -6, 3, -2, 0, 4, 8, 4], [-12, -12, 0, 11, 2, 19, 2, 5], [-2, 17, 3, -4, 2, -1, 1, 9], [-1, 13, -1, -18, 5, 0, 9, -5], [0, -3, -5, -12, 8, -1, 4, -10], [-11, 11, 2, 12, -4, 2, -6, 7], [0, 1, -2, -4, -3, -8, 6, -10], [20, -13, 1, -6, -14, -2, -5, 0], [11, 13, -2, 5, 10, 2, 1, -7], [-5, 8, -7, 12, 0, -1, -11, 7], [14, 5, 8, -17, 5, -7, 0, -16], [4, 6, -3, -4, -7, -4, 4, -6], [7, 10, -1, 14, -6, 8, -7, 16], [16, -7, -5, 10, 10, 8, -5, -5], [2, 0, 3, -6, -3, -4, 6, 0], [17, 4, 6, -4, -3, -10, -2, 2], [18, 1, 5, 3, -6, 8, -7, 9], [-7, 2, 14, -11, -7, -1, 10, -8], [-7, -19, 5, 8, 3, -5, -9, -4], [14, -6, 5, -3, -1, -8, 1, 0], [-22, 10, 6, -3, -8, -3, 1, 7], [-2, 2, 13, 1, -1, 6, 9, -6], [16, -17, 6, 8, 11, 2, 4, -1], [-11, 8, -1, -6, -4, -10, 8, -2], [-8, 15, -3, -4, 4, -4, 8, -1], [2, 15, -5, -6, 4, -5, 8, -8], [-12, 6, 4, 5, -12, -5, -12, 5], [9, 4, 3, 6, -6, -1, -5, 15], [-5, 4, 13, -15, -4, -7, 11, -3], [-5, -9, -9, -10, 5, -7, 7, -8], [20, -18, -5, 4, 3, 0, -5, -6], [1, 13, 0, 3, 4, 1, -8, 3], [-6, -11, 1, 16, -2, 14, -7, 15], [15, 4, 12, 6, -1, 9, -2, 1], [7, 10, 3, 13, -2, 7, -2, 1], [5, -15, -12, -8, 14, -10, 10, -9], [-26, 8, -11, 4, -7, -11, -11, 7], [25, -2, 13, -2, -4, -1, -3, -2], [-9, 13, 0, -6, 16, -3, 14, 2], [-23, 7, 11, -9, 11, -8, -3, -11], [-10, -10, 4, -1, -8, -8, 0, 2], [4, 2, -8, 7, -10, 2, -4, 14], [5, -1, -3, -9, 1, -2, -9, -10], [17, 7, 1, -11, 7, -4, -5, -13], [28, -8, 15, -14, -13, -3, 3, -3], [-37, 13, -2, 5, -8, -1, -10, 10], [4, -18, 1, 20, -1, 8, 2, 10], [7, -1, 4, -7, 4, 4, -10, -1], [6, 8, -10, -6, 10, -8, 6, 4], [-3, 8, -8, -3, -5, 0, 0, 2], [-8, 13, -5, -16, 12, -8, 0, -5], [14, -21, 3, 16, 2, 3, 5, 2], [14, 14, 6, -8, 6, -11, 3, -1], [9, 5, -4, -11, 18, -5, 8, -4], [18, -14, -12, 1, 8, -1, -2, -5], [18, 5, 17, -11, -2, -1, 3, -8], [14, 10, -3, 2, 11, 5, 3, 11], [-8, -1, -14, 2, -9, 1, -2, -2], [-16, 23, 0, 6, 1, 4, 3, 9], [-6, 13, 9, -8, 10, -7, 0, -8], [6, 0, -8, 3, -14, 0, -3, 4], [-26, 8, -9, 2, 1, -8, -5, 6], [20, 5, 10, 7, 3, 1, -12, 2], [6, -11, -8, 6, 7, 2, 6, 9], [8, 4, 8, 1, 0, 7, -4, -1], [14, -7, 1, 3, 2, 12, 5, 3], [2, 9, 8, -3, 1, -1, 4, -13], [-8, 26, 14, -2, -8, 2, -3, 6], [11, -1, 3, 2, 7, 15, 9, -6], [-24, 10, -2, 1, -14, -6, 2, 12], [19, 2, 18, -13, -5, 2, 2, -6], [10, 1, 3, -10, 2, -10, -5, -6], [12, 16, -2, -26, 4, -9, 8, -1], [-6, -7, -5, 9, -8, 2, -14, 5], [6, -2, 3, 8, -1, 13, 7, 9], [3, 18, 0, -7, -3, 3, 8, -4], [12, -8, -17, 5, 3, -6, -4, 10], [2, 9, 8, -1, 9, 13, 16, 0], [-29, -7, 0, 13, 6, 12, -6, 10], [26, -11, 6, 5, -3, 11, 9, 2], [-3, -16, 7, 15, -14, 7, -7, 11], [3, -11, 12, 9, -6, 10, 5, 5], [-3, 16, 17, -14, -8, -4, -7, 2], [-5, 1, 4, -15, 0, -13, -4, -16], [17, -1, 6, -3, 6, 10, 0, -7], [-8, -11, -3, -6, 6, 10, -3, -8], [-31, 7, 8, -12, -12, -5, 4, -1], [-25, 10, -11, 9, 2, -5, 7, 3], [37, -11, 6, -4, -6, -2, -4, -1], [-24, 4, 10, 8, -4, 14, -3, 10], [-26, 2, 0, 13, -12, 0, -12, 14], [-4, 8, 0, 8, -8, 2, -8, 10], [-21, -15, 1, 23, 1, 17, -1, 10], [-4, 9, -12, 0, 1, -21, -7, -2], [29, -10, -17, 13, -2, -2, -13, 0], [19, -6, -5, 9, -4, 14, -1, 6], [11, 2, 4, -14, 1, -1, -6, 3], [1, -3, -1, 2, -3, -17, -7, -6], [-23, 2, -8, 18, 3, 3, 2, 17], [14, -17, -7, 16, 18, 16, 3, 5], [4, 20, -2, -19, 2, -22, 2, -2], [4, -10, -16, 13, -12, 12, -10, 12], [7, -2, -7, 8, 4, 7, 3, 1], [19, 7, -4, 13, 0, 9, -6, 18], [21, -26, 3, 0, 2, 5, 11, -9], [18, -19, -12, 11, 7, 8, 3, -3], [-2, -16, 4, 9, -6, 14, 0, 10], [9, -15, 6, -7, -4, 8, 18, -9], [13, 3, 9, -21, 1, -4, 9, -17], [-2, -3, 4, -16, 7, -6, 8, -11], [-23, 13, -7, 15, 9, 8, 1, 7], [-25, 10, -12, 2, 5, 5, -1, 5], [-10, -5, 2, -3, -17, 4, -8, 1], [-16, 11, -6, 3, -5, 6, 0, 15], [7, -21, 6, -1, -4, 8, 6, 3], [21, -6, -5, 2, -10, 10, 2, -4], [-24, 4, -8, 8, -2, 7, 3, 1], [3, -6, 16, 10, -5, 3, 4, 1], [10, 15, 4, -13, -5, -9, 4, -4], [25, -18, 4, -5, -7, 0, -6, 2], [17, 10, -1, -12, -8, 0, 11, -3], [8, 8, -15, -8, 7, -17, 3, -1], [11, -2, 15, -8, 4, 2, 5, -1], [21, 25, 1, -18, 3, -7, 7, -8], [-5, 24, -8, -2, -7, -2, 1, 10], [-23, -18, -12, 8, -5, 6, 0, 3], [-20, 7, 1, -17, 2, -6, 4, -1], [-21, 9, 10, 13, -16, 4, -2, 5], [0, -4, -1, 8, 5, 1, -3, 1], [21, -12, 16, 7, -3, -5, -1, 1], [-4, 1, -3, -18, 14, 2, 9, -9], [5, -16, 18, 0, -1, -9, 4, -7], [21, 1, -3, -3, 1, -12, 7, -15], [24, 2, -4, 11, -10, 3, -10, -3], [-1, -18, 3, 18, 6, 1, 0, 3], [-15, -16, 5, -7, 6, -8, 2, -14], [2, -11, -28, 16, -9, 9, -2, 2], [-44, 25, 5, 3, -8, 4, -1, 9], [51, -11, -4, 4, 6, 9, 2, -12], [-9, 3, -1, -13, -1, -11, 19, -8], [-17, -1, -20, 12, -4, -8, -10, 5], [11, 5, 12, 1, -2, -1, 4, 10], [-1, -7, 12, -12, -2, -10, 14, -11], [-21, 17, -9, 8, -3, -16, -7, -7], [-19, -1, -6, 1, 6, -8, -13, -7], [15, -29, -10, 4, 2, -5, 1, -8], [27, -12, 8, -4, -7, 15, -8, -5], [-11, -10, -10, -4, 1, 11, 3, -1], [-16, -14, -9, 1, 13, -9, 3, -3], [-13, -3, 3, -5, -19, 3, -9, 10], [23, -4, -14, -12, 7, -18, 3, -12], [1, 4, 4, -19, -1, -1, 18, -16], [19, -9, -17, 6, 3, 8, 7, 1], [-27, 8, 13, -2, -6, -15, -2, -5], [-16, 0, -20, 7, 4, 2, 12, 6], [-19, -3, 5, 5, 5, 6, -7, 6], [-13, 8, 15, -9, 4, -14, -12, -6], [-1, -3, 10, 6, -14, 4, 14, 9], [-2, 0, 11, -2, -3, 5, 20, -5], [23, -11, 4, -7, 0, -9, -3, -8], [-24, 0, 10, -3, 14, 9, -2, 3], [3, 9, 14, -1, -8, 10, -2, 1], [6, -11, 2, 2, 13, 7, -7, -14], [-12, -1, -14, 19, -11, -2, -10, 9], [-16, 18, 22, -12, -4, -5, -8, 1], [-27, 27, 7, 6, 3, 9, 4, 2], [-29, 2, -2, 5, -1, -12, -10, 6], [19, -11, 1, 4, -5, 11, 7, 4], [-29, -7, 8, 4, 6, 2, -6, 7], [-5, 2, 4, 7, 15, 12, 4, -3], [13, 2, 16, 5, 3, 16, 17, -2], [-14, -6, 5, 5, 1, 12, 11, 0], [-16, -15, -9, 10, 2, 9, -13, 0], [5, 0, 3, -11, 18, 0, 9, -20], [-19, -10, -22, -4, -5, -8, -11, 0], [30, 14, 13, 8, 1, -12, -1, 4], [2, 1, 5, 20, 6, 14, -3, 5], [-18, 11, 1, 2, 10, 9, -3, 8], [25, -3, -7, 7, 11, 13, 2, -4], [32, 3, 14, 3, -5, 10, -6, 1], [-9, 8, 2, 21, -1, 8, 0, 4], [32, 3, -16, 3, 7, 0, 4, -3], [-9, 20, -22, -4, 13, -6, -6, -6], [-17, 7, -1, -26, -5, -8, 3, 9], [1, -6, -2, -13, 9, -5, 14, -19], [-1, 9, -12, 7, -6, 9, -15, 12], [-7, -8, -13, -12, 16, -20, 11, -24], [-29, 16, 2, 12, -3, 10, -6, 12], [-6, -7, 11, 8, 4, 20, -3, 17], [-24, 15, 10, 1, -7, 9, -16, 14], [-11, 2, -5, 23, 0, -2, -22, 10], [-26, 10, -9, 17, -11, -4, -14, 22], [-15, -19, 14, 15, -4, 20, -10, 7], [32, -16, -4, 19, 2, -4, 2, -2], [6, -3, 3, -5, -12, -2, 8, 3], [44, -23, -4, -4, -3, -7, -9, -6], [24, 18, 10, -5, 2, 0, -1, 12], [15, 9, -14, -6, 0, 2, 15, 9], [-16, 1, -4, 7, -3, -1, 4, 0], [4, -4, -6, 18, 10, 16, 1, 14], [-14, 12, 15, 0, -9, 9, 0, -1], [-21, 5, 7, -16, -13, 2, 3, -11], [24, -10, -3, 9, 21, 3, 3, -5], [10, 2, 3, 8, 1, 2, 14, 2], [-45, -7, -9, 2, 3, -9, -15, -1], [20, -3, 7, 23, 6, 10, -9, 17], [-10, -24, -10, 10, 8, 17, -4, 1], [-5, -2, 7, -4, 8, 9, 11, 7], [-5, -15, -7, -6, -5, -7, -15, -10], [-6, 8, -24, 11, -12, 1, -4, 3], [-14, 4, -11, -14, -13, -6, -3, -2], [13, -10, 2, 6, -5, 9, -6, 4], [-16, 0, 3, 20, 13, 14, 2, 6], [25, 24, -17, -4, 6, -6, -5, -10], [25, -4, 1, 10, -16, 6, -3, 7], [16, 7, -19, -10, 4, -11, 1, -4], [-34, -4, -7, 3, 5, 8, 3, 3], [6, 10, -20, -5, 14, -9, -7, -9], [19, 5, -4, -13, -10, -27, 0, -17], [-6, 5, 0, -16, 7, -11, -6, -2], [-30, 19, 5, -2, 2, -7, 7, 16], [-2, 1, 7, -13, 6, -11, -5, -18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8016_2_a_x();".
// 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_8016_2_a_x(:prec:=8)
    chi := MakeCharacter_8016_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_8016_2_a_x();".
// 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_8016_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-18, -124, 196, 9, -125, 50, 7, -7, 1]>,<7,R![-16, 48, 84, -255, 63, 65, -19, -4, 1]>,<11,R![-1596, 1316, 1804, -435, -691, -104, 41, 13, 1]>,<13,R![56, 412, 994, 973, 320, -52, -37, 0, 1]>],Snew);
    return Vf;
end function;