
// Make newform 8016.2.a.z in Magma, downloaded from the LMFDB on 25 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_z();".
// 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_z();".
// 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, 14, -13, -31, 19, 15, -8, -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, 1, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0], [0, -4, 0, 1, 0, 0, 0, 0], [3, 0, -5, 0, 1, 0, 0, 0], [-1, -5, -8, 13, 6, -7, -1, 1], [-3, -13, 16, 23, -8, -9, 1, 1], [-4, 9, 13, -18, -7, 8, 1, -1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 2, 2, 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_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], [0, 1, 1, 0, 0, -1, 0, 0], [-1, 1, -1, 1, -1, -1, -1, -1], [1, -2, 0, -1, 1, 2, 0, 1], [0, -1, -1, -1, 2, 1, 1, 1], [0, -2, 1, -1, 0, 0, 1, -1], [-5, 1, -1, 1, 0, -1, 1, 0], [0, 0, -1, 0, -1, 3, 1, 0], [-1, 1, 2, 1, -1, 0, 0, 0], [-1, -4, -1, -2, 1, 4, 0, 2], [3, -3, 1, -1, 1, 0, 0, 2], [0, -1, 1, 2, -2, -2, 1, -1], [-4, 2, -1, -1, -1, -1, 0, 1], [2, -2, 3, -1, -1, -2, 0, -4], [-5, 2, 0, 3, 0, -2, -1, 0], [-4, -1, 1, -3, 2, 2, 2, 2], [0, 1, -2, 2, -2, 0, 0, 1], [-3, 2, -2, 1, 0, 3, 0, 3], [-1, -1, 5, -1, -2, -4, 0, -1], [-2, 5, -1, 1, -1, -1, -3, -2], [0, 4, -1, -1, -3, 1, -4, 1], [-1, 0, 1, 0, 3, -3, -1, -1], [1, 5, -3, 1, 0, 1, -1, 2], [-6, 4, -2, 4, -4, -7, -1, -3], [0, -3, -1, -4, 5, 6, 3, 0], [-5, 3, -2, 4, -1, 1, 2, -2], [4, 0, 1, -3, 2, 3, -3, 2], [-3, 0, 2, 2, 1, -6, 0, -2], [-4, -3, 0, -2, 4, 5, 2, 2], [-1, -8, 0, 0, -1, 3, -1, -1], [1, 3, -2, -3, 2, 5, -1, 2], [-8, -3, 1, 1, 1, -3, 3, 0], [-11, 4, 0, 5, 1, -1, 4, 0], [5, -9, 3, -6, 6, 7, 2, 3], [-10, 0, 0, 5, 1, -9, 0, -1], [4, -3, -7, -5, 0, 4, 2, 0], [-2, 1, 0, -2, 0, 3, -2, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [8, -3, 5, -1, 4, 2, 2, 6], [-5, -4, 2, 6, -3, -7, -1, -3], [8, 2, -6, -3, 2, 8, -2, 1], [-2, -4, 4, 3, 4, -7, 1, 0], [0, 1, 0, -1, -6, -3, -1, -3], [3, -8, 2, -3, 2, 5, 2, 3], [2, 1, -5, -1, -1, 1, -1, 0], [1, 2, -2, 0, -3, -2, -8, -4], [-4, 1, 2, -3, 4, 4, 0, 2], [6, 1, 5, -3, 0, -3, -3, 3], [4, -4, -3, -5, 2, 2, -2, 1], [-11, 11, -1, 7, -9, -8, -4, -8], [0, 5, 4, -2, 5, -2, 0, 4], [-4, 4, 5, 8, -3, -12, 0, -3], [-17, 12, 0, 2, -8, -7, -3, -4], [-8, 1, -7, 0, -1, 1, -2, 1], [-4, 3, 1, 1, 1, 0, 2, 3], [-1, 7, -1, 2, -7, -5, -6, -4], [9, -5, 2, -4, -2, 2, -1, 0], [-1, 6, -4, -2, -2, 2, -4, 3], [-7, 0, 2, 2, 1, -9, -5, -5], [0, -3, 3, 5, 2, 0, 4, -2], [-4, 7, 0, 5, 1, -1, -3, -6], [-10, 4, 0, 1, -1, -4, -1, -2], [-9, 5, -1, 3, -5, -2, 2, -4], [-10, -1, -2, 5, -1, -6, 4, -7], [-11, 7, -1, -3, -3, -2, 0, 2], [0, 3, 8, -6, -3, -7, 0, -3], [0, -2, -3, -2, -5, 3, -5, -2], [6, 2, 8, 8, 1, -7, -3, -6], [-3, -2, -1, 2, -4, -3, 3, -8], [-10, -5, 7, 7, -1, -7, 7, 0], [-5, 9, -3, 11, -4, 0, 2, -5], [2, -6, 5, 4, -3, -7, 3, -4], [-4, 8, 8, 2, -3, -5, 1, -4], [3, 6, 2, -9, 2, 2, -1, 6], [-1, 3, 0, -2, 5, -4, 0, 1], [-6, 4, -11, -3, 2, -1, 6, 0], [3, -2, 0, 3, -1, 3, -5, 0], [-4, -8, -4, -7, -1, 6, -3, 0], [1, 1, -4, -4, 3, 2, 0, 9], [-8, -7, 5, 0, 0, -6, -2, -5], [8, -3, -8, -7, 2, 14, 2, 8], [-6, 3, 3, 3, -2, -5, 9, -1], [3, 7, 3, 4, -2, 0, -4, -4], [11, 1, -8, -2, -3, 14, 3, 3], [7, 1, 6, -1, -5, -5, -1, -1], [-5, -4, 1, 2, 7, 3, 1, 1], [-16, 6, -7, 14, -6, -5, -5, 1], [0, 5, 1, -3, -2, -7, 3, 1], [2, 3, 7, 2, -4, -14, -7, -3], [-5, -4, -4, 0, -2, 6, 4, -3], [7, -3, -8, -3, 3, 13, 11, 11], [3, 8, -3, -3, 8, -1, -6, 1], [8, -12, -4, -9, 11, 19, 3, 3], [-3, -8, 0, 1, 3, 6, 11, 7], [-17, 5, -9, 4, 3, -7, -1, 2], [2, -5, 10, -1, -3, -8, 4, -9], [-8, 1, 2, -4, 3, 1, 0, 5], [-14, -9, 12, 3, 0, -13, 5, -1], [-6, 8, 3, 12, 4, -10, 0, -4], [1, -7, -1, 0, 4, 3, 0, -1], [6, 3, -5, -3, 6, 11, -3, 3], [7, -11, 9, -11, 6, 6, 2, 1], [15, -15, 1, -18, 3, 15, 2, 10], [-3, 3, -9, -6, 4, 17, 0, 7], [12, -10, 1, -7, 1, 11, 7, 1], [10, -6, -12, -10, 1, 16, 0, 7], [0, 0, -6, 1, -1, 13, -4, -5], [2, 0, 4, -2, 5, -8, -4, 1], [-2, -2, 5, 2, 9, -4, -2, -1], [4, 5, 1, -12, 9, 8, -1, 8], [10, 11, -11, 0, -2, 1, -5, 2], [-18, 5, -10, -7, 4, 8, -6, 6], [2, -4, 5, -3, -1, 1, 4, 3], [1, 1, 5, 7, 3, -10, 2, -2], [-2, 7, 11, 6, 2, -10, -3, -9], [-1, -16, 11, 1, 10, 3, 4, -1], [5, 0, 10, 2, 2, 0, 4, -1], [3, -3, -8, -3, 2, 15, 15, 10], [1, 8, 1, 6, -10, 8, 0, -3], [11, 3, -5, 9, -9, 1, 1, -7], [11, -7, 10, 1, 6, 0, 4, 3], [19, -19, 0, -11, 16, 17, 9, 10], [-1, 5, -12, 3, -5, 8, 2, 8], [-1, 7, 13, 0, -9, -22, -9, -7], [11, -1, 10, 3, -1, -2, -4, -10], [7, 11, 3, 1, 3, -16, -2, -4], [15, -10, -3, -8, 11, 14, 0, 2], [-9, 16, -8, -3, -2, 5, -2, 13], [-5, 16, -8, 7, -13, 0, -12, -1], [-5, 8, -2, -3, -2, -1, 4, 5], [-5, -4, -9, 0, -7, -3, 1, -7], [-18, 1, -2, 6, -2, -4, 7, -13], [1, 2, -9, 4, -2, 15, 3, 10], [14, -10, -2, -8, -3, 9, 5, -4], [18, -8, -1, -8, 7, 16, 6, 13], [0, 8, -10, 11, -9, -7, -12, -7], [-11, -2, -11, 7, 2, 17, 4, 13], [9, -8, 2, -7, 1, -1, 10, 4], [-2, -8, -5, 2, 0, 6, -2, -6], [1, 3, 2, 10, -10, -10, -1, -4], [-12, -3, -12, -2, 1, -3, 0, -3], [7, -20, -1, -9, 13, 4, 11, 13], [-6, -11, -2, 7, 8, -5, 5, -3], [-13, 4, -2, -4, 2, -9, 5, 4], [-6, 2, 5, 7, -4, -19, -9, -12], [-5, -4, 9, 1, -10, -9, 1, -11], [23, 1, 3, -5, -6, -12, -10, -5], [-12, 3, -8, -1, 5, 7, 3, 14], [-13, -4, 14, 0, 5, -13, -1, -3], [-3, 6, 5, 0, -2, -11, -1, 0], [2, 1, 1, -4, -7, -15, -4, -7], [13, 1, -2, -5, 6, 10, -2, -5], [8, -2, 11, 6, -3, -11, 1, -12], [8, 6, 9, -8, 7, -4, 4, 7], [-24, 5, 1, 7, -6, -9, 7, -5], [15, -3, 8, 9, 3, -19, -9, -5], [6, -7, -6, -1, 5, 22, 8, 11], [11, -5, 16, -7, 10, 0, -4, 1], [22, -17, -1, -8, 12, 16, 1, 1], [-7, -6, 0, 9, 5, -10, -4, -5], [-16, 7, -4, 18, -10, -14, -7, -1], [21, -8, 12, 0, -11, 1, -3, -15], [-7, -5, -9, -5, 2, 11, 3, -4], [-7, -11, -4, 0, 4, 10, 11, 6], [6, 7, 4, 7, -10, -20, -4, -10], [-9, -7, -16, 12, 2, 9, 2, -1], [33, 1, 5, 2, 1, 9, 1, 10], [11, -5, 7, -2, -8, -2, 2, -6], [-11, 3, 3, -5, -8, 1, -7, 2], [-6, -7, -4, -1, -8, 7, -7, -7], [-13, 7, -13, 1, -6, 5, -7, 0], [21, 6, 10, 8, -5, -3, -9, -11], [29, -26, 0, -17, 12, 24, 12, 12], [3, 0, -4, 13, 2, -5, 2, -1], [-1, -5, 1, 5, 0, 1, 7, 2], [22, -19, 1, -13, 7, 9, 7, 4], [-15, 3, -2, 6, -5, -12, -3, -9], [11, -4, -2, -4, -8, 1, 1, 2], [-13, 0, 8, 0, 4, -10, 2, -3], [8, 8, 10, 0, -6, -1, 7, 1], [4, -21, 2, -11, 10, 6, 6, 16], [-5, -6, -13, 0, -4, 10, 6, 9], [-13, 7, 4, 0, -3, -15, 0, -12], [16, 12, 10, 2, -2, 0, -2, 0], [25, -15, 1, 3, 9, 6, 2, 2], [-8, -4, 1, -1, -4, -3, 1, -6], [4, -15, -9, -9, 7, 0, 4, 1], [22, -10, 8, -11, 5, 10, 0, 0], [1, 11, -2, -11, 1, 8, 2, 0], [12, -14, 5, -4, 7, 5, 5, 6], [36, 9, 7, -13, -3, 14, -4, 9], [-28, 24, -4, 10, -1, -20, -4, 5], [-18, 5, -2, 13, -6, -3, 9, 5], [3, -8, 2, 1, 2, 21, 3, 3], [20, -9, 11, -11, -3, 2, 8, 7], [-7, 16, -2, 9, 6, 3, -1, 9], [-25, 21, -2, 18, -9, -6, -4, -7], [11, -18, 4, -14, 10, 6, 0, 7], [18, -9, 4, -12, -6, 7, 1, 8], [-6, -3, -10, 6, -9, 8, 3, 8], [9, 4, 8, 10, -8, -5, -7, -8], [-10, 11, 6, 19, -6, -23, -5, -7], [-6, 3, -5, -1, -5, -8, 6, 11], [2, 7, -6, -1, -10, -3, 1, -5], [-8, -7, 7, 5, 7, -10, -2, -9], [6, 4, 11, 9, 1, 10, -9, 2], [-11, -1, -13, -12, 7, 22, 0, 19], [2, -7, -5, 4, 8, -3, -4, 4], [20, 2, -1, -6, 2, 3, -3, -1], [24, -8, -5, -13, 11, 30, -2, 10], [14, -2, -14, -6, -1, 13, 3, 4], [7, 13, -8, -14, -1, 2, -9, -3], [-1, 5, 2, -1, 11, -2, 2, 6], [-9, 6, -12, 17, -11, -8, -14, -5], [2, -5, -12, 4, -9, 9, 1, -7], [4, 8, 11, 5, -10, -20, 5, -1], [9, 2, -1, 6, -1, 10, 12, 4], [7, -7, 11, -13, 2, -9, -3, 4], [6, -23, 9, -2, 7, 5, -4, -9], [-3, -7, 12, 1, -5, 7, 7, -3], [24, -10, -4, -8, 10, 9, -9, 5], [16, -5, 9, -2, 25, -7, 4, 7], [0, -10, -8, -6, 9, 34, 8, 11], [6, -5, 2, 3, 14, 15, 9, 17], [0, 17, 11, 4, -12, -9, -15, -10], [-16, 31, -5, 25, -21, -28, -10, -17], [11, -6, -14, -8, 4, 13, 5, 8], [27, 8, 4, -7, -1, 7, 0, 4], [5, 0, 10, -6, 4, 3, 7, -12], [5, -21, 0, -25, 14, 20, 4, 9], [-6, 1, -23, -8, -6, 3, 4, -2], [-3, 0, 7, -5, -9, -15, -4, -6], [7, 9, 10, 10, -3, -10, -11, -9], [-8, -4, 5, -8, 2, -5, 3, 3], [-5, -18, -17, -10, 5, 28, 2, 20], [23, 3, 1, -11, 21, 16, -4, 14], [-10, -13, 7, 3, 19, -4, 4, -9], [-5, -15, 4, -23, 4, -5, 3, -2], [-31, 6, 9, 17, -8, -27, 13, -13], [0, -6, -14, 6, -2, 1, -5, -5], [-18, 13, -5, 7, -15, -3, -7, -6], [24, 7, -8, 0, 6, 2, -3, 11], [23, -31, 2, -29, 9, 22, 8, 10], [-1, -22, 5, 8, 0, -10, 2, 1], [-7, 1, 5, -12, 8, 2, 8, 6], [7, 2, -15, -8, 1, 21, 7, 11], [-34, 15, -4, 11, -11, -11, -5, -6], [22, -9, 8, -5, 4, 2, -10, 6], [40, 5, 16, -13, -10, 10, -10, 4], [-16, 2, -2, -7, 0, -15, -1, 0], [-4, -7, -8, 3, 5, -3, 7, 8], [-15, 2, -3, -2, -4, -10, 6, 3], [-11, 7, 5, 17, -5, -33, -5, -19], [34, -15, 3, -14, 5, 19, 12, 5], [9, 30, -18, -4, -13, 6, -20, -2], [4, 16, -12, 16, -8, -2, -6, -8], [-26, 7, -6, 9, 5, 8, 14, -1], [35, -30, 7, -15, 19, 19, 2, 2], [-20, 11, 11, 21, -6, -30, -2, 0], [19, -21, 6, -4, 1, 9, 4, 0], [13, -7, 1, -15, 9, -5, 7, -1], [4, -4, -7, -24, 12, 21, 7, 9], [-34, 20, -3, 18, -12, -18, 2, -6], [7, 20, -5, 2, 8, -6, 0, -3], [-5, 2, 13, 4, -2, 1, -1, 0], [-26, 19, -7, 16, -3, 0, 17, 6], [3, -15, 2, -10, 15, 16, 5, 1], [11, -19, -3, -4, 17, 30, 13, 27], [-11, -7, 2, 7, -4, 1, -7, -2], [-23, -7, 3, 15, -4, -3, 5, 4], [-17, 7, 6, 19, 2, -34, -2, -9], [-24, 8, -13, 4, -2, 5, -3, 3], [10, 15, -4, 13, -10, -35, -13, -11], [19, 12, -1, 7, -11, -5, -7, -14], [17, -16, -3, -9, 3, -6, -5, 1], [-4, -13, 21, 9, 9, -14, 8, -7], [10, 15, 2, 3, -10, -16, -16, -4], [20, -13, 16, 1, -5, 0, 10, 1], [-12, 19, -8, 8, -13, -15, -13, -19], [16, -5, 25, -11, 6, -5, 13, -5], [7, -27, 5, -8, 27, -5, 18, 10], [19, -12, 9, 2, 0, -6, 10, -5], [-14, 7, -1, -2, 2, 1, 19, 12], [-10, 8, -25, -7, -3, 1, 8, 1], [3, 13, -1, -1, 4, 5, 11, 12], [29, -11, -2, -20, 2, 16, -6, 6], [40, -23, 2, -26, 7, 27, 0, 15], [17, -1, 3, 5, -3, -22, -2, -16], [-4, -16, -1, -11, 6, -12, 5, -7], [-11, 27, -10, 9, 3, -7, 3, -3], [-3, 5, 9, 5, -5, -21, 15, -3], [-23, 6, -4, 12, -1, 14, -4, 8], [-14, -9, 0, 14, -1, 11, 1, 7], [7, -9, 13, 0, -4, 3, 7, -11], [7, -3, 7, 21, -3, -32, -8, -12], [-13, -9, -4, -11, 19, 10, 4, 10], [17, -18, -5, -27, 23, 30, 13, 7], [-4, 31, -18, 3, -10, -16, -4, 0], [8, -10, -10, -18, 12, 18, 10, 6], [4, 8, 17, -26, 9, 0, 2, 11], [2, 6, 10, -15, 3, -15, 0, -1], [15, -14, -18, -14, 3, 19, -11, 7], [39, -20, -12, -6, 17, 21, -5, 9], [-24, 3, 5, 3, -21, -5, -9, -14], [10, -3, -1, 25, -11, -8, 2, -19], [-2, 5, -15, 3, -9, 5, 1, -8], [4, -4, -2, 12, 8, 11, -5, -3], [-24, -3, -12, 17, -4, 24, 8, 6], [-2, 7, 14, 19, -4, -22, 4, -16], [-15, 8, -3, 18, -15, -10, 10, -6], [-13, 17, -3, 1, -5, 5, -5, 15], [17, 12, 21, 0, 5, -30, -8, -18], [-10, 8, 11, 17, -1, -10, 9, -10], [30, -29, 4, -17, 6, 18, 14, 20], [-33, 10, -4, 16, -1, -14, 10, -6], [-22, 3, -2, -1, -7, -5, 11, -8], [-19, -7, 0, -5, 4, -12, 10, -1], [4, -19, 2, -9, 23, 21, 13, 10], [-11, 15, -6, -10, 0, 7, 0, 19], [-9, -6, -20, -6, -8, 16, 4, 5], [29, 10, 14, 10, -12, -26, -6, -21], [15, 6, -8, -14, -1, 8, -16, 6], [14, -17, -9, -24, 20, 51, 2, 26], [-23, -1, 8, 6, 1, 17, 4, 2], [0, -23, -1, 9, -15, -13, 5, -6], [-23, 21, -13, 31, -14, -13, -15, -10], [-14, 1, 0, -9, -3, -4, 0, 11], [-21, 12, -16, 9, -6, 11, 5, 5], [-15, -2, -11, 8, -14, 8, -6, 7], [5, 16, -12, 1, 14, 1, -11, 3], [-24, 5, -11, 13, 1, 18, 8, -7], [1, 26, -3, 9, -6, -24, -3, -18], [4, -27, 16, -26, 16, 0, 6, 1], [-9, 7, -27, 2, 13, 13, 0, 6], [-17, 2, 0, 2, 5, 28, 8, 14], [-19, 6, 3, -4, -4, 12, -6, 9], [-3, -4, -30, -5, -6, 29, 14, 9], [-14, -12, -2, 0, -8, 7, 5, -9], [-24, 1, -35, 21, -3, 9, -7, 2], [-8, -15, 12, -5, 15, -5, 1, -4], [-8, -10, 1, 4, -2, -1, 5, -5], [4, -24, 0, -5, 0, 30, 21, 13], [-20, 18, 7, 11, 0, -33, 0, 6], [17, -14, -5, -4, 6, -19, 5, 0], [-6, 5, 8, 3, 6, -5, 3, -13], [13, -2, -11, -2, -7, 9, -9, -3], [-30, 8, -23, 8, -15, 12, 0, 9], [15, -3, 22, 13, -9, -18, 0, -4], [-29, 37, -11, 11, -7, -13, -9, 5], [-35, 3, -4, 11, -9, 8, -6, -4], [28, -2, 28, 4, 13, -13, -15, -8], [2, 6, 21, -3, -4, -18, -9, -17], [-36, 6, 12, 3, -2, -21, -4, -8], [-6, -21, 1, 7, -11, -3, 7, 2], [5, -7, -2, -18, 4, 27, 2, 9], [-49, 25, -7, 17, -24, -24, 0, -23], [26, -20, -7, -2, 13, -2, -2, 1], [-21, 10, -15, -1, -2, -24, -5, -14], [6, 17, 3, 26, 2, -19, -11, -8], [-14, -2, 2, 0, 7, -13, 11, 4], [-18, 22, -8, 25, -28, -18, -19, -23], [-10, -7, -20, -3, 4, 34, 22, 26], [30, -14, 10, -21, 24, 15, 7, 20], [0, 26, -7, 11, 0, -11, -6, 2], [7, -13, -10, -1, 13, 6, 6, 8], [16, -18, 20, -1, 13, -6, 7, -4], [-4, -11, 6, -8, 10, 7, 6, 18], [2, 32, -17, 2, -30, 3, -25, -11], [35, -14, 8, -4, -3, 14, 14, -6], [-20, -14, 10, 7, -15, -5, -6, -9], [-1, 8, 6, 4, 2, -4, -18, -5], [-8, 7, -13, 9, -12, -5, -19, -1], [37, -21, 0, -14, 12, 18, -3, 16], [12, 17, 2, 1, -15, 6, -4, -13], [-40, -7, -7, -5, 1, 28, 14, 11], [17, -9, 28, 1, 13, -8, -2, 0], [40, -10, -10, -7, 14, 12, -6, 9], [-17, 7, 17, 8, 10, -10, 17, 8], [-1, -6, 0, -6, 6, -3, 7, -6], [18, -21, -5, 1, 10, 10, -4, 10], [-9, -12, -10, -6, 17, 29, -7, -5], [-5, -4, 1, 3, -20, -8, -1, -26], [19, -5, -20, -13, -2, 27, -3, 22], [-2, -28, 18, -18, 0, -7, 3, 1], [10, 17, 5, 1, -2, -2, 0, 2], [13, -16, 8, -13, 8, 16, 3, 4], [35, 0, 6, -16, -10, -4, -6, -7], [-42, 15, -4, 21, -21, -38, -2, -25], [-47, 42, -14, 32, -18, -38, -8, -27], [6, -4, 4, 16, 1, -16, -10, 1], [38, -12, -12, -10, 1, 25, -3, -4], [-20, 36, 7, 28, -21, -37, -11, -20], [-46, 7, -16, 10, 0, -21, -2, -8], [-21, -3, -8, 1, 1, 15, 17, 15], [42, 2, 8, -8, 7, 8, -2, 9], [-30, 22, -2, 0, -4, -23, 1, -3], [19, 7, 9, -9, -16, -3, -11, 0], [-40, 30, -1, 23, -14, -23, -8, -12], [-18, 16, -9, 1, 17, 11, 6, 25], [-21, -4, -12, -18, -1, 10, 14, 10], [18, -1, 20, 3, -4, 6, -12, 0], [-43, 25, 7, 8, -17, -41, -12, -24], [-28, 34, -2, 17, 7, -13, -4, -1], [23, -17, -7, -34, -1, 24, 5, 13], [-4, -23, 1, -6, -8, 6, -7, -17], [-15, 29, 23, 5, -26, -27, -13, -16], [23, -36, 17, -21, 13, 14, 21, 1], [-4, -20, 26, -17, -3, -14, 1, -20], [-6, -30, 2, -20, 11, 19, 15, 14], [15, -11, -23, -7, 20, 42, 24, 33], [52, -33, 7, -18, 19, 29, 8, 19], [8, 0, 1, -9, 6, -26, -4, -1], [-8, 19, -6, -11, -16, -7, -17, -3], [-4, 13, 25, 11, -10, -39, 15, -5], [-7, -12, 1, 8, 15, -2, -4, -4], [22, -19, 13, -3, 11, -2, 10, -1], [30, 4, 11, -10, 6, 9, -3, 13], [-1, -21, 0, -17, 22, -7, 5, 8], [-26, 6, 9, 5, -10, -25, -15, -22], [-30, -13, -13, 24, -1, -3, 8, -5], [-3, 5, -20, -11, -3, 33, 19, 13], [-1, 18, -6, 13, -12, 3, -15, 5], [-16, 5, 11, 13, -6, -20, -26, -22], [3, -3, -28, 6, 15, 29, 3, 28], [31, -4, 12, 1, -7, 17, -6, 0], [-3, -6, 5, 1, 10, 5, 13, 21], [-55, 2, -1, 24, 2, -19, 5, -12], [-20, 5, -19, 3, 16, 22, -2, 0], [28, -22, -13, -18, -2, 16, -22, 0]];
    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_z();".
// 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_z(: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_z();".
// 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_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 48, -18, -117, 91, 18, -21, -1, 1]>,<7,R![-16, 160, -224, -59, 175, -7, -31, 0, 1]>,<11,R![500, -700, -460, 547, 151, -104, -23, 5, 1]>,<13,R![9224, 3828, -6922, -237, 1088, -10, -61, 0, 1]>],Snew);
    return Vf;
end function;