
// Make newform 4016.2.a.f in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4016_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_4016_2_a_f();".
// 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_4016_2_a_f();".
// 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 := [-5, -12, 20, 15, -8, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [1, -9, -15, 4, 4, -1], [-17, 5, 19, -4, -4, 1], [-3, -27, 17, 6, -6, 1], [-4, 15, 12, -7, -3, 1]];
    Rf_basisdens := [1, 1, 4, 4, 4, 2];
    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_4016_a();"
function MakeCharacter_4016_a()
    N := 4016;
    order := 1;
    char_gens := [2511, 3013, 257];
    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_4016_a_Hecke(Kf)
    return MakeCharacter_4016_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, 0, 0, 1, 0], [-1, 1, 0, -1, 0, 0], [-1, 0, 0, 0, 0, 1], [-1, 1, -1, -1, 0, 0], [-1, 1, 0, 1, 0, 0], [2, -1, -1, 0, 1, 0], [0, 0, -1, 2, -1, -1], [-3, -1, 1, -1, -1, 0], [0, 0, 3, 0, -1, -1], [0, -2, 1, 0, 0, -1], [-1, -1, 0, 2, 0, 0], [1, 0, -3, 1, 0, -1], [-1, 3, 0, 0, -2, 0], [0, -2, 0, 2, -2, -2], [1, -1, -1, 3, 0, -2], [-1, 1, 4, -4, 0, 2], [-11, 1, -1, -1, 0, 0], [3, -1, -2, 1, 0, -4], [-2, 0, -2, 4, 2, 0], [1, -1, 4, 0, -3, 2], [-5, -1, 3, -1, -3, -1], [-4, 4, 0, 2, -2, -2], [5, -1, 0, 4, 1, 2], [2, -2, 0, -4, -2, 0], [-2, -4, 4, 2, -3, 0], [1, -2, 5, -5, -2, 3], [-4, 2, 3, -2, 1, 1], [-5, -1, -4, 4, 0, -4], [4, 0, 2, 1, 0, 2], [0, 0, -4, 2, 2, -2], [3, -5, 2, 1, 2, 2], [8, -6, 0, 0, 0, 0], [-3, 3, 4, 2, -4, -2], [-4, 2, 0, -4, 5, 2], [-2, 0, -4, 2, 4, 4], [-9, -1, -5, 1, 4, 0], [-6, 2, -1, -8, 1, 3], [-6, -2, 0, 0, 0, -2], [6, 4, -2, 0, 2, 2], [2, -5, 0, -2, 0, 3], [-2, 1, 2, 2, -6, -1], [-10, 6, -2, -2, 2, -4], [0, 2, 2, 0, -6, -4], [1, 3, 4, -5, 0, 0], [0, 2, -4, -4, 2, 2], [-9, 3, 0, -1, 2, 2], [-4, 2, 0, -4, 4, 4], [0, 1, 4, -4, 2, 5], [5, -1, 0, 2, 6, 6], [0, 2, 1, -4, 0, 0], [-2, -6, 6, -2, -2, 0], [7, -4, -5, 3, 0, -3], [1, 0, 0, 0, 0, 0], [-2, 2, 2, -6, -4, 0], [0, -1, 1, 0, 3, 4], [4, -4, 4, -6, -6, 2], [-5, 4, 3, -3, 6, 3], [2, 4, -9, 2, 5, 1], [4, 4, -2, 0, 6, 2], [12, -10, 4, 4, 2, -2], [1, 5, 4, -4, 0, 0], [3, 3, -3, 3, 2, -6], [-5, 3, 0, 0, 1, 4], [2, 4, -2, -2, 6, 0], [18, -3, -2, -2, 2, -1], [2, -2, 6, -6, 1, 4], [-10, 1, 3, -4, -3, -6], [9, -1, -12, 8, 2, 2], [-10, -2, -4, 0, 2, 2], [10, -2, 4, -2, -2, 2], [-7, -2, 1, 7, -2, -1], [8, 0, 0, 0, -4, 0], [4, 2, 0, -2, -5, -4], [16, -6, 4, 8, 0, 2], [2, 2, -6, -8, 4, 4], [2, 0, 1, 2, -3, -1], [2, -6, 1, -2, -1, -7], [-10, 0, -6, 2, -2, 2], [0, -6, 4, -2, 2, 2], [7, -1, 5, -5, 6, 10], [-6, -10, 5, 6, -1, 1], [4, -2, 7, 8, -4, 1], [0, -10, 6, 2, -2, 4], [6, 8, -8, -4, 6, 0], [-16, -4, 2, 6, -8, -4], [2, -1, -9, 4, 1, -4], [-20, 6, 8, -10, 0, 4], [3, -7, 4, 2, -2, 2], [-6, -2, -2, 6, -6, 0], [10, -6, 5, 8, -3, -3], [6, 4, 0, 0, -4, -10], [4, -6, -2, 8, 4, 2], [9, 1, -12, 3, 0, 0], [9, -3, -1, -7, 6, 2], [9, -9, 2, 3, 0, 4], [-23, 7, 4, -7, 2, 6], [10, -8, 6, 0, -4, 0], [0, 0, -2, 4, -7, -2], [8, 2, 0, 10, 1, 0], [4, 4, -2, 0, -7, -6], [1, 5, 1, 3, -6, 4], [7, 1, -8, 0, -6, -2], [9, 3, 0, -4, 4, 2], [8, 0, 2, -2, 12, 4], [-3, 5, -1, -5, -8, 0], [-9, 7, 2, -5, 0, 8], [-1, -5, -1, -5, 3, 4], [0, -2, 4, 2, 0, -2], [-2, 6, 0, 2, 4, 2], [-17, 5, -1, 3, -8, -4], [5, 3, -3, -9, 12, 10], [7, -9, -1, 3, 3, -4], [5, 1, 2, 1, -6, 2], [-14, 2, 14, -10, -4, 4], [0, 12, -2, 2, 2, 2], [0, 10, -8, -8, 8, 2], [-4, 0, -4, -2, -6, -10], [15, -5, 4, 10, -2, -4], [4, -12, -3, 14, -3, -7], [-16, 12, -5, -8, 1, -5], [-8, 0, 8, 2, -10, -8], [-2, 0, -10, 4, 13, 0], [-3, 5, -2, -7, 0, -4], [-11, -7, 4, 1, -12, 0], [-14, 12, -4, -8, -4, -4], [14, 1, 2, 4, 8, -3], [-7, 2, -8, 8, 0, -3], [-24, 3, -4, -2, -2, -9], [16, -2, -5, 4, 3, 1], [0, 9, 4, -4, -2, -3], [4, -6, -7, 8, 0, 3], [-9, 4, 8, 0, 0, 5], [10, -8, 4, -2, 3, 8], [8, 0, 0, 4, 8, 2], [-3, -1, 1, -1, -7, -5], [-20, 6, 0, -8, -1, 6], [-7, 9, 1, -7, 4, -2], [-2, 6, -5, 2, -1, -1], [-4, 2, -10, 8, -4, -4], [-6, -1, -8, 4, -10, 1], [1, -5, -2, 3, -8, -12], [-4, -2, -2, -4, 8, 14], [16, 12, 0, -12, 8, 8], [-8, 6, 0, 6, 2, 0], [12, 4, -11, -2, 8, -4], [-23, 3, 0, -9, 4, -4], [0, 4, -4, 4, 10, 2], [-14, 1, -2, -2, 2, -9], [-12, -5, -4, 2, 10, -1], [10, 3, 2, -6, 6, 1], [6, -4, 0, 14, -6, -10], [-7, 5, 9, -7, 4, 2], [24, 0, -2, 8, -4, -6], [8, -16, 4, 6, 6, -2], [28, 6, -6, 2, 6, -2], [-2, -2, 0, -4, -4, 4], [-2, 0, -8, 10, 0, -10], [8, 2, -2, -6, 10, 10], [-20, 0, -1, -4, -11, 3], [-4, 1, 12, 0, -10, -3], [8, -2, 0, 6, -12, -4], [-33, 7, 4, -4, 1, -4], [7, -5, 0, 9, -12, -12], [4, -6, 8, -6, 8, 6], [-16, -4, 6, -4, 2, 8], [4, -14, 0, 12, 2, 2], [-17, 1, 0, 0, 8, 10], [17, -10, 12, 0, 4, 1], [-4, -6, 0, 4, -8, 4], [5, 9, -4, 1, -16, -4], [-11, -9, 10, -1, -8, -4], [-3, 3, 14, -17, 0, 12], [4, 4, 0, -6, 10, 10], [4, -14, 7, 0, 2, 9], [-10, -2, 1, -12, 16, 5], [34, -10, -8, 10, -6, -8], [-1, -3, -4, 0, 2, 2], [-2, 2, 12, 0, -4, 4], [-6, 4, 4, -10, 9, 4], [-19, -3, -5, -3, 13, 3], [11, -7, 7, 7, -6, -8], [-21, -3, 10, -15, 2, 2], [2, 14, 9, -12, -4, -3], [-20, 4, -2, -2, -2, 4], [-33, 9, 8, -7, 6, 2], [0, -10, -8, 20, -8, -12], [8, 0, 10, 0, -1, -6], [-1, 11, -5, 7, 5, -8], [7, 3, -12, -2, 18, -4], [4, 4, 2, 7, -12, -2], [1, 9, 7, 5, -8, 2], [-13, 15, -8, 12, 6, -4], [-5, 15, -9, 5, -4, 0], [-13, 11, -4, -14, -2, 0], [6, 1, -5, 8, 1, -6], [19, -1, -1, 11, -5, -8], [-2, 4, 16, -2, -13, 4], [4, -6, 2, 14, -10, 0], [-6, 20, -10, 3, 6, -4], [0, 2, -4, -6, 10, 6], [-7, 7, -18, 11, 0, -4], [14, -19, -2, 8, -4, 9], [9, -4, 8, 4, 4, -5], [-22, -2, 2, 4, -5, 6], [22, -4, 0, 8, -9, -14], [10, -6, -10, -6, 10, 0], [7, -1, 4, 3, 12, 12], [6, 0, 6, -6, 8, 6], [-14, -11, -4, 12, -2, -1], [-12, 14, -6, -14, 6, 0], [-4, -4, 5, -4, 3, 5], [-9, -4, -8, 0, 16, 5], [-12, 8, -2, -6, 9, 4], [-1, 14, -8, -4, 8, -5], [12, 12, -5, 0, 5, -1], [2, -8, -8, 0, 12, 2], [-14, 2, -2, -1, -16, -2], [-9, -14, 7, -11, -2, -3], [-16, 14, 2, -6, -5, 10], [14, -10, 7, -10, 9, 7], [-11, -5, 1, -5, -3, -4], [10, -16, -1, 2, -8, 0], [10, 12, -8, -4, 12, 8], [-7, -7, -13, -3, 12, 2], [-26, 3, 6, 6, -6, -3], [-28, 10, 2, 8, -6, -4], [32, 6, 4, -2, -2, 0], [-5, 3, 3, -21, 1, 3], [16, -4, -2, 10, -7, -12], [17, -17, 0, -9, -2, 2], [12, -8, 0, 6, 9, 2], [-10, -2, -4, -10, 10, 6], [-6, 0, 0, -12, 8, 2], [15, -5, -7, 7, -10, 2], [10, -10, -3, 12, -8, -12], [-6, 0, 8, 6, -10, -2], [1, 5, -20, 12, 0, 0], [2, -6, -7, 4, -1, 5], [12, -3, -11, 0, 3, -14], [-36, -5, 0, -2, 8, 1], [-12, 7, -12, 12, 2, -9], [-10, -4, -6, -2, 4, 0], [-21, -5, -3, 1, -6, -4], [-18, -4, -8, -8, 14, 2], [0, -6, 0, -14, 6, 0], [16, -6, 8, -2, -10, -4], [16, -4, 2, 10, -8, 0], [-7, 13, -6, -13, 4, 8], [-2, -4, 0, -2, -6, 0], [-19, -1, 16, -1, -14, 2], [-8, -8, 14, 12, 4, 4], [-1, -5, -11, 15, 7, -3], [5, 3, 19, -17, -13, -3], [19, -11, -5, 13, -8, -10], [-26, 6, 6, 0, 15, 10], [0, 2, 2, -4, -10, 6], [-8, 0, 16, 10, -3, -2], [6, 7, 12, -14, -4, 11], [6, 2, -10, 8, 0, -12], [-14, 10, -2, 2, -8, -6], [-18, -5, -4, 10, 4, -1], [-3, 19, -8, -16, 2, 6], [-24, -6, 3, 0, 5, 9], [2, -10, -2, 10, -20, -6], [-15, 1, 8, 6, 4, -4], [-4, 0, -8, 10, 4, -2], [-18, -6, 12, -6, -8, 2], [0, 15, -6, 8, 0, -15], [0, 14, 0, 0, 4, 0], [38, 18, -17, -6, 17, 3], [0, -8, -12, 16, -15, -8], [-18, -2, 18, -20, -6, 0], [-20, 2, -4, 8, -14, -10], [-9, -3, -16, 8, -1, -2], [16, -24, 4, 4, 0, 8], [-20, -14, -11, 6, 17, 3], [6, 0, -1, -14, -13, 1], [10, -2, -14, 16, 0, 2], [13, -17, 7, 5, -2, 8], [0, 0, -9, 10, -12, -8], [0, -2, -9, 0, 6, -11], [-6, 6, -24, 8, -8, -6], [3, 11, -5, 13, 4, 0], [-28, 8, -2, -14, 11, 4], [0, 4, 0, -10, 7, 4], [2, -8, 10, 0, -4, 20], [19, -11, -3, -7, -8, -2], [-28, 16, 7, -2, 8, 16], [0, -12, 12, 2, 2, 10], [-7, -5, -16, 3, 14, 10], [14, 0, 19, 0, -1, -3], [0, 12, -2, -7, 4, 14], [39, 3, 6, 9, -16, 0], [-40, -6, 0, -2, 6, 0], [-20, 18, -8, -26, 6, 6], [20, -14, -18, 7, 6, 0], [9, 15, 0, -4, -6, -6], [-28, 8, -10, -12, 5, 2], [-4, 10, -11, -2, 3, -5], [-6, 14, -4, -2, 12, 4], [-32, 6, 2, -10, -9, -2], [-16, -12, 0, 4, -16, -10], [-18, -22, 13, 4, -9, -7], [-20, 19, -16, -16, 2, -1], [-24, 9, -4, 4, -8, 3], [5, 3, 17, 9, -14, 4], [-4, -9, -24, 16, 10, -9], [-20, 2, 2, -19, 14, 8], [-34, 1, 4, 2, -4, -3], [-14, 8, 4, 2, -2, 0], [36, -24, 1, 0, -3, -3], [-9, 9, -5, -3, -9, -3], [-14, -9, 18, 6, -10, 5], [-19, 19, -11, -9, -7, 4], [-32, 10, 0, -16, 13, 8], [-16, 3, -14, -10, 10, 7], [-3, -7, 4, 0, 15, 0], [-2, -10, -5, 0, 16, -1], [-4, -7, 20, -20, -12, -1], [-19, 17, -14, 21, 8, -8], [-24, 2, 22, -22, -10, 4], [11, -9, 16, -6, 2, -4], [18, 14, -7, 0, 12, -12], [12, 18, -14, -4, 14, 8], [-10, -10, 14, -13, 8, 2], [-5, 3, 4, -8, 2, 8], [-9, 3, -12, -1, -2, -2], [-30, 0, 10, -16, -4, 6], [-4, -1, -18, 0, 4, 1], [-19, 5, -19, 3, -6, -16], [13, 13, -19, 27, 16, -4], [-20, 2, 20, -18, -8, 10], [-25, 15, -21, 11, -2, -2], [-17, -1, 12, -37, -8, 12], [8, -10, 20, 12, -4, 6], [1, -7, -15, 11, -6, -8], [-16, -8, -8, 14, -4, -12], [-16, 12, 10, -1, -4, -6], [-1, 13, -13, 15, 4, 0], [-17, 25, -9, -7, 0, -14], [-3, 13, -10, -23, 12, 8], [-16, 14, 4, 6, -8, 4], [-15, -13, 1, -1, 3, 0], [12, -3, 14, -24, 12, 15], [7, -17, 2, 7, -4, 4], [-22, -4, -5, 2, 11, -7], [30, -14, 26, -6, -5, 12], [32, -25, 0, -2, 2, 7], [2, -2, -4, 12, -16, 2], [-36, 8, 6, -12, -2, -2], [26, -9, -9, 12, 1, -16], [32, -14, -10, 32, -2, -10], [8, -22, 2, -3, -6, -12], [-29, 3, 1, -3, -14, -4], [-4, 4, -16, -14, 8, 10], [16, -9, -13, 0, 1, 6], [21, -19, 0, 13, 2, -14], [-44, 10, 6, -6, 0, 8], [-8, 10, 1, 0, -8, -9], [-31, -3, -7, 11, -2, 0], [-22, 2, -26, 20, 10, -4], [22, 2, 8, -18, 4, 4], [-6, -10, 16, 6, 6, 4], [9, -7, 2, 7, -14, -14], [39, -12, -12, 16, 8, -19], [0, 18, -6, 6, -3, 2], [34, 2, -6, 10, 0, -8], [-10, 2, 6, 0, 12, 10], [34, -2, -18, 8, -11, -6], [17, -17, -7, 13, 0, 8], [-42, 16, -2, 2, -6, -10], [16, 14, -20, -2, 13, 0], [-8, -9, 11, -12, -11, -6], [-4, -18, 17, -4, -5, 7], [-12, 12, -8, 2, 11, 6], [-11, 2, -4, 0, -20, -15], [-51, 2, 5, -3, -8, 5], [2, 10, 10, -20, 6, 6], [-8, 22, -6, 4, -10, -12], [26, -4, -1, 14, -3, 1], [28, 14, -8, 4, -1, 0], [-8, -14, 8, -2, -6, -2], [-31, 10, -11, 17, 0, -3], [8, -12, -12, 16, 8, 8], [-9, -10, 16, -8, -12, 19], [-51, 1, 0, -12, 9, 8], [28, 2, 17, -12, -15, -9], [14, -14, 20, 8, -1, 12], [-30, 14, 4, 4, -10, -12], [-16, 10, 1, -4, 20, 12], [18, -8, 17, 6, -3, -9], [-21, 23, 4, 7, -12, -16], [-30, -1, 0, 0, 14, 5], [16, 7, -8, 2, 18, -1], [14, -16, 16, 10, 0, 8], [-6, 8, 14, -21, -10, 12], [5, -7, -19, 29, -1, -16], [-20, 32, -14, -18, -1, -4], [-43, 11, 20, -20, 6, 6], [-13, 7, 3, -15, 20, 8], [-5, 2, -8, 0, -8, -13], [36, -6, 12, 26, -16, -10], [-1, 9, 0, 4, -15, -10], [0, 23, 10, -12, -8, 5], [-32, 24, 0, -10, 4, -2], [40, 2, 6, 8, -5, -8], [-18, -2, 4, -6, -16, -6], [11, -7, 1, -5, 12, -4], [-42, -2, 4, -8, 20, 4], [-16, 20, -16, -8, -6, 4], [18, -18, 0, -4, 8, 14], [24, 28, -4, -16, 4, -4], [12, -10, 11, 2, 5, 5], [-4, 8, -12, 12, 10, 2], [-18, 5, 2, 20, 0, -7], [9, -15, 1, -11, -1, 17], [48, -4, 10, 16, -14, -8], [22, -24, 10, 10, -2, -2], [-12, -4, -4, 10, -17, -12], [-21, -21, 0, 12, 16, 12], [-6, -12, -4, 8, -4, 4], [-22, 12, -4, 6, -12, -26], [9, -3, 12, 2, 22, 8], [29, -9, -7, 7, -7, 3], [31, 9, -14, 9, 16, 8], [-8, 12, -3, 2, 15, 1], [-23, -3, 2, -7, 12, 0], [-7, -9, 9, -27, 2, 12], [-18, 16, -10, 18, -4, -20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4016_2_a_f();".
// 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_4016_2_a_f(:prec:=6)
    chi := MakeCharacter_4016_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_4016_2_a_f();".
// 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_4016_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-88, 8, 74, -9, -16, 1, 1]>],Snew);
    return Vf;
end function;