
// Make newform 1950.2.y.i in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1950_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1950_y_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_2_y_i();".
// 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_1950_2_y_i();".
// 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, 0, 0, 0, -1, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_1950_y();"
function MakeCharacter_1950_y()
    N := 1950;
    order := 6;
    char_gens := [1301, 1327, 301];
    v := [6, 3, 5];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1950_y_Hecke();"
function MakeCharacter_1950_y_Hecke(Kf)
    N := 1950;
    order := 6;
    char_gens := [1301, 1327, 301];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := [[-1, 0, 0, 0, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, -1, -1, -1, 0, 1], [-1, 0, -2, -1, -1, -1, 0, 1], [0, -3, 0, 0, 0, 4, 0, 0], [0, -3, 1, -3, 0, 3, -1, 3], [-2, 2, 0, 1, 1, -1, 0, -2], [-1, 1, 1, 4, -1, 3, 0, -3], [-1, 4, 1, 1, 1, -5, -2, -5], [-2, 3, 0, -3, 4, 2, 1, 5], [-4, 3, -2, -5, 4, 2, 4, 2], [0, 1, 5, -3, 0, -4, 0, 4], [4, -3, 1, -5, -2, 5, -1, 3], [4, 3, -4, 3, 0, 1, 2, -4], [-2, 1, 0, -1, 4, -2, -1, -1], [2, 6, -3, 4, -1, -4, 3, -6], [0, 3, -4, -3, -4, -3, -4, 3], [6, 4, 2, -4, -6, 0, -4, 0], [10, 4, 1, 1, -5, -1, -1, -4], [-3, 0, -4, 0, 0, 4, 2, -4], [1, 6, 0, 6, 0, -3, 0, -3], [-1, 1, -4, 1, 0, 3, 2, -4], [-2, -1, 1, 5, -2, 6, 0, -6], [0, 8, -1, -2, 0, -2, -1, 8], [-11, 0, 1, 1, 11, -1, -2, -1], [0, 3, 0, -3, 0, -4, 3, -1], [-1, 0, 7, 6, -1, 6, 0, -6], [-4, -4, 0, 4, 8, 1, 10, -3], [0, 4, 4, 0, 0, 0, -4, -4], [-2, 8, 0, 8, -2, 0, 0, 0], [-5, -4, -10, -4, 0, 2, 5, 2], [0, -1, -3, 4, -9, 4, -3, -1], [0, -1, -3, 4, 3, 4, -3, -1], [0, 4, 1, 5, 0, -5, -1, -4], [-5, -2, 0, 2, 10, 7, -4, 5], [-4, 7, 0, -7, 8, 4, 0, 11], [0, -2, 2, -1, 3, -1, 2, -2], [-5, -7, -3, 13, 5, -6, 6, -6], [4, 10, -2, 4, -2, -4, 2, -10], [3, 10, 2, -3, -3, -7, -4, -7], [0, 3, 4, 3, 0, -10, -2, 7], [0, 2, 3, 1, 3, 1, 3, 2], [-3, 0, 0, -4, 3, 4, 0, 4], [11, -8, 1, 5, -11, 3, -2, 3], [0, 5, 2, 1, -5, 1, 2, 5], [-1, -5, 0, -2, 1, 7, 0, 7], [-6, -1, 7, 2, 6, -1, -14, -1], [0, -9, 1, 6, 8, 6, 1, -9], [6, -1, 0, 1, -12, 10, 8, 9], [6, -8, 0, 8, -12, 6, -4, -2], [1, 6, 0, -6, -2, 1, 15, 7], [10, 4, -3, 6, -5, -6, 3, -4], [0, -11, 4, 2, -3, 2, 4, -11], [-4, -3, -1, -7, -4, -4, 0, 4], [9, -7, -3, -5, 9, 2, 0, -2], [0, -1, -6, -3, 13, -3, -6, -1], [-1, 10, -5, 7, -1, -3, 0, 3], [16, 2, 4, 3, -8, -3, -4, -2], [-8, 2, 0, -2, 16, -3, 11, -1], [1, -7, 4, -2, 1, 5, 0, -5], [0, -2, 7, 1, 2, 1, 7, -2], [9, 9, -22, 9, 0, 1, 11, -10], [1, -11, -2, -11, 0, 12, 1, -1], [-2, -4, 0, 4, 4, -8, -7, -12], [-3, -2, -16, -2, 0, -11, 8, 13], [12, -1, 13, -3, -6, 3, -13, 1], [-5, 16, 0, -16, 10, -14, -3, 2], [-18, -3, 0, 4, 9, -4, 0, 3], [2, 9, -4, -3, 2, -12, 0, 12], [-11, -7, 0, 12, 11, -5, 0, -5], [4, 8, 0, -8, -8, -12, -4, -4], [-17, -2, 3, -1, -17, 1, 0, -1], [24, -1, 0, -9, -12, 9, 0, 1], [10, 10, 6, 2, 10, -8, 0, 8], [0, 4, -3, 9, -7, 9, -3, 4], [13, 3, 6, 3, 0, -11, -3, 8], [0, 12, 4, -6, -12, -6, 4, 12], [6, -2, -10, 2, 6, 4, 0, -4], [-6, -8, -17, -8, 3, 8, 17, 8], [11, 10, 12, -17, -11, 7, -24, 7], [-6, -13, 0, 13, 12, -4, 16, -17], [-3, 3, 11, 16, -3, 13, 0, -13], [6, 4, 6, 8, -3, -8, -6, -4], [4, 0, 0, -8, -4, 8, 0, 8], [0, 1, 0, -1, 0, 5, -23, 6], [-14, 4, 6, 7, 7, -7, -6, -4], [16, 4, -7, -8, -16, 4, 14, 4], [8, 0, -11, 1, -4, -1, 11, 0], [-2, 3, -10, 3, 0, -14, 5, 11], [8, -3, 0, 3, -16, -7, -1, -10], [0, 8, 2, -6, 0, -14, 0, 14], [0, -5, 2, -5, 17, -5, 2, -5], [-17, -4, 8, -1, 17, 5, -16, 5], [1, -7, 0, 7, -2, 12, 0, 5], [2, -12, 5, 9, -1, -9, -5, 12], [-1, -1, 0, -21, -1, -20, 0, 20], [-14, 6, 0, 6, 0, 4, 0, -10], [0, 6, 20, 0, 0, -6, 0, 6], [-4, 6, 0, -6, 8, -17, -6, -11], [-9, 3, 0, -3, 18, -20, -12, -17], [-16, -4, 4, -14, 16, 18, -8, 18], [-4, -7, 3, 0, 2, 0, -3, 7], [-9, -12, -7, 9, 9, 3, 14, 3], [5, 11, 14, 11, 0, -9, -7, -2], [3, -20, -12, -20, 0, 10, 6, 10], [24, 8, -11, 3, -24, -11, 22, -11], [-1, 17, 0, 17, 0, -20, 0, 3], [11, 12, 2, 12, 0, -13, -1, 1], [4, -20, 8, 10, -4, 10, -16, 10], [6, 15, -3, 0, -3, 0, 3, -15], [-12, -8, 12, 1, 12, 7, -24, 7], [0, 10, -2, 4, 8, 4, -2, 10], [8, 2, 0, -2, -16, -2, 12, 0], [0, 2, 6, -4, 0, 4, -6, -2], [0, 11, 3, 8, -7, 8, 3, 11], [0, 4, -8, -2, 4, -2, -8, 4], [-1, 3, -9, 7, -1, 4, 0, -4], [5, 17, -12, 15, 5, -2, 0, 2], [0, 5, -14, -2, 3, -2, -14, 5], [12, -4, 2, -6, 12, -2, 0, 2], [-5, 10, -6, 10, -5, 0, 0, 0], [-14, 4, 0, -4, 28, 2, -8, 6], [0, 9, 8, -16, 17, -16, 8, 9], [-2, -8, 28, -4, -2, 4, 0, -4], [16, 15, -22, 15, 0, 0, 11, -15], [-4, 17, 4, 15, 2, -15, -4, -17], [0, -15, 13, 9, -5, 9, 13, -15], [-15, -17, 0, 17, 30, 15, 3, -2], [-16, -12, -12, -12, 0, 9, 6, 3], [1, -12, -19, 5, 1, 17, 0, -17], [4, 16, 0, -20, -4, 4, 0, 4], [3, 24, -1, -5, -3, -19, 2, -19], [-20, 4, 4, 3, -20, -1, 0, 1], [-6, -6, 20, -13, 3, 13, -20, 6], [13, -20, 21, 6, 13, 26, 0, -26], [0, -12, -20, 2, -4, 2, -20, -12], [0, -6, 8, 17, -21, 17, 8, -6], [22, -11, 15, 12, -11, -12, -15, 11], [15, -21, 8, 26, -15, -5, -16, -5], [2, -4, 0, 4, -4, 18, 6, 14], [-10, -10, 6, -10, -10, 0, 0, 0], [-12, -18, 13, -25, 6, 25, -13, 18], [1, 5, -24, 5, 0, 5, 12, -10], [8, 4, 16, 6, -8, -10, -32, -10], [-26, -4, 23, -5, 13, 5, -23, 4], [16, -4, 4, -4, 0, 4, -2, 0], [5, 15, 0, -15, -10, 19, 5, 34], [6, -28, 6, -28, 0, 35, -3, -7], [5, -12, -6, -12, 0, -7, 3, 19], [0, -11, -12, -1, 10, -1, -12, -11], [-11, -11, -2, 12, 11, -1, 4, -1], [-6, 14, 0, -14, 12, 22, -6, 36], [-12, 8, 16, 18, -12, 10, 0, -10], [-8, -7, -11, 9, -8, 16, 0, -16], [13, -14, 2, -14, 0, 13, -1, 1], [0, -20, -3, 21, 0, 21, -3, -20], [0, 2, -6, -2, 0, 2, 6, -2], [-8, -12, 0, 12, 16, 20, -21, 8], [-4, 18, 0, -18, 8, 0, 26, 18], [-8, 12, 11, 5, 8, -17, -22, -17], [-12, -9, 5, 13, 6, -13, -5, 9], [24, 12, -4, 12, 0, -8, 2, -4], [0, -22, -5, 16, 5, 16, -5, -22], [44, 4, 0, -8, -44, 4, 0, 4], [-20, 8, 16, 8, 0, 12, -8, -20], [-9, 33, 2, -15, 9, -18, -4, -18], [12, -20, -26, -8, -6, 8, 26, 20], [1, -8, 0, 8, -2, -10, 28, -18], [-2, 13, 0, -13, 4, 4, -1, 17], [4, 2, 0, -2, -8, 2, 14, 4], [8, -23, 12, -1, -4, 1, -12, 23], [0, -2, 2, -6, -16, -6, 2, -2], [0, -8, -5, 28, 11, 28, -5, -8], [-9, 15, 6, 15, 0, 19, -3, -34], [0, -1, 1, -3, 20, -3, 1, -1], [-8, -30, -10, -14, -8, 16, 0, -16], [-5, 5, 0, -5, 10, 3, 11, 8], [9, 12, 3, -3, 9, -15, 0, 15], [18, 1, 10, 8, -18, -9, -20, -9], [23, -5, 22, -5, 0, 5, -11, 0], [4, -1, -14, -1, 0, -13, 7, 14], [8, 27, 0, -27, -16, -7, -12, 20], [33, 19, -18, 19, 0, -5, 9, -14], [34, 0, 0, 0, -34, 0, 0, 0], [0, 1, 4, -1, -19, -1, 4, 1], [-12, -22, 0, 22, 24, 5, 16, -17], [5, 33, 0, 33, 0, -8, 0, -25], [-9, -14, -26, -10, -9, 4, 0, -4], [-18, -5, 35, 1, 9, -1, -35, 5], [34, 6, -9, 14, -17, -14, 9, -6], [0, 13, -14, -12, -21, -12, -14, 13], [12, 12, -30, 12, 0, 1, 15, -13], [5, 19, -2, -1, 5, -20, 0, 20], [8, 19, 13, -16, -4, 16, -13, -19], [9, 0, 17, 23, 9, 23, 0, -23], [5, 0, 0, 0, -10, -12, 25, -12], [-32, 20, 12, 7, 16, -7, -12, -20], [2, 4, 26, 4, 0, -19, -13, 15], [3, -16, 0, 16, -6, -19, -3, -35], [-16, -20, 9, -3, 8, 3, -9, 20], [0, -8, -26, 2, 10, 2, -26, -8], [-2, -14, -2, -8, 2, 22, 4, 22], [-19, 22, -14, 22, 0, -14, 7, -8], [0, 4, 2, 22, 0, 18, 0, -18], [-24, -3, -3, 4, -24, 7, 0, -7], [-18, 10, 0, -10, 36, -10, 12, 0], [-34, 17, -2, -8, 17, 8, 2, -17], [2, 9, 25, 9, 2, 0, 0, 0], [0, 13, -3, -9, 16, -9, -3, 13], [-3, -16, 0, -28, -3, -12, 0, 12], [10, 27, 18, 27, 0, -36, -9, 9], [12, 11, 17, -4, 12, -15, 0, 15], [0, 11, 1, 8, 8, 8, 1, 11], [-36, -8, 16, -8, 18, 8, -16, 8], [-11, -22, 0, 22, 22, 28, -12, 6], [14, -7, 0, 7, -28, 12, -7, 5], [0, 18, 0, -12, -4, -12, 0, 18], [27, 1, 5, -8, -27, 7, -10, 7], [3, -4, 20, -4, 0, 7, -10, -3], [-8, 19, 12, 12, 8, -31, -24, -31], [20, -9, 0, 9, -40, -2, 7, -11], [-15, -5, 0, 5, 30, 26, -18, 21], [0, 23, 4, -15, 29, -15, 4, 23], [2, -30, 7, -19, 2, 11, 0, -11], [24, 12, -8, 12, 0, -11, 4, -1], [23, 6, -23, 11, 23, 5, 0, -5], [0, -10, -10, -4, 0, -4, -10, -10], [22, -31, -3, -10, -11, 10, 3, 31], [1, -24, 0, 24, -2, 0, 13, -24], [-2, -17, 0, -27, -2, -10, 0, 10], [-31, 21, -4, -18, 31, -3, 8, -3], [-6, 8, 40, 6, -6, -2, 0, 2], [15, -7, -10, -7, 0, -9, 5, 16], [26, 8, 0, -8, -52, -14, 2, -6], [11, -5, -34, -5, 0, 12, 17, -7], [22, 2, 35, -10, -11, 10, -35, -2], [1, -4, -33, 5, -1, -1, 66, -1], [0, 29, -9, -12, -24, -12, -9, 29], [15, -19, 5, 24, -15, -5, -10, -5], [8, 0, 11, 3, -8, -3, -22, -3], [-12, 4, -18, -20, 12, 16, 36, 16], [28, -10, -8, 2, -14, -2, 8, 10], [6, 17, 30, -2, 6, -19, 0, 19], [0, -10, -24, 17, 1, 17, -24, -10], [19, -13, -6, -13, 0, -8, 3, 21], [0, -13, -8, -3, -27, -3, -8, -13], [-23, 2, -15, 2, -23, 0, 0, 0], [-24, -13, -9, 15, 12, -15, 9, 13], [12, -10, 14, -16, 12, -6, 0, 6], [-2, -15, -12, 17, 2, -2, 24, -2], [-6, -9, -5, 5, -6, 14, 0, -14], [9, 1, -34, 1, 0, 2, 17, -3], [-8, 22, 1, -8, 8, -14, -2, -14], [10, 4, 0, -4, -20, -12, -10, -8], [-8, 21, 17, 4, 4, -4, -17, -21], [0, 15, -28, -13, -16, -13, -28, 15], [-3, -4, -3, 17, 3, -13, 6, -13], [9, -13, 0, 13, -18, 9, -35, -4], [0, 14, -16, 14, 39, 14, -16, 14], [-4, -19, 0, 19, 8, 2, 27, -17], [-9, -6, 6, -29, -9, -23, 0, 23], [-6, 12, 0, -12, 12, 26, -16, 38], [-20, -21, -10, -15, -20, 6, 0, -6], [0, 8, 0, -27, -23, -27, 0, 8], [0, 28, 4, -12, 20, -12, 4, 28], [-16, -4, 24, 8, 8, -8, -24, 4], [9, 13, 0, -13, -18, 13, -11, 26], [0, -3, -13, 1, 30, 1, -13, -3], [-12, -4, -18, -42, 6, 42, 18, 4], [-26, 20, 20, 20, 0, 14, -10, -34], [5, -19, -8, -19, 0, -18, 4, 37], [-21, -4, 5, 26, 21, -22, -10, -22], [0, 0, -3, 7, 2, 7, -3, 0], [14, -24, 13, -10, -7, 10, -13, 24], [-26, 10, -6, -24, 26, 14, 12, 14], [0, 14, 17, -10, 25, -10, 17, 14], [-8, 34, 0, -34, 16, -20, 22, 14], [-14, 12, -12, -22, 7, 22, 12, -12], [0, -23, 11, 24, 42, 24, 11, -23], [2, -16, -16, 22, 2, 38, 0, -38], [-9, 5, 44, -9, -9, -14, 0, 14], [-9, -24, 0, 24, 18, -3, 11, -27], [8, 17, 32, 8, 8, -9, 0, 9], [7, -1, -4, -1, 0, -12, 2, 13], [19, 8, -14, 8, 0, 11, 7, -19], [12, -14, 0, 14, -24, 26, 24, 12], [10, 21, -38, 21, 0, -24, 19, 3], [-6, 9, -10, 3, 3, -3, 10, -9], [0, 3, 9, 15, -50, 15, 9, 3], [-41, 18, 11, -9, 41, -9, -22, -9], [-14, 3, 14, -16, 7, 16, -14, -3], [17, 3, -16, 6, -17, -9, 32, -9], [-10, -36, 25, -6, 5, 6, -25, 36], [16, 11, -6, -25, -8, 25, 6, -11], [39, -18, 0, -18, 0, 7, 0, 11], [40, -5, 10, -5, 0, 12, -5, -7], [12, -3, 27, 0, 12, 3, 0, -3], [18, -22, 1, -7, -18, 29, -2, 29], [-38, 13, -1, 9, 38, -22, 2, -22], [28, -12, 22, -4, -14, 4, -22, 12], [-26, 23, 22, 23, 0, -20, -11, -3], [44, -1, 15, -8, -44, 9, -30, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1950_y_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_2_y_i();".
// 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_1950_2_y_i(:prec:=8)
    chi := MakeCharacter_1950_y();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1950_2_y_i();".
// 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_1950_2_y_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 8, 20, 8, 22, 16, 14, 4, 1]>],Snew);
    return Vf;
end function;