
// Make newform 1950.2.y.l 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_l();".
// 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_l();".
// 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], [1, 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 := [[0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, -1, -1, 0, 0, 0], [-2, -1, -2, 0, 1, 0, 2, 1], [0, 0, 0, 3, 0, 0, 0, 1], [0, 3, -1, 3, 0, 0, 0, 0], [-1, 1, 0, 2, -1, 1, 0, -1], [2, -4, -1, -1, -1, 1, 1, 4], [0, 1, 1, 4, -1, 4, 1, 1], [2, -3, 0, 3, -4, -2, -1, -5], [0, 5, 2, -3, 4, -3, 2, 5], [0, -3, 5, 1, 0, -1, -5, 3], [-2, 5, -1, 3, -2, -2, 0, 2], [-4, -3, 4, -3, 0, -1, -2, 4], [-2, 1, 0, -1, 4, -2, -1, -1], [1, 4, -3, 6, 1, 2, 0, -2], [-4, -3, -4, 3, 4, 0, 8, 0], [0, 4, -2, -4, -6, -4, -2, 4], [5, 1, 1, 4, 5, 3, 0, -3], [3, 0, 4, 0, 0, -4, -2, 4], [1, 6, 0, 6, 0, -3, 0, -3], [1, -1, 4, -1, 0, -3, -2, 4], [-4, 5, 1, -1, 2, 1, -1, -5], [0, 2, 1, -8, 0, 6, -2, 6], [0, 1, 1, 0, -11, 0, 1, 1], [0, 3, 0, -3, 0, -4, 3, -1], [2, -6, -7, 0, -1, 0, 7, 6], [4, 4, 0, -4, -8, -1, -10, 3], [0, 0, -4, -4, 0, -4, 0, 4], [4, -8, 0, -8, -2, 8, 0, 8], [-5, -4, -10, -4, 0, 2, 5, 2], [9, -4, 3, 1, -9, 3, -6, 3], [3, 4, -3, -1, -3, -3, 6, -3], [0, 5, 1, 4, 0, -1, 0, 1], [5, 2, 0, -2, -10, -7, 4, -5], [-4, 7, 0, -7, 8, 4, 0, 11], [-3, 1, -2, 2, 3, -3, 4, -3], [0, -13, 3, 7, 5, 7, 3, -13], [-2, -4, 2, -10, -2, -6, 0, 6], [0, -3, 2, 10, 3, 10, 2, -3], [0, 3, 4, 3, 0, -10, -2, 7], [3, 1, 3, 2, -3, -3, -6, -3], [0, 4, 0, 0, 3, 0, 0, 4], [0, -5, -1, 8, -11, 8, -1, -5], [-5, 1, 2, 5, 5, -6, -4, -6], [0, -2, 0, -5, -1, -5, 0, -2], [0, -2, -7, 1, 6, 1, -7, -2], [-8, -6, -1, 9, 8, -3, 2, -3], [-6, 1, 0, -1, 12, -10, -8, -9], [6, -8, 0, 8, -12, 6, -4, -2], [-1, -6, 0, 6, 2, -1, -15, -7], [5, 6, -3, 4, 5, -2, 0, 2], [-3, 2, 4, -11, 3, 9, -8, 9], [8, 7, 1, 3, -4, -3, -1, -7], [-18, 5, 3, 7, 9, -7, -3, -5], [13, -3, -6, -1, -13, 4, 12, 4], [-2, 7, -5, 10, 1, -10, 5, -7], [-8, -3, -4, -2, -8, 1, 0, -1], [8, -2, 0, 2, -16, 3, -11, 1], [-2, 2, -4, 7, 1, -7, 4, -2], [-2, -1, -7, 2, 2, -1, 14, -1], [-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], [6, -3, 13, -1, 6, 2, 0, -2], [-5, 16, 0, -16, 10, -14, -3, 2], [9, -4, 0, 3, 9, 7, 0, -7], [4, -3, -4, 9, -2, -9, 4, 3], [0, -12, 0, 7, 11, 7, 0, -12], [-4, -8, 0, 8, 8, 12, 4, 4], [34, 1, -3, 2, -17, -2, 3, -1], [-12, 9, 0, 1, -12, -8, 0, 8], [20, 2, 6, 10, -10, -10, -6, -2], [7, -9, 3, -4, -7, 13, -6, 13], [13, 3, 6, 3, 0, -11, -3, 8], [12, 6, -4, -12, -12, 6, 8, 6], [12, 2, -10, -2, -6, 2, 10, -2], [-3, -8, -17, -8, -3, 0, 0, 0], [0, -17, 12, 10, 11, 10, 12, -17], [6, 13, 0, -13, -12, 4, -16, 17], [-6, 16, 11, 3, 3, -3, -11, -16], [-3, -8, -6, -4, -3, 4, 0, -4], [0, -8, 0, 0, 4, 0, 0, -8], [0, 1, 0, -1, 0, 5, -23, 6], [-7, 7, 6, 4, -7, -3, 0, 3], [0, 8, 7, -4, -16, -4, 7, 8], [4, 1, -11, 0, 4, -1, 0, 1], [2, -3, 10, -3, 0, 14, -5, -11], [8, -3, 0, 3, -16, -7, -1, -10], [0, -6, 2, 8, 0, -8, -2, 6], [-17, 5, -2, 5, 17, -10, 4, -10], [0, -1, 8, -4, -17, -4, 8, -1], [-1, 7, 0, -7, 2, -12, 0, -5], [-1, -9, -5, 12, -1, 21, 0, -21], [-2, -21, 0, -1, 1, 1, 0, 21], [-14, 6, 0, 6, 0, 4, 0, -10], [0, 0, -20, -6, 0, 6, 20, 0], [4, -6, 0, 6, -8, 17, 6, 11], [-9, 3, 0, -3, 18, -20, -12, -17], [0, 14, -4, 4, 16, 4, -4, 14], [2, 0, -3, 7, 2, 7, 0, -7], [0, 9, -7, -12, -9, -12, -7, 9], [5, 11, 14, 11, 0, -9, -7, -2], [-3, 20, 12, 20, 0, -10, -6, -10], [0, -3, 11, -8, -24, -8, 11, -3], [1, -17, 0, -17, 0, 20, 0, -3], [11, 12, 2, 12, 0, -13, -1, 1], [0, 10, 8, -20, 4, -20, 8, 10], [-3, 0, 3, -15, -3, -15, 0, 15], [0, -1, -12, 8, 12, 8, -12, -1], [-8, -4, 2, -10, 8, 14, -4, 14], [-8, -2, 0, 2, 16, 2, -12, 0], [0, -4, 6, 2, 0, 6, 0, -6], [-7, 8, 3, 11, 7, -19, -6, -19], [-4, 2, 8, -4, 4, 2, -16, 2], [2, -7, 9, -3, -1, 3, -9, 7], [-10, -15, 12, -17, 5, 17, -12, 15], [3, -2, -14, 5, -3, -3, 28, -3], [24, -6, 2, -4, -12, 4, -2, 6], [10, -10, 6, -10, -5, 10, -6, 10], [-14, 4, 0, -4, 28, 2, -8, 6], [-17, 16, -8, -9, 17, -7, 16, -7], [-4, -4, 28, -8, 2, 8, -28, 4], [16, 15, -22, 15, 0, 0, 11, -15], [-2, 15, 4, 17, -2, 2, 0, -2], [-5, 9, 13, -15, 5, 6, -26, 6], [-15, -17, 0, 17, 30, 15, 3, -2], [16, 12, 12, 12, 0, -9, -6, -3], [2, 5, -19, -12, -1, 12, 19, -5], [0, 20, 0, -16, -4, -16, 0, 20], [0, -5, -1, 24, 3, 24, -1, -5], [40, -3, -4, -4, -20, 4, 4, 3], [-3, -13, 20, -6, -3, 7, 0, -7], [26, 6, 21, -20, -13, 20, -21, -6], [4, -2, 20, 12, -4, -10, -40, -10], [21, -17, -8, 6, -21, 11, 16, 11], [-11, -12, -15, 11, -11, 23, 0, -23], [0, 26, 8, -21, 15, -21, 8, 26], [-2, 4, 0, -4, 4, -18, -6, -14], [-20, -10, 6, -10, 10, 10, -6, 10], [6, 25, -13, 18, 6, -7, 0, 7], [-1, -5, 24, -5, 0, -5, -12, 10], [0, 6, 16, 4, 8, 4, 16, 6], [-13, -5, 23, -4, -13, 1, 0, -1], [-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], [-10, 1, 12, 11, 10, -12, -24, -12], [0, 12, -2, -11, -11, -11, -2, 12], [-6, 14, 0, -14, 12, 22, -6, 36], [24, -18, -16, -8, -12, 8, 16, 18], [16, -9, 11, 7, -8, -7, -11, 9], [13, -14, 2, -14, 0, 13, -1, 1], [0, 21, -3, -20, 0, -1, 6, -1], [0, -2, -6, 2, 0, 4, 0, -4], [-8, -12, 0, 12, 16, 20, -21, 8], [4, -18, 0, 18, -8, 0, -26, -18], [0, -5, -11, -12, 8, -12, -11, -5], [6, -13, -5, 9, 6, 22, 0, -22], [-24, -12, 4, -12, 0, 8, -2, 4], [5, 16, -5, -22, -5, 6, 10, 6], [0, 8, 0, -4, -44, -4, 0, 8], [20, -8, -16, -8, 0, -12, 8, 20], [0, -15, 2, 33, -9, 33, 2, -15], [-6, 8, 26, 20, -6, 12, 0, -12], [-1, 8, 0, -8, 2, 10, -28, 18], [-2, 13, 0, -13, 4, 4, -1, 17], [-4, -2, 0, 2, 8, -2, -14, -4], [4, -1, 12, -23, 4, -22, 0, 22], [-16, -6, 2, -2, 16, 8, -4, 8], [-11, -28, 5, 8, 11, 20, -10, 20], [-9, 15, 6, 15, 0, 19, -3, -34], [20, -3, 1, -1, -20, 4, -2, 4], [-16, -14, -10, -30, 8, 30, 10, 14], [5, -5, 0, 5, -10, -3, -11, -8], [-18, 3, -3, -12, 9, 12, 3, -3], [0, 8, 10, 1, 18, 1, 10, 8], [-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], [0, 0, 0, 0, -34, 0, 0, 0], [-19, -1, 4, 1, 19, 0, -8, 0], [-12, -22, 0, 22, 24, 5, 16, -17], [-5, -33, 0, -33, 0, 8, 0, 25], [-18, -10, -26, -14, 9, 14, 26, 10], [-9, 1, 35, -5, -9, -6, 0, 6], [-17, -14, 9, -6, -17, 8, 0, -8], [21, 12, 14, -13, -21, 1, -28, 1], [12, 12, -30, 12, 0, 1, 15, -13], [10, -1, -2, 19, -5, -19, 2, 1], [-4, 16, -13, -19, -4, -35, 0, 35], [-18, -23, -17, 0, 9, 0, 17, 23], [-5, 0, 0, 0, 10, 12, -25, 12], [16, -7, -12, -20, 16, -13, 0, 13], [-2, -4, -26, -4, 0, 19, 13, -15], [3, -16, 0, 16, -6, -19, -3, -35], [-8, -3, 9, -20, -8, -17, 0, 17], [10, 2, -26, -8, -10, 6, 52, 6], [0, 8, 2, 14, 2, 14, 2, 8], [-19, 22, -14, 22, 0, -14, 7, -8], [0, 22, 2, 4, 0, -4, -2, -22], [48, -4, 3, 3, -24, -3, -3, 4], [18, -10, 0, 10, -36, 10, -12, 0], [17, 8, 2, -17, 17, -25, 0, 25], [4, 9, 25, 9, -2, -9, -25, -9], [16, -9, -3, 13, -16, -4, 6, -4], [6, 28, 0, 16, -3, -16, 0, -28], [10, 27, 18, 27, 0, -36, -9, 9], [-24, 4, -17, -11, 12, 11, 17, -4], [-8, -8, -1, -11, 8, 19, 2, 19], [-18, -8, 16, -8, -18, 0, 0, 0], [11, 22, 0, -22, -22, -28, 12, -6], [14, -7, 0, 7, -28, 12, -7, 5], [-4, -12, 0, 18, 4, -6, 0, -6], [0, 8, -5, -1, -27, -1, -5, 8], [-3, 4, -20, 4, 0, -7, 10, 3], [0, 12, 12, 19, -8, 19, 12, 12], [-20, 9, 0, -9, 40, 2, -7, 11], [15, 5, 0, -5, -30, -26, 18, -21], [-29, 15, -4, -23, 29, 8, 8, 8], [-4, 19, -7, 30, 2, -30, 7, -19], [24, 12, -8, 12, 0, -11, 4, -1], [-46, -11, 23, -6, 23, 6, -23, 11], [0, -4, -10, -10, 0, 14, 20, 14], [-11, 10, 3, 31, -11, 21, 0, -21], [-1, 24, 0, -24, 2, 0, -13, 24], [4, 27, 0, 17, -2, -17, 0, -27], [0, -18, -4, 21, -31, 21, -4, -18], [-12, 6, 40, 8, 6, -8, -40, -6], [15, -7, -10, -7, 0, -9, 5, 16], [26, 8, 0, -8, -52, -14, 2, -6], [-11, 5, 34, 5, 0, -12, -17, 7], [11, -10, 35, 2, 11, 12, 0, -12], [0, -5, 33, 4, -1, 4, 33, -5], [-24, -12, -9, 29, 24, -17, 18, -17], [0, 24, 5, -19, 15, -19, 5, 24], [0, -3, -11, 0, -8, 0, -11, -3], [0, -20, -18, 4, -12, 4, -18, -20], [-14, -2, 8, 10, -14, 12, 0, -12], [12, -2, 30, 17, -6, -17, -30, 2], [-1, -17, 24, 10, 1, 7, -48, 7], [19, -13, -6, -13, 0, -8, 3, 21], [27, 3, 8, 13, -27, -16, -16, -16], [-46, 2, -15, 2, 23, -2, 15, -2], [-12, 15, -9, -13, -12, -28, 0, 28], [-24, 16, -14, 10, 12, -10, 14, -16], [0, -17, 12, 15, 2, 15, 12, -17], [-12, 5, -5, -9, 6, 9, 5, -5], [-9, -1, 34, -1, 0, -2, -17, 3], [0, -8, 1, 22, -8, 22, 1, -8], [10, 4, 0, -4, -20, -12, -10, -8], [-4, 4, 17, 21, -4, 17, 0, -17], [-16, -13, -28, 15, 16, -2, 56, -2], [0, -17, 3, 4, 3, 4, 3, -17], [9, -13, 0, 13, -18, 9, -35, -4], [-39, -14, 16, -14, 39, 28, -32, 28], [-4, -19, 0, 19, 8, 2, 27, -17], [18, 29, -6, 6, -9, -6, 6, -29], [6, -12, 0, 12, -12, -26, 16, -38], [40, 15, 10, 21, -20, -21, -10, -15], [23, 27, 0, -8, -23, -19, 0, -19], [20, -12, 4, 28, -20, -16, -8, -16], [-8, 8, 24, -4, -8, -12, 0, 12], [-9, -13, 0, 13, 18, -13, 11, -26], [-30, -1, 13, 3, 30, -2, -26, -2], [6, 42, 18, 4, 6, -38, 0, 38], [-26, 20, 20, 20, 0, 14, -10, -34], [-5, 19, 8, 19, 0, 18, -4, -37], [0, -26, -5, 4, 21, 4, -5, -26], [2, 7, -3, 0, -2, -7, 6, -7], [-7, 10, -13, 24, -7, 14, 0, -14], [0, 24, 6, -10, 26, -10, 6, 24], [-25, 10, -17, -14, 25, 4, 34, 4], [8, -34, 0, 34, -16, 20, -22, -14], [-7, -22, -12, 12, -7, 34, 0, -34], [42, 24, 11, -23, -42, -1, -22, -1], [-4, -22, 16, 16, 2, -16, -16, 22], [-18, -9, 44, 5, 9, -5, -44, 9], [-9, -24, 0, 24, 18, -3, 11, -27], [16, 8, 32, 17, -8, -17, -32, -8], [-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], [-3, 3, -10, 9, -3, 6, 0, -6], [-50, 15, 9, 3, 50, -18, -18, -18], [0, -9, 11, 18, -41, 18, 11, -9], [7, 16, -14, -3, 7, -19, 0, 19], [0, -6, 16, -3, -17, -3, 16, -6], [-5, -6, 25, -36, -5, -30, 0, 30], [-8, 25, 6, -11, -8, -36, 0, 36], [39, -18, 0, -18, 0, 7, 0, 11], [40, -5, 10, -5, 0, 12, -5, -7], [-24, 0, -27, 3, 12, -3, 27, 0], [0, -7, 1, -22, 18, -22, 1, -7], [0, -9, 1, -13, 38, -13, 1, -9], [-14, 4, -22, 12, -14, 8, 0, -8], [26, -23, -22, -23, 0, 20, 11, 3], [0, -8, 15, -1, 44, -1, 15, -8]];
    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_l();".
// 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_l(: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_l();".
// 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_l( : 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;