
// Make newform 9900.2.c.x in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9900_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_9900_c_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_9900_2_c_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_9900_2_c_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 := [25, 0, 11, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 6, 0, 1], [6, 0, 1, 0]];
    Rf_basisdens := [1, 1, 5, 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_9900_c();"
function MakeCharacter_9900_c()
    N := 9900;
    order := 2;
    char_gens := [4951, 5501, 2377, 4501];
    v := [2, 2, 1, 2];
    // 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_9900_c_Hecke();"
function MakeCharacter_9900_c_Hecke(Kf)
    N := 9900;
    order := 2;
    char_gens := [4951, 5501, 2377, 4501];
    char_values := [[1, 0, 0, 0], [1, 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], [0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 3, 0], [1, 0, 0, 0], [0, 0, -1, 0], [0, -1, -2, 0], [-3, 0, 0, 2], [0, -1, 1, 0], [5, 0, 0, -1], [-5, 0, 0, 2], [0, -2, -3, 0], [5, 0, 0, 2], [0, 0, -10, 0], [0, -2, -7, 0], [0, 3, -3, 0], [-7, 0, 0, 2], [6, 0, 0, 1], [0, 0, 4, 0], [0, 0, 0, -6], [0, 1, -5, 0], [3, 0, 0, -7], [0, -5, -4, 0], [1, 0, 0, 1], [0, 1, 9, 0], [12, 0, 0, -3], [0, -5, -8, 0], [0, -6, 3, 0], [3, 0, 0, -1], [0, -2, -7, 0], [0, 3, -8, 0], [6, 0, 0, 3], [0, 7, 8, 0], [1, 0, 0, 6], [-4, 0, 0, -4], [5, 0, 0, 0], [0, 4, -6, 0], [0, 3, -13, 0], [0, 2, -11, 0], [0, -6, -3, 0], [6, 0, 0, -9], [2, 0, 0, 3], [-19, 0, 0, -1], [0, 4, 4, 0], [0, -7, 7, 0], [7, 0, 0, 3], [5, 0, 0, 0], [0, 2, -15, 0], [0, -7, -8, 0], [13, 0, 0, 3], [0, 1, -13, 0], [-4, 0, 0, -1], [-7, 0, 0, 9], [-1, 0, 0, -7], [0, 0, 18, 0], [0, 4, -13, 0], [-1, 0, 0, 11], [5, 0, 0, -6], [0, -2, 3, 0], [3, 0, 0, -6], [0, 11, 0, 0], [0, 4, 14, 0], [0, 1, 30, 0], [13, 0, 0, 4], [0, -6, -10, 0], [0, 3, 0, 0], [3, 0, 0, -8], [0, 0, 22, 0], [0, 3, 21, 0], [11, 0, 0, -2], [0, 4, -7, 0], [-6, 0, 0, 0], [0, -3, -17, 0], [0, 8, 3, 0], [31, 0, 0, 0], [0, 0, 18, 0], [-12, 0, 0, 0], [0, -1, -10, 0], [22, 0, 0, 4], [-17, 0, 0, 0], [27, 0, 0, 0], [20, 0, 0, -3], [7, 0, 0, 10], [0, 8, 9, 0], [-3, 0, 0, 5], [0, -2, -28, 0], [28, 0, 0, 1], [0, 3, -5, 0], [3, 0, 0, -12], [0, -6, -19, 0], [0, -2, 5, 0], [-7, 0, 0, -4], [0, -6, 13, 0], [11, 0, 0, 2], [-6, 0, 0, 5], [0, -2, -22, 0], [-6, 0, 0, -9], [2, 0, 0, 2], [0, 4, -14, 0], [-16, 0, 0, -3], [0, 3, 1, 0], [0, 16, -1, 0], [0, -9, 15, 0], [10, 0, 0, -5], [4, 0, 0, -13], [0, 5, 2, 0], [0, -5, 2, 0], [0, -2, 17, 0], [25, 0, 0, -11], [16, 0, 0, -7], [0, 0, 34, 0], [0, -17, -14, 0], [0, 6, -12, 0], [-17, 0, 0, -6], [2, 0, 0, -9], [31, 0, 0, 4], [0, 12, -10, 0], [0, -8, -1, 0], [0, 11, 4, 0], [13, 0, 0, 7], [27, 0, 0, -2], [0, 10, -11, 0], [0, 2, 4, 0], [0, 10, -7, 0], [-16, 0, 0, 9], [-28, 0, 0, -4], [-8, 0, 0, 12], [-15, 0, 0, -12], [0, -9, -2, 0], [0, -18, -19, 0], [-26, 0, 0, 9], [0, 1, -37, 0], [15, 0, 0, 1], [0, -8, -15, 0], [-16, 0, 0, 8], [31, 0, 0, -6], [0, -1, -29, 0], [0, -6, 1, 0], [0, -13, -2, 0], [-24, 0, 0, 12], [-27, 0, 0, 16], [-20, 0, 0, 16], [0, 0, 8, 0], [0, -2, -16, 0], [29, 0, 0, -5], [-4, 0, 0, 11], [0, -9, -22, 0], [0, 12, 3, 0], [20, 0, 0, 10], [0, -8, 5, 0], [0, -16, -28, 0], [-18, 0, 0, -9], [0, -4, 15, 0], [0, 8, -17, 0], [0, 16, 12, 0], [-2, 0, 0, -8], [-36, 0, 0, -4], [3, 0, 0, -6], [0, 4, -24, 0], [25, 0, 0, -14], [0, 10, 23, 0], [0, 0, 42, 0], [0, 7, -24, 0], [-10, 0, 0, 5], [0, 6, -9, 0], [0, -16, 4, 0], [29, 0, 0, -15], [0, 3, -32, 0], [37, 0, 0, 0], [0, 18, 24, 0], [34, 0, 0, 1], [22, 0, 0, -7], [39, 0, 0, -3], [0, -9, -7, 0], [-23, 0, 0, 12], [18, 0, 0, -12], [0, 0, 0, 4], [-47, 0, 0, 4], [0, 6, 26, 0], [-4, 0, 0, -8], [0, 6, 28, 0], [-4, 0, 0, 11], [0, 6, 35, 0], [0, 0, -18, 0], [0, 2, -47, 0], [5, 0, 0, -4], [0, -6, -38, 0], [0, -9, 41, 0], [7, 0, 0, 18], [-56, 0, 0, 4], [0, 5, -36, 0], [0, 0, 36, 0], [15, 0, 0, 10], [-5, 0, 0, -5], [0, 7, 41, 0], [0, -6, 3, 0], [-22, 0, 0, 6], [0, 22, 1, 0], [0, -5, -13, 0], [0, 7, 8, 0], [-32, 0, 0, 13], [-2, 0, 0, 11], [0, 1, -12, 0], [9, 0, 0, -13], [-46, 0, 0, 8], [0, -1, 37, 0], [12, 0, 0, 8], [0, 16, -1, 0], [10, 0, 0, -8], [-22, 0, 0, 3], [0, 12, 13, 0], [3, 0, 0, 6], [0, -3, -31, 0], [0, 10, 20, 0], [37, 0, 0, -2], [-16, 0, 0, -6], [0, -20, -12, 0], [-15, 0, 0, -15], [0, -9, 33, 0], [0, 5, 49, 0], [-53, 0, 0, 5], [-11, 0, 0, 0], [-4, 0, 0, 8], [0, -3, 2, 0], [0, 6, 15, 0], [1, 0, 0, 12], [0, 21, 9, 0], [6, 0, 0, 6], [0, -12, -8, 0], [19, 0, 0, -29], [0, -9, 11, 0], [-23, 0, 0, -12], [-5, 0, 0, 8], [-15, 0, 0, 12], [0, -5, 10, 0], [0, 12, 3, 0], [-28, 0, 0, -2], [0, 1, 47, 0], [29, 0, 0, -1], [-40, 0, 0, -1], [0, 16, 41, 0], [-16, 0, 0, 6], [0, -2, 10, 0], [-23, 0, 0, 0], [0, -18, 6, 0], [12, 0, 0, -9], [0, -3, 25, 0], [-2, 0, 0, -5], [-29, 0, 0, -3], [0, 2, -32, 0], [0, -6, -44, 0], [1, 0, 0, -2], [0, -10, -38, 0], [37, 0, 0, 0], [0, -3, 21, 0], [-44, 0, 0, 10], [-37, 0, 0, 18], [0, 0, 61, 0], [0, -1, 49, 0], [0, 12, 10, 0], [0, 12, 8, 0], [0, -6, -18, 0], [-39, 0, 0, 14], [0, 25, 4, 0], [0, 14, 13, 0], [-8, 0, 0, -6], [-39, 0, 0, 24], [60, 0, 0, 3], [0, -18, -7, 0], [0, -12, 45, 0], [50, 0, 0, -18], [0, -8, -51, 0], [0, -18, 8, 0], [13, 0, 0, -18], [0, 17, 14, 0], [0, 14, -15, 0], [0, 0, -12, 0], [1, 0, 0, 3], [-12, 0, 0, -17], [-19, 0, 0, -16], [0, -19, -2, 0], [-52, 0, 0, 0], [0, -19, 19, 0], [35, 0, 0, -12], [0, 21, 13, 0], [-8, 0, 0, 28], [0, -15, -16, 0], [0, 9, -33, 0], [-30, 0, 0, 11], [13, 0, 0, -23], [55, 0, 0, -8], [0, -22, -17, 0], [0, -12, -48, 0], [3, 0, 0, -9], [0, 0, 35, 0], [22, 0, 0, 19], [-17, 0, 0, 2], [0, 6, 63, 0], [-38, 0, 0, 4], [0, -4, -31, 0], [0, 2, 39, 0], [0, 12, -6, 0], [-40, 0, 0, 7], [0, -10, 1, 0], [23, 0, 0, 15], [0, -10, -13, 0], [0, 10, 8, 0], [24, 0, 0, -19], [-11, 0, 0, 7], [0, -4, 57, 0], [0, -7, 34, 0], [13, 0, 0, -11], [-14, 0, 0, 7], [0, 18, 17, 0], [0, 10, 53, 0], [24, 0, 0, -10], [-26, 0, 0, 4], [-12, 0, 0, -6], [0, -22, -48, 0], [12, 0, 0, 15], [-42, 0, 0, -5], [0, 21, -17, 0], [-63, 0, 0, 6], [0, -20, -47, 0], [0, -25, -14, 0], [42, 0, 0, 16], [-29, 0, 0, -15], [0, 5, -39, 0], [0, 26, 28, 0], [0, 8, 22, 0], [-4, 0, 0, 6], [0, -8, -55, 0], [7, 0, 0, -3], [0, 2, -11, 0], [44, 0, 0, 6], [0, 0, -54, 0], [-10, 0, 0, 34], [0, -10, 46, 0], [-24, 0, 0, -5], [0, -21, -41, 0], [0, -6, 74, 0], [0, -20, 29, 0], [1, 0, 0, -26], [-14, 0, 0, 17], [0, -8, -61, 0], [-77, 0, 0, -5], [-15, 0, 0, -11], [0, -28, 5, 0], [-19, 0, 0, 29], [0, -8, 71, 0], [8, 0, 0, 3], [0, -23, -15, 0], [-3, 0, 0, -18], [0, -21, 32, 0], [-8, 0, 0, 0], [0, -12, -3, 0], [10, 0, 0, 4], [-52, 0, 0, -13], [0, -18, -57, 0], [0, 0, -36, 0], [0, 17, -1, 0], [25, 0, 0, 22], [0, -8, -13, 0], [65, 0, 0, 11], [0, 7, 15, 0], [0, 6, 5, 0], [0, 30, 24, 0], [0, -6, 29, 0], [-4, 0, 0, -24], [59, 0, 0, -7], [60, 0, 0, 8], [0, 15, 33, 0], [-57, 0, 0, 15], [-37, 0, 0, -9], [0, 16, -3, 0], [-6, 0, 0, -9], [-6, 0, 0, -21], [0, 24, -10, 0], [10, 0, 0, -11], [0, -1, -73, 0], [-18, 0, 0, 30], [0, 2, 97, 0], [0, 14, 2, 0], [0, -21, -6, 0], [-37, 0, 0, 4], [0, 9, -48, 0], [20, 0, 0, -9], [0, -12, 40, 0], [0, 12, 5, 0], [0, 8, -50, 0], [-36, 0, 0, -7], [0, -32, 8, 0], [-17, 0, 0, -14], [0, 30, 16, 0], [-39, 0, 0, 24], [0, 18, 11, 0], [55, 0, 0, 9], [12, 0, 0, -18], [9, 0, 0, -2], [-22, 0, 0, -7], [-20, 0, 0, 24], [0, 11, -23, 0], [0, 3, 25, 0], [0, -4, -65, 0], [75, 0, 0, 6], [-37, 0, 0, -12], [0, 14, 26, 0], [-41, 0, 0, 16], [0, 0, -28, 0], [70, 0, 0, -14], [0, 3, 2, 0], [0, -32, -13, 0], [0, -20, 5, 0], [-52, 0, 0, -3], [0, 0, 22, 0], [1, 0, 0, -2], [-39, 0, 0, 0], [0, 5, 92, 0], [0, 18, 27, 0], [0, -3, -72, 0], [51, 0, 0, -6], [0, 26, -13, 0], [0, 24, -3, 0], [25, 0, 0, 16], [0, 18, 14, 0], [0, 12, -36, 0], [0, 20, -14, 0], [-11, 0, 0, 25], [-6, 0, 0, -32], [-27, 0, 0, 27], [41, 0, 0, 24], [-33, 0, 0, 33], [31, 0, 0, 18], [0, 21, 3, 0], [0, -6, 13, 0], [41, 0, 0, -7], [-20, 0, 0, 36], [-52, 0, 0, 18], [0, -15, -32, 0], [49, 0, 0, -6], [0, 25, -16, 0], [-21, 0, 0, -6], [-3, 0, 0, -7], [11, 0, 0, 14], [31, 0, 0, -1], [0, 32, -2, 0], [0, -20, -41, 0], [0, 6, 57, 0], [57, 0, 0, -16], [-53, 0, 0, 2], [0, -41, -30, 0], [64, 0, 0, -17], [0, -23, 23, 0], [22, 0, 0, 22], [0, 0, -77, 0], [3, 0, 0, 6], [-8, 0, 0, -9], [80, 0, 0, -7], [0, -6, -22, 0], [0, -26, -10, 0], [7, 0, 0, -9], [23, 0, 0, -6], [22, 0, 0, 7], [-7, 0, 0, -9], [0, -23, -51, 0], [0, 9, 26, 0], [-10, 0, 0, -26], [0, -5, 56, 0], [1, 0, 0, -2], [-31, 0, 0, -15], [0, -4, -45, 0], [0, 4, 14, 0], [10, 0, 0, 1], [44, 0, 0, -6], [16, 0, 0, 7], [0, -32, -11, 0], [-14, 0, 0, -17], [39, 0, 0, -32], [0, 0, -78, 0], [0, -8, -19, 0], [0, 4, -2, 0], [-25, 0, 0, 2], [0, -12, 16, 0], [-36, 0, 0, -24], [0, -6, -19, 0], [0, 1, 29, 0], [-3, 0, 0, -25], [-63, 0, 0, -12], [-44, 0, 0, 12], [-45, 0, 0, 10], [0, -11, 6, 0], [0, 20, -8, 0], [83, 0, 0, 7], [0, 6, 84, 0], [-13, 0, 0, 8], [68, 0, 0, -12], [0, 22, -6, 0], [0, -22, -71, 0], [-66, 0, 0, -4], [-28, 0, 0, 18], [-43, 0, 0, -7], [0, -22, 12, 0], [0, 15, 48, 0], [0, -4, 8, 0], [0, -12, -34, 0], [0, -15, 0, 0], [0, -27, -24, 0], [40, 0, 0, -4], [0, -6, 52, 0], [0, 18, -37, 0], [-79, 0, 0, 11], [20, 0, 0, 8], [0, -6, -28, 0], [0, -22, 7, 0], [-99, 0, 0, 4], [0, -10, 2, 0], [90, 0, 0, -6], [-17, 0, 0, 42], [26, 0, 0, -25], [0, -30, -8, 0], [0, 21, -4, 0], [5, 0, 0, -11], [-17, 0, 0, -11], [0, -1, -32, 0], [44, 0, 0, -2], [-14, 0, 0, 10], [0, 12, 17, 0], [0, -32, 23, 0], [0, 26, 4, 0], [-47, 0, 0, 25], [0, 48, 14, 0], [0, -14, -58, 0], [0, 2, 11, 0], [5, 0, 0, 20], [0, -27, -52, 0], [0, -2, 98, 0], [0, 9, -32, 0], [-3, 0, 0, 15], [36, 0, 0, -25], [0, -21, -8, 0], [-47, 0, 0, 22], [0, 7, 2, 0], [59, 0, 0, 13], [0, -23, 23, 0], [27, 0, 0, -30], [-18, 0, 0, 25], [0, 15, 59, 0], [0, -14, 47, 0], [0, 12, -47, 0], [-86, 0, 0, 16], [36, 0, 0, -39], [0, -8, -65, 0], [0, 4, 62, 0], [0, -45, -24, 0], [44, 0, 0, -16], [77, 0, 0, 6], [0, 7, -24, 0], [-95, 0, 0, 16], [56, 0, 0, -18], [0, -8, -69, 0], [0, 0, -87, 0], [89, 0, 0, -10], [102, 0, 0, 0], [0, -8, -59, 0], [-34, 0, 0, -26], [14, 0, 0, 6], [0, -32, -19, 0], [66, 0, 0, 2], [0, 4, 20, 0], [-19, 0, 0, 38], [0, -33, -64, 0], [0, -13, -47, 0], [25, 0, 0, -3], [0, 26, 8, 0], [11, 0, 0, -27], [29, 0, 0, -22], [0, 28, 23, 0], [61, 0, 0, -6], [-23, 0, 0, 22], [-25, 0, 0, 51], [5, 0, 0, 26], [0, -30, 26, 0], [0, 1, 50, 0], [-44, 0, 0, 4], [-40, 0, 0, 33], [87, 0, 0, -3], [0, 19, -23, 0], [0, -34, -41, 0], [76, 0, 0, -5], [0, 17, -16, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9900_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9900_2_c_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_9900_2_c_x(:prec:=4)
    chi := MakeCharacter_9900_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(4297) - 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_9900_2_c_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_9900_2_c_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9900_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 0, 23, 0, 1]>,<13,R![1, 0, 1]>,<17,R![9, 0, 15, 0, 1]>,<29,R![15, -9, 1]>,<41,R![15, -12, 1]>,<59,R![15, 12, 1]>],Snew);
    return Vf;
end function;