
// Make newform 3150.2.bp.a in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3150_bp();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3150_bp_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_3150_2_bp_a();".
// 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_3150_2_bp_a();".
// 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_3150_bp();"
function MakeCharacter_3150_bp()
    N := 3150;
    order := 6;
    char_gens := [2801, 127, 451];
    v := [3, 3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3150_bp_Hecke();"
function MakeCharacter_3150_bp_Hecke(Kf)
    N := 3150;
    order := 6;
    char_gens := [2801, 127, 451];
    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 := [[-1, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, -2], [-4, 0, 0, -1, 2, 1, 0, 0], [2, -1, 0, -1, 0, 0, 0, 1], [-4, -2, 0, -2, 2, 2, 0, 2], [0, 2, 1, 0, 0, -2, 0, 2], [2, 0, 1, 0, -2, 0, -2, 0], [0, -2, 0, 2, 0, -2, 4, -4], [0, -2, -2, 0, 0, 0, 2, 2], [0, 1, -6, 1, 0, 0, 0, 0], [-4, 2, 0, 2, 0, -3, 0, 1], [0, 2, 0, -2, 0, 0, -2, 2], [-1, 4, -2, 0, -1, -4, 0, 4], [0, 2, -2, -6, -1, -6, -2, 2], [0, -3, 2, -2, -6, -2, 2, -3], [0, 2, -2, 6, 0, 4, 0, -4], [8, -6, -6, -6, -4, 6, 6, 6], [0, 4, 0, -4, 0, 2, 4, 6], [0, -4, 0, 0, -6, 0, 0, -4], [6, -2, 4, 6, -6, -4, -8, -4], [0, -6, 0, 6, 0, 8, 2, 2], [-4, -4, 0, 7, 4, -3, 0, -3], [-6, -2, -12, -2, 0, 6, 6, -4], [0, -4, 6, 0, 2, 0, 6, -4], [4, -6, 0, 5, -4, 1, 0, 1], [2, -6, 2, 0, -2, 6, -4, 6], [0, -2, 4, 8, -6, 8, 4, -2], [-6, -6, 8, -6, 0, -2, -4, 8], [8, 2, 0, -2, -16, -1, 4, 1], [8, -5, -2, 1, -8, 4, 4, 4], [0, 4, 0, 2, 4, 2, 0, 4], [6, 0, 0, 0, -12, 8, 4, 8], [-6, 6, -8, 0, -6, -6, 0, 6], [0, 10, 2, -4, 0, -4, 2, 10], [0, 7, -6, -3, -8, -3, -6, 7], [-6, -4, -6, 6, -6, 10, 0, -10], [2, 8, 0, -8, -4, -8, 1, 0], [-1, 10, 2, 2, -1, -8, 0, 8], [0, -5, -6, 3, 0, -3, 6, 5], [-4, 4, 0, -4, 8, -6, -14, -2], [-2, -8, -8, -2, -2, 6, 0, -6], [-4, 0, -6, 6, 2, -6, 6, 0], [2, -10, -2, -10, 0, 12, 1, -2], [12, -4, 8, -6, -6, 6, -8, 4], [-5, -8, 8, -8, 0, -2, -4, 10], [4, -11, -8, -11, 0, 1, 4, 10], [4, 0, 0, 10, -2, -10, 0, 0], [0, 14, 4, 2, 0, -12, 0, 12], [10, 10, -2, -4, -10, -6, 4, -6], [6, 4, 0, -4, -12, -12, 8, -8], [0, -4, -3, -12, 0, 12, 3, 4], [-2, 9, -8, 9, 0, -8, 4, -1], [0, 10, -6, 4, 0, -6, 0, 6], [0, 0, -6, 4, -4, 4, -6, 0], [0, 2, 10, 6, 0, 6, 10, 2], [2, 0, 14, 0, 2, 0, 0, 0], [4, 7, 6, 8, -2, -8, -6, -7], [-4, -2, 0, 2, 8, 7, -4, 5], [0, 0, -6, -10, 2, -10, -6, 0], [-2, -2, 0, 2, 4, 4, -13, 2], [2, 6, -8, 6, 0, -8, 4, 2], [0, -4, -4, 6, -10, 6, -4, -4], [-2, -14, 4, -2, 2, 16, -8, 16], [-4, -8, -4, 8, 4, 0, 8, 0], [-12, -2, -5, -4, 12, 6, 10, 6], [0, 0, 0, 0, 0, 6, 10, 6], [0, 0, 10, 2, 2, 2, 10, 0], [4, -4, 0, 4, -8, 4, 12, 0], [-4, 0, 6, 12, 2, -12, -6, 0], [-4, 0, -16, -2, -4, -2, 0, 2], [0, -10, 0, -3, 0, -3, 0, -10], [6, -20, 6, -5, 6, 15, 0, -15], [-9, 4, -8, 4, 0, 6, 4, -10], [7, 0, 2, -4, 7, -4, 0, 4], [4, -2, 2, 6, -2, -6, -2, 2], [-6, 8, 10, -7, 6, -1, -20, -1], [8, 1, -4, -6, 8, -7, 0, 7], [0, -16, 0, -20, 0, 20, 0, 16], [-4, 0, -4, 0, 0, 15, 2, -15], [16, 2, 8, 2, 0, 4, -4, -6], [16, 16, -6, 6, -8, -6, 6, -16], [-2, 2, 12, 2, 0, -2, -6, 0], [4, 16, -4, 4, 4, -12, 0, 12], [4, -2, 2, -4, -4, 6, -4, 6], [0, 13, 0, -13, 0, -10, 0, 3], [6, 14, 6, 8, 6, -6, 0, 6], [2, -16, 4, -16, 0, 4, -2, 12], [-4, -11, 0, 11, 8, -6, 4, -17], [0, -2, 6, 2, 0, 4, 0, -4], [0, 4, 6, 8, -4, 8, 6, 4], [-24, -11, 4, -6, 12, 6, -4, 11], [6, -7, 0, 7, -12, 5, 6, -2], [-12, -8, -14, 12, 12, -4, 28, -4], [-2, 12, 0, -12, 4, -20, 0, -8], [-16, -10, 8, 0, 16, 10, -16, 10], [0, 9, 8, 5, 4, 5, 8, 9], [-12, -16, 6, 2, 12, 14, -12, 14], [2, -10, 2, 22, -2, -12, -4, -12], [-14, 10, 0, -10, 28, -8, 4, 2], [0, -10, -14, -2, -3, -2, -14, -10], [-20, -2, -4, 16, 10, -16, 4, 2], [-4, -5, -16, 18, -4, 23, 0, -23], [0, 16, 6, -8, -4, -8, 6, 16], [0, 6, 2, 4, 24, 4, 2, 6], [12, -20, 0, 20, -24, 4, -22, -16], [6, -10, 12, 6, 6, 16, 0, -16], [-8, 18, 6, 4, 4, -4, -6, -18], [8, 4, 0, -4, -16, -12, 22, -8], [-28, 1, 4, 2, 28, -3, -8, -3], [8, -9, -6, -5, -4, 5, 6, 9], [4, 6, -28, 6, 0, 0, 14, -6], [-22, -6, -8, -18, 11, 18, 8, 6], [0, -4, -28, -4, 0, -12, 14, 16], [-16, -5, 16, -7, 8, 7, -16, 5], [-8, 0, 16, 0, 0, -6, -8, 6], [6, 20, -14, 16, -3, -16, 14, -20], [18, -18, -5, 8, -18, 10, 10, 10], [6, 9, 0, -9, -12, 5, 6, 14], [4, 2, 10, 4, -2, -4, -10, -2], [6, 6, 0, -6, -12, -10, -6, -4], [2, 14, -15, 16, 2, 2, 0, -2], [0, -24, 2, 0, 6, 0, 2, -24], [-4, 12, -12, -4, -4, -16, 0, 16], [-6, 8, 0, -8, 12, -18, 6, -10], [-2, 12, -2, 14, 2, -26, 4, -26], [6, -2, 10, -10, -6, 12, -20, 12], [0, 11, -8, 11, 0, -22, 4, 11], [12, -21, -2, 13, -12, 8, 4, 8], [0, -8, 11, 6, 12, 6, 11, -8], [3, 12, -20, 12, 0, -8, 10, -4], [8, -6, 12, 6, -8, 0, -24, 0], [-6, -15, 0, 15, 12, 11, 14, -4], [-12, -19, 8, 10, 12, 9, -16, 9], [-8, 4, 0, -4, 16, 4, -27, 8], [-20, 24, 9, 6, 10, -6, -9, -24], [0, -16, -4, 14, 8, 14, -4, -16], [-16, -16, 0, 16, 32, 8, 16, -8], [8, -10, 16, -21, -4, 21, -16, 10], [-5, -8, 0, 8, 10, 22, -20, 14], [0, -2, 16, -4, 0, -2, 0, 2], [-16, -11, -8, -6, 8, 6, 8, 11], [-6, -14, 16, -14, 0, 10, -8, 4], [-12, 4, -16, -2, 6, 2, 16, -4], [2, 28, 8, 28, 0, -18, -4, -10], [-4, 12, -4, 12, 0, -15, 2, 3], [-32, 12, 12, 2, 16, -2, -12, -12], [-2, 28, 16, 12, -2, -16, 0, 16], [14, 16, 3, -12, -14, -4, -6, -4], [-18, 1, 8, 12, -18, 11, 0, -11], [-16, 5, 32, 5, 0, 2, -16, -7], [8, 16, 0, -16, -16, -12, 10, 4], [-12, 0, 8, 8, -12, 8, 0, -8], [-20, -8, 16, -12, 10, 12, -16, 8], [0, -26, 0, 26, 0, 0, 8, -26], [-18, 4, 2, 26, 18, -30, -4, -30], [8, 5, 16, 1, -8, -6, -32, -6], [22, -14, 16, -14, 0, 20, -8, -6], [0, -10, 0, 20, 26, 20, 0, -10], [4, -2, -16, -6, -4, 8, 32, 8], [12, -6, 16, -6, 0, 30, -8, -24], [-4, -9, 0, 9, 8, 11, -12, 2], [4, 19, 10, -11, -4, -8, -20, -8], [0, 6, -2, -12, 0, -12, -2, 6], [-6, 8, -2, 16, 3, -16, 2, -8], [0, 0, -8, -2, -4, -2, -8, 0], [0, 29, 2, 0, -2, 0, 2, 29], [-4, -20, 14, -20, 0, 4, -7, 16], [9, 18, -2, 26, 9, 8, 0, -8], [20, -21, -2, -22, -10, 22, 2, 21], [-6, -6, 0, 6, 12, 10, 28, 4], [-4, 2, -28, -10, -4, -12, 0, 12], [0, 8, 4, 4, 0, -4, -4, -8], [20, -10, -20, -10, -10, 10, 20, 10], [32, 13, 16, 13, 0, -13, -8, 0], [7, -16, 8, -16, 0, 6, -4, 10], [-8, 28, 2, 4, 4, -4, -2, -28], [12, -3, 16, -3, 0, -2, -8, 5], [-2, 10, -30, 12, -2, 2, 0, -2], [-4, -9, 8, -23, -4, -14, 0, 14], [24, -16, -20, -16, 0, 13, 10, 3], [0, 12, 0, -12, 0, 9, -6, 21], [8, 6, -8, 0, 8, -6, 0, 6], [0, -4, 2, 0, 19, 0, 2, -4], [0, -4, 20, -6, 6, -6, 20, -4], [-4, 20, 0, 3, 2, -3, 0, -20], [0, -8, -10, -4, -6, -4, -10, -8], [16, 8, -1, 12, -16, -20, 2, -20], [0, 18, -10, -14, -22, -14, -10, 18], [-16, -12, 22, -2, 16, 14, -44, 14], [-10, 8, -12, 8, 0, -6, 6, -2], [23, 26, -12, -6, -23, -20, 24, -20], [-16, -16, 2, 16, 16, 0, -4, 0], [0, -6, 8, 2, -18, 2, 8, -6], [8, -6, -26, 2, -4, -2, 26, 6], [0, -12, 20, 4, -5, 4, 20, -12], [-2, -8, 2, 25, -2, 33, 0, -33], [4, 14, 0, -14, -8, 18, -6, 32], [-2, -24, -4, -16, -2, 8, 0, -8], [-20, -24, 2, -12, 10, 12, -2, 24], [-12, 18, 0, -18, 24, 2, 10, 20], [6, -8, 2, -3, -6, 11, -4, 11], [24, -4, 5, -28, -12, 28, -5, 4], [-2, -1, 20, -1, 0, -9, -10, 10], [36, -12, -19, -6, -18, 6, 19, 12], [-26, -6, -4, 8, -26, 14, 0, -14], [2, -6, -4, 0, -2, 6, 8, 6], [-12, 10, 0, -10, 24, -17, -24, -7], [8, -16, 1, 2, -4, -2, -1, 16], [10, 14, -24, 8, 10, -6, 0, 6], [2, 6, 44, 6, 0, -8, -22, 2], [12, -3, 0, 3, -24, -11, -12, -14], [-6, -6, 6, -32, -6, -26, 0, 26], [0, -22, -6, 20, -2, 20, -6, -22], [-3, -16, 12, -36, -3, -20, 0, 20], [24, 9, -20, 3, -12, -3, 20, -9], [0, -26, 8, 29, 20, 29, 8, -26], [15, 0, 2, -24, -15, 24, -4, 24], [-23, 4, 28, 4, 0, -6, -14, 2], [-20, 12, 20, -4, 20, -8, -40, -8], [-2, 22, 0, -22, 4, -12, 30, 10], [12, -7, -32, -13, 12, -6, 0, 6], [-4, 19, -4, 22, -4, 3, 0, -3], [4, -10, 0, 10, -8, 26, 16, 16], [-10, -8, 4, -8, 0, 14, -2, -6], [18, 20, 12, -4, 18, -24, 0, 24], [-12, 4, 0, 18, 6, -18, 0, -4], [20, 15, -16, -5, -20, -10, 32, -10], [-18, -10, -26, -5, -18, 5, 0, -5], [-20, 0, -32, -9, 10, 9, 32, 0], [14, 14, 32, -18, -7, 18, -32, -14], [30, -10, 20, -10, 0, -4, -10, 14], [40, 14, 4, 7, -20, -7, -4, -14], [-8, -2, -8, -2, 0, 20, 4, -18], [0, -20, 12, -4, 0, 4, -12, 20], [4, -4, 44, -12, 4, -8, 0, 8], [-10, 22, -17, -8, 10, -14, 34, -14], [4, -12, 0, 12, -8, 1, 2, -11], [38, -10, 4, -10, 0, -8, -2, 18], [0, -2, -8, 6, -14, 6, -8, -2], [20, 14, 21, -4, 20, -18, 0, 18], [0, 27, -4, -10, 0, -10, -4, 27], [34, -10, 2, 26, -34, -16, -4, -16], [6, 6, 0, -6, -12, 18, 16, 24], [30, 2, 2, 6, -30, -8, -4, -8], [0, 14, -8, 14, 0, -21, 4, 7], [0, -8, -16, 3, 6, 3, -16, -8], [0, -4, -22, 0, -24, 0, -22, -4], [10, 0, -42, 0, 0, -4, 21, 4], [22, -3, 0, 3, -44, -1, -30, -4], [-4, 11, 8, 6, 4, -17, -16, -17], [0, -8, -1, 16, -46, 16, -1, -8], [3, 16, 0, -16, -6, -12, 16, 4], [-32, -28, -8, 4, 16, -4, 8, 28], [14, -11, 0, 4, 14, 15, 0, -15], [0, -20, 2, 30, 2, 30, 2, -20], [0, 4, -16, -12, -26, -12, -16, 4], [-1, 24, 0, -24, 2, 6, 10, 30], [18, 10, 0, -24, -18, 14, 0, 14], [16, -17, -36, -15, -8, 15, 36, 17], [8, 22, 32, 22, 0, -28, -16, 6], [-36, 6, 24, 8, 18, -8, -24, -6], [18, -15, 12, -15, 0, -1, -6, 16], [-4, -12, -30, -8, 2, 8, 30, 12], [8, 22, 33, 12, 8, -10, 0, 10], [-8, -22, 0, 22, 16, 4, 6, -18], [-8, -18, -24, -18, 0, -19, 12, 37], [-6, 0, 0, 0, 12, -10, -14, -10], [0, -8, -9, 2, 10, 2, -9, -8], [-2, -4, -34, 14, -2, 18, 0, -18], [8, 2, -38, 0, -4, 0, 38, -2], [0, -6, -2, -16, -28, -16, -2, -6], [0, -16, -4, 26, 0, -10, 8, -10], [0, -2, 16, -2, 0, 8, -8, -6], [-8, 15, 12, 2, 8, -17, -24, -17], [-4, 6, 6, -30, 4, 24, -12, 24], [0, 12, 8, -20, 0, -20, 8, 12], [31, -16, -24, 0, -31, 16, 48, 16], [2, -31, 0, 4, -2, 27, 0, 27], [12, 0, 21, 24, -6, -24, -21, 0], [0, -12, 2, 40, -22, 40, 2, -12], [1, 32, 0, -32, -2, -12, -6, 20], [-4, 2, 0, -2, 8, -6, -36, -4], [-18, -4, -24, -2, 18, 6, 48, 6], [-6, 24, -2, 18, -6, -6, 0, 6], [12, -18, -10, 0, -6, 0, 10, 18], [2, 14, -6, 14, 0, -12, 3, -2], [-20, -28, 34, -8, 10, 8, -34, 28], [24, -30, 8, -30, 0, 15, -4, 15], [-12, -2, 36, 10, 6, -10, -36, 2], [-32, 16, -14, -12, 16, 12, 14, -16], [18, 28, 14, -20, -18, -8, -28, -8], [-30, 7, -16, 7, 0, -4, 8, -3], [-8, 16, 0, -16, 16, 23, 10, 39], [0, 36, 2, -28, -18, -28, 2, 36], [0, -10, 36, -24, 0, -14, 0, 14], [4, 8, 0, -8, -8, 4, -32, 12], [16, 41, 2, -17, -16, -24, -4, -24], [-22, 10, -4, 10, 0, -8, 2, -2], [30, 32, -2, -10, -30, -22, 4, -22], [-34, -14, -1, 16, 34, -2, 2, -2], [0, 18, 10, -38, -14, -38, 10, 18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3150_bp_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3150_2_bp_a();".
// 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_3150_2_bp_a(:prec:=8)
    chi := MakeCharacter_3150_bp();
    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_3150_2_bp_a();".
// 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_3150_2_bp_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3150_bp();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![9409, 23280, 25796, 16320, 6447, 1632, 260, 24, 1]>,<13,R![1, -16, 20, -8, 1]>,<17,R![1024, -3072, 1280, 5376, 3936, 1344, 248, 24, 1]>],Snew);
    return Vf;
end function;