// Make newform 3072.2.d.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3072_d();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3072_d_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_3072_2_d_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_3072_2_d_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 := [2, -12, 30, -44, 43, -28, 14, -4, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-5, 6, -7, 2, -1, 0, 0, 0], [4, -12, 19, -15, 10, -3, 1, 0], [-5, 16, -24, 17, -11, 3, -1, 0], [31, -126, 223, -256, 175, -98, 28, -8], [-42, 166, -285, 325, -220, 123, -35, 10], [-43, 168, -285, 325, -220, 123, -35, 10], [109, -440, 774, -890, 610, -342, 98, -28]]; Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 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_3072_d();" function MakeCharacter_3072_d() N := 3072; order := 2; char_gens := [2047, 2053, 1025]; v := [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_3072_d_Hecke();" function MakeCharacter_3072_d_Hecke(Kf) N := 3072; order := 2; char_gens := [2047, 2053, 1025]; char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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 := 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 := 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, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, 0], [1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -1, -1], [0, 0, 0, 0, -2, 0, 1, 1], [0, 1, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -1, 1], [0, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 2, 1, 0], [-3, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 2, -1, 1], [0, -1, 3, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 3, 1], [0, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, 2, 1, 0], [0, 0, 0, 0, 0, 4, 0, 0], [0, 0, 0, 0, -4, -2, -1, 1], [0, 0, 0, 0, -4, -2, -2, -2], [-2, 2, 0, 0, 0, 0, 0, 0], [2, 2, 4, 0, 0, 0, 0, 0], [-3, -1, -4, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, 3, -1], [2, -2, 6, 2, 0, 0, 0, 0], [0, 2, 6, -2, 0, 0, 0, 0], [0, 0, 0, 0, 5, 0, -1, -2], [1, -1, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 0, 4], [0, 0, 0, 0, -6, 0, -1, -1], [-2, 0, 4, -4, 0, 0, 0, 0], [-3, -1, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 2, -2, -2], [0, -1, 3, 3, 0, 0, 0, 0], [0, 0, 0, 0, 4, -6, -2, -2], [0, 0, 0, 0, 7, -4, -1, 0], [1, -1, 8, 4, 0, 0, 0, 0], [0, 0, 0, 0, -8, 4, -1, -3], [0, 0, 0, 0, 0, -1, 1, 3], [-10, -2, 2, -4, 0, 0, 0, 0], [0, 0, 0, 0, -9, -6, 1, 2], [0, 0, 0, 0, 0, 2, -6, 2], [0, 0, 0, 0, 6, 2, -1, 3], [10, -2, -6, 0, 0, 0, 0, 0], [-4, -6, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 9, 2, -3, 0], [7, 1, -4, -4, 0, 0, 0, 0], [0, 0, 0, 0, -4, -10, 2, 2], [3, 1, 8, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, 3, -1], [0, 0, 0, 0, -2, 6, 5, -3], [2, 6, 2, -2, 0, 0, 0, 0], [14, -2, 10, 4, 0, 0, 0, 0], [-4, 4, -2, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, 5, -3], [10, 2, 2, -2, 0, 0, 0, 0], [-16, -4, -4, 4, 0, 0, 0, 0], [0, 0, 0, 0, 5, 8, -1, 6], [-13, 1, -8, -4, 0, 0, 0, 0], [0, 0, 0, 0, 2, 2, 1, 5], [-10, 2, -2, 2, 0, 0, 0, 0], [0, 0, 0, 0, -12, -4, 0, 0], [0, 0, 0, 0, 1, -6, 5, 8], [0, 0, 0, 0, -4, -2, 6, -2], [-4, 0, -12, 4, 0, 0, 0, 0], [4, -6, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, -8, 1, 6], [0, 0, 0, 0, -12, -6, -2, 6], [10, -2, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 16, 9, -1, 3], [0, 0, 0, 0, 8, -2, 5, 7], [-2, -2, -6, 6, 0, 0, 0, 0], [-6, 6, 4, 8, 0, 0, 0, 0], [-7, 3, 12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 4, 5, -5], [0, 0, 0, 0, 0, 3, -7, -1], [10, 6, -10, 0, 0, 0, 0, 0], [0, 0, 0, 0, -11, 10, -1, 2], [0, 0, 0, 0, 8, -4, 3, 1], [0, -5, -5, -5, 0, 0, 0, 0], [-4, 6, -2, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 7, -7, -3], [0, 0, 0, 0, -2, 6, -7, 1], [4, 8, 10, -4, 0, 0, 0, 0], [4, 0, -8, 0, 0, 0, 0, 0], [11, 5, -12, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 13, -5, -1], [-16, -1, -1, -5, 0, 0, 0, 0], [16, 2, -2, -2, 0, 0, 0, 0], [0, 0, 0, 0, 5, 10, 3, -2], [9, 3, -12, 0, 0, 0, 0, 0], [0, 0, 0, 0, -16, 9, -9, -5], [6, -2, 4, 12, 0, 0, 0, 0], [19, -3, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 0, -4], [0, 0, 0, 0, 12, 0, 4, 4], [-18, 2, -12, -8, 0, 0, 0, 0], [0, 0, 0, 0, 19, -2, -5, -4], [0, -1, -17, 3, 0, 0, 0, 0], [0, 0, 0, 0, 8, 15, 1, -1], [0, 0, 0, 0, 2, -10, -9, -1], [0, 0, 0, 0, 0, -7, 3, -7], [0, 0, 0, 0, 9, -8, -3, 4], [0, 0, 0, 0, -16, 13, 3, 3], [16, -1, 3, 7, 0, 0, 0, 0], [0, 0, 0, 0, 4, -12, 0, 0], [-2, -2, -16, -4, 0, 0, 0, 0], [0, 0, 0, 0, -20, 10, -2, 2], [-2, 8, -12, -4, 0, 0, 0, 0], [-12, 4, 14, 0, 0, 0, 0, 0], [2, -2, -4, 0, 0, 0, 0, 0], [1, -5, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 1, 3], [6, -8, -4, 4, 0, 0, 0, 0], [0, 0, 0, 0, -12, 0, 0, -8], [15, 1, -16, 0, 0, 0, 0, 0], [8, 1, -11, -3, 0, 0, 0, 0], [0, 0, 0, 0, 8, -7, 7, 5], [12, 4, -2, -8, 0, 0, 0, 0], [0, 0, 0, 0, -3, 10, -1, -6], [0, 0, 0, 0, 20, -4, 4, 0], [0, 0, 0, 0, -16, 0, 3, 1], [-12, 4, -12, -4, 0, 0, 0, 0], [0, 0, 0, 0, -9, -20, 3, -4], [0, 0, 0, 0, 16, 7, 1, -3], [0, 0, 0, 0, -8, -5, -3, 7], [0, 0, 0, 0, -3, -4, 7, -2], [0, 0, 0, 0, -18, -6, -5, 7], [10, 10, -8, -4, 0, 0, 0, 0], [-1, -7, 8, -4, 0, 0, 0, 0], [0, 0, 0, 0, 10, -10, 9, -3], [0, 0, 0, 0, -4, 4, -8, 0], [-22, 6, 2, 0, 0, 0, 0, 0], [3, 1, 4, 12, 0, 0, 0, 0], [0, 0, 0, 0, 2, -10, 11, 3], [-8, 5, -11, 1, 0, 0, 0, 0], [12, -2, 8, 12, 0, 0, 0, 0], [0, 0, 0, 0, -7, 24, -1, 2], [0, 0, 0, 0, 8, -9, -3, -5], [0, 0, 0, 0, 3, 8, 7, 12], [8, 5, 1, -11, 0, 0, 0, 0], [0, 0, 0, 0, 8, -3, -1, 9], [0, 0, 0, 0, 7, 4, 5, -6], [-25, -7, -12, 8, 0, 0, 0, 0], [0, 0, 0, 0, 16, -6, 2, -2], [0, 0, 0, 0, 18, 16, -1, 7], [-14, 2, -4, 12, 0, 0, 0, 0], [0, 0, 0, 0, 16, 12, 5, 7], [8, -3, -7, -7, 0, 0, 0, 0], [0, 0, 0, 0, 32, -9, 1, 3], [6, 10, 10, -8, 0, 0, 0, 0], [0, 0, 0, 0, -8, 22, -5, -7], [2, -10, -2, 10, 0, 0, 0, 0], [0, 0, 0, 0, 8, 11, -7, -5], [8, 8, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 32, -13, -3, -1], [-2, -2, -6, 12, 0, 0, 0, 0], [19, -3, 0, 8, 0, 0, 0, 0], [8, 1, 1, -3, 0, 0, 0, 0], [-10, 0, 0, -16, 0, 0, 0, 0], [0, 0, 0, 0, -11, -14, 7, 2], [0, 0, 0, 0, -8, 2, 10, -6], [8, -1, -9, 3, 0, 0, 0, 0], [-23, -1, -8, 0, 0, 0, 0, 0], [0, 0, 0, 0, -8, -11, -5, -1], [-16, -3, -3, 9, 0, 0, 0, 0], [-12, 8, 0, 4, 0, 0, 0, 0], [-5, -7, -4, 12, 0, 0, 0, 0], [0, 0, 0, 0, -4, 6, 15, 1], [-12, 2, 10, -6, 0, 0, 0, 0], [0, 0, 0, 0, 5, -2, -9, 2], [0, 0, 0, 0, -20, -16, -8, 4], [0, 0, 0, 0, -12, 12, 17, 7], [10, 2, 6, -4, 0, 0, 0, 0], [-22, -4, -12, -4, 0, 0, 0, 0], [3, -7, -8, -4, 0, 0, 0, 0], [-26, -2, 18, -2, 0, 0, 0, 0], [0, 0, 0, 0, -28, -8, 4, 4], [0, 0, 0, 0, 11, -20, 7, 4], [15, 9, -8, 4, 0, 0, 0, 0], [0, 0, 0, 0, 16, -6, 13, -1], [5, -1, -12, 4, 0, 0, 0, 0], [0, 0, 0, 0, -24, -15, 7, -5], [0, 0, 0, 0, 18, -4, -7, -11], [-10, -2, -6, 14, 0, 0, 0, 0], [14, 2, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, -11, -10, 3, 14], [0, 0, 0, 0, -6, -2, 11, -5], [0, 0, 0, 0, 4, 20, 8, 0], [4, 0, 34, 6, 0, 0, 0, 0], [-24, 8, 8, 8, 0, 0, 0, 0], [-6, 0, -4, -20, 0, 0, 0, 0], [0, 0, 0, 0, 24, -16, -12, -4], [0, 0, 0, 0, 28, 16, -4, -4], [0, 0, 0, 0, 3, 10, -3, -6], [0, 0, 0, 0, 4, -2, 2, 2], [-18, -4, -16, 16, 0, 0, 0, 0], [-2, 10, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -42, 6, -1, -5], [-14, -2, 10, 6, 0, 0, 0, 0], [8, 0, 4, -16, 0, 0, 0, 0], [0, 0, 0, 0, 13, 16, 1, 12], [-1, -3, 4, -8, 0, 0, 0, 0], [0, 0, 0, 0, 2, 4, -3, -7], [10, -6, 4, 8, 0, 0, 0, 0], [0, 0, 0, 0, -8, -3, 19, 3], [0, 0, 0, 0, 5, -18, -11, 4], [-21, 1, -4, -4, 0, 0, 0, 0], [0, 0, 0, 0, 32, -7, -9, -1], [18, 8, 20, -4, 0, 0, 0, 0], [0, 0, 0, 0, 8, -1, -11, -9], [-12, -6, 12, -8, 0, 0, 0, 0], [0, 0, 0, 0, 23, -4, 3, -4], [-25, 1, -8, -12, 0, 0, 0, 0], [0, 0, 0, 0, -32, 0, -4, -4], [30, -6, 8, 8, 0, 0, 0, 0], [16, 4, 8, -8, 0, 0, 0, 0], [25, -5, -8, 0, 0, 0, 0, 0], [-8, -9, 3, -9, 0, 0, 0, 0], [38, 2, -14, 0, 0, 0, 0, 0], [0, 0, 0, 0, -35, 4, -7, 4], [0, 0, 0, 0, 42, 0, -7, 5], [3, -3, 12, 8, 0, 0, 0, 0], [-8, -1, 35, 3, 0, 0, 0, 0], [-11, 7, 8, 20, 0, 0, 0, 0], [0, 0, 0, 0, 32, 1, -9, 3], [0, 0, 0, 0, 4, 4, -7, -17], [-6, 8, 0, 8, 0, 0, 0, 0], [-2, 2, -4, -8, 0, 0, 0, 0], [-15, -1, 4, 8, 0, 0, 0, 0], [0, 0, 0, 0, 20, -8, 8, 4], [0, 0, 0, 0, -10, 24, 3, -1], [0, 0, 0, 0, -8, 33, 11, 1], [17, 3, 20, -8, 0, 0, 0, 0], [-18, 18, 14, -6, 0, 0, 0, 0], [0, 0, 0, 0, -16, -1, -19, 7], [-50, -2, 0, -4, 0, 0, 0, 0], [-8, -10, 18, -6, 0, 0, 0, 0], [0, 0, 0, 0, 51, -8, -5, 0], [0, 0, 0, 0, 16, -17, 17, -3], [-8, 8, -10, -8, 0, 0, 0, 0], [0, 0, 0, 0, 32, -3, 3, -5], [0, 0, 0, 0, -20, 22, -10, -2], [-13, 5, 4, -4, 0, 0, 0, 0], [0, 0, 0, 0, -8, -4, 13, -9], [26, -4, 0, 16, 0, 0, 0, 0], [10, -10, -12, -8, 0, 0, 0, 0], [-21, 9, 8, 12, 0, 0, 0, 0], [0, 0, 0, 0, 0, -4, 0, 4], [0, 0, 0, 0, -4, -10, 6, 6], [-20, -4, -4, 16, 0, 0, 0, 0], [0, 0, 0, 0, -10, -8, -11, -3], [-2, -12, -8, 16, 0, 0, 0, 0], [24, 8, -10, 8, 0, 0, 0, 0], [-6, -8, 20, 4, 0, 0, 0, 0], [0, 0, 0, 0, -17, 18, 3, 0], [0, 0, 0, 0, -16, -22, 2, -10], [0, 0, 0, 0, 18, -4, 11, 11], [0, 0, 0, 0, 28, 0, -16, 0], [0, 0, 0, 0, 19, -10, 7, 16], [4, 0, 8, -16, 0, 0, 0, 0], [15, -11, 8, 8, 0, 0, 0, 0], [0, 0, 0, 0, 8, 0, -4, 4], [0, 0, 0, 0, -2, -4, 15, 11], [0, 0, 0, 0, -48, 0, -7, 3], [8, 1, -3, 1, 0, 0, 0, 0], [0, 0, 0, 0, -4, 6, 14, -10], [0, 0, 0, 0, 11, -24, -5, -8], [10, -4, -8, -16, 0, 0, 0, 0], [0, 0, 0, 0, 0, -15, -13, -3], [0, 0, 0, 0, 35, 20, 7, -4], [0, 0, 0, 0, -20, -18, -7, 15], [0, 0, 0, 0, -28, 12, -4, 4], [-26, 0, -12, 4, 0, 0, 0, 0], [15, 5, -8, -8, 0, 0, 0, 0], [6, 8, 8, 8, 0, 0, 0, 0], [11, 5, 12, 0, 0, 0, 0, 0], [0, 0, 0, 0, -24, -7, 7, -1], [0, 0, 0, 0, 40, 12, 3, 1], [-14, 2, 8, 20, 0, 0, 0, 0], [0, 0, 0, 0, 8, -22, -6, -10], [6, -14, -12, 16, 0, 0, 0, 0], [-21, -3, -12, 20, 0, 0, 0, 0], [16, 8, 22, -8, 0, 0, 0, 0], [0, 0, 0, 0, 16, -24, 5, -1], [0, 0, 0, 0, -40, -7, 3, -3], [-24, -16, 20, -8, 0, 0, 0, 0], [20, 4, -10, 18, 0, 0, 0, 0], [0, 0, 0, 0, 13, 8, -11, 0], [1, 7, -16, 8, 0, 0, 0, 0], [-10, 10, 22, -6, 0, 0, 0, 0], [0, 0, 0, 0, -11, -10, 15, 2], [0, 0, 0, 0, -8, 10, 2, 18], [-30, -8, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 48, -2, 6, -2], [0, 0, 0, 0, -24, 18, 11, 1], [0, 0, 0, 0, 37, 6, 11, -10], [9, -5, -16, 4, 0, 0, 0, 0], [0, 0, 0, 0, 21, 8, 11, 10], [0, 0, 0, 0, 8, -21, -3, 5], [0, 0, 0, 0, -8, -13, 9, 23], [36, -10, -28, -8, 0, 0, 0, 0], [0, 0, 0, 0, -25, 2, 5, -2], [-37, -15, 16, -4, 0, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3072_d_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3072_2_d_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_3072_2_d_i(:prec:=8) chi := MakeCharacter_3072_d(); 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_3072_2_d_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_3072_2_d_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3072_d(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![64, 0, 320, 0, 160, 0, 24, 0, 1]>,<7,R![4, 24, -8, -4, 1]>],Snew); return Vf; end function;