// Make newform 1152.3.g.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1152_g();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1152_g_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_1152_3_g_f();". // 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_1152_3_g_f();". // 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; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, -6, 0, -2, -2, 4], [0, -2, -4, -2, 0, 2, 2, 0], [-4, 0, 0, 0, 8, 0, 0, 0], [0, 2, 0, -2, 0, -6, 2, -4], [0, 4, 0, 4, 0, 4, 0, -8], [0, 4, -8, 4, 0, -4, 4, 0], [0, 0, 0, 0, 0, 0, 8, 0]]; 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_1152_g();" function MakeCharacter_1152_g() N := 1152; order := 2; char_gens := [127, 901, 641]; v := [1, 2, 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_1152_g_Hecke();" function MakeCharacter_1152_g_Hecke(Kf) N := 1152; order := 2; char_gens := [127, 901, 641]; 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 := 3; raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [2, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 0, 0, -1, 0, 0, 1], [6, 0, 0, 0, 0, 1, -1, 0], [-2, 0, 0, 0, 0, -2, -1, 0], [0, -2, 0, 1, 0, 0, 0, 1], [0, -2, 0, 0, 0, 0, 0, 0], [10, 0, 5, 0, 0, -2, 0, 0], [0, 2, 0, -2, -3, 0, 0, 2], [-2, 0, 6, 0, 0, 3, 1, 0], [-10, 0, 0, 0, 0, -2, 5, 0], [0, -4, 0, -3, 2, 0, 0, -5], [0, -4, 0, -2, 2, 0, 0, 0], [-22, 0, -3, 0, 0, -2, -4, 0], [0, -2, 0, -3, -2, 0, 0, -8], [-34, 0, 2, 0, 0, -1, -7, 0], [0, -6, 0, 1, -6, 0, 0, 0], [0, -2, 0, 6, 0, 0, 0, -8], [-2, 0, -20, 0, 0, -2, 2, 0], [0, -10, 0, 2, 1, 0, 0, -4], [0, -5, 0, 0, 9, 0, 0, 5], [30, 0, 8, 0, 0, 4, 2, 0], [50, 0, -24, 0, 0, 0, 10, 0], [66, 0, 5, 0, 0, 2, -10, 0], [0, -12, 0, -16, 1, 0, 0, -10], [0, -6, 0, 5, 6, 0, 0, -10], [70, 0, 4, 0, 0, -3, -9, 0], [-42, 0, -8, 0, 0, 8, 6, 0], [0, -10, 0, -2, -9, 0, 0, 10], [0, -12, 0, -3, -12, 0, 0, 8], [22, 0, -8, 0, 0, 2, -1, 0], [0, 2, 0, 11, -14, 0, 0, 4], [-78, 0, 15, 0, 0, 8, 0, 0], [0, -4, 0, 16, 7, 0, 0, -18], [-42, 0, 6, 0, 0, 3, 5, 0], [0, 8, 0, -9, -18, 0, 0, 15], [0, -2, 0, 32, -10, 0, 0, 0], [74, 0, -9, 0, 0, 4, -10, 0], [0, 10, 0, -5, 10, 0, 0, 16], [-34, 0, -28, 0, 0, 13, 3, 0], [0, -12, 0, 18, -4, 0, 0, 12], [-18, 0, -44, 0, 0, 2, -2, 0], [74, 0, -7, 0, 0, -6, 20, 0], [0, 12, 0, -28, -11, 0, 0, 6], [0, -10, 0, -7, 14, 0, 0, 0], [0, -2, 0, -18, 15, 0, 0, 0], [0, 17, 0, 2, -5, 0, 0, 15], [94, 0, -16, 0, 0, 9, -9, 0], [38, 0, -16, 0, 0, 8, -18, 0], [0, -8, 0, -30, 28, 0, 0, 0], [70, 0, 12, 0, 0, 26, -12, 0], [0, 25, 0, 4, 7, 0, 0, -15], [62, 0, 8, 0, 0, -8, 16, 0], [0, -6, 0, 26, 18, 0, 0, -40], [98, 0, -27, 0, 0, -22, 2, 0], [0, 34, 0, 2, -13, 0, 0, 0], [-210, 0, -12, 0, 0, 5, -5, 0], [118, 0, -16, 0, 0, -12, -4, 0], [0, 16, 0, -3, 24, 0, 0, 0], [218, 0, 37, 0, 0, -2, -8, 0], [0, 2, 0, 21, 2, 0, 0, 0], [0, -42, 0, -22, -14, 0, 0, -24], [142, 0, 36, 0, 0, 2, -10, 0], [-222, 0, -23, 0, 0, -2, -2, 0], [0, 18, 0, 11, 30, 0, 0, -22], [-114, 0, -4, 0, 0, 6, -6, 0], [0, 7, 0, -30, 13, 0, 0, -15], [30, 0, -46, 0, 0, -1, 21, 0], [-74, 0, -16, 0, 0, 16, 6, 0], [0, 38, 0, 38, 16, 0, 0, 4], [0, 22, 0, 14, 25, 0, 0, 30], [-106, 0, -22, 0, 0, -41, 1, 0], [0, 26, 0, -9, 12, 0, 0, 21], [0, 20, 0, -14, -36, 0, 0, 0], [298, 0, -25, 0, 0, 4, 10, 0], [-154, 0, 26, 0, 0, -9, 13, 0], [46, 0, -80, 0, 0, 10, 5, 0], [-14, 0, 24, 0, 0, -16, -34, 0], [0, -19, 0, 62, 15, 0, 0, 19], [206, 0, 52, 0, 0, -3, 23, 0], [0, 28, 0, -2, 46, 0, 0, -12], [66, 0, 8, 0, 0, -44, 8, 0], [0, 8, 0, 16, 5, 0, 0, -22], [0, -61, 0, 14, 41, 0, 0, -15], [-66, 0, 88, 0, 0, 14, -53, 0], [-142, 0, 24, 0, 0, -32, -6, 0], [74, 0, 71, 0, 0, 4, 34, 0], [0, -54, 0, 10, 23, 0, 0, -30], [0, 29, 0, 18, 15, 0, 0, -45], [0, -4, 0, 28, -2, 0, 0, 0], [0, 0, 0, -20, 31, 0, 0, 0], [0, -12, 0, 43, -24, 0, 0, 34], [0, 40, 0, 35, 0, 0, 0, 0], [0, 66, 0, -18, -28, 0, 0, 20], [-102, 0, 105, 0, 0, -22, -40, 0], [158, 0, 72, 0, 0, -22, 13, 0], [0, 6, 0, 71, -16, 0, 0, 45], [-98, 0, -36, 0, 0, -19, -49, 0], [0, -4, 0, -69, 14, 0, 0, 25], [50, 0, -53, 0, 0, -60, 12, 0], [0, -63, 0, -10, -33, 0, 0, -15], [-58, 0, -112, 0, 0, 38, 37, 0], [0, -60, 0, 9, 72, 0, 0, 6], [-90, 0, 52, 0, 0, 50, 6, 0], [0, -8, 0, -49, -28, 0, 0, 30], [374, 0, -24, 0, 0, 24, -26, 0], [0, -82, 0, 20, 52, 0, 0, -44], [-26, 0, -108, 0, 0, 26, -22, 0], [0, 22, 0, 2, -41, 0, 0, -10], [54, 0, 114, 0, 0, -41, -43, 0], [-154, 0, -104, 0, 0, -4, 48, 0], [0, -8, 0, -47, 32, 0, 0, 20], [0, 4, 0, -44, -61, 0, 0, 64], [-106, 0, -56, 0, 0, -38, 11, 0], [0, -20, 0, 83, -50, 0, 0, -35], [0, -14, 0, -32, 76, 0, 0, 20], [90, 0, 123, 0, 0, 40, -54, 0], [0, 20, 0, 33, -16, 0, 0, -18], [-362, 0, 150, 0, 0, 67, 5, 0], [-270, 0, -8, 0, 0, -60, 44, 0], [450, 0, 107, 0, 0, -60, 16, 0], [0, 17, 0, 64, 59, 0, 0, -25], [0, -108, 0, 47, -6, 0, 0, -1], [-126, 0, -47, 0, 0, -34, 62, 0], [414, 0, -112, 0, 0, 25, -13, 0], [0, 8, 0, 4, 54, 0, 0, -64], [0, 12, 0, -16, -89, 0, 0, 50], [-386, 0, 120, 0, 0, 9, -21, 0], [0, -44, 0, -37, 8, 0, 0, -90], [0, -2, 0, -56, -118, 0, 0, 80], [0, -18, 0, -70, -7, 0, 0, -52], [-290, 0, 0, 0, 0, -15, -81, 0], [-258, 0, -56, 0, 0, -66, 51, 0], [-274, 0, 36, 0, 0, -22, -46, 0], [1090, 0, 9, 0, 0, 30, 10, 0], [0, -50, 0, -23, 28, 0, 0, -5], [-62, 0, -29, 0, 0, 28, -64, 0], [-42, 0, -80, 0, 0, 10, -105, 0], [0, -14, 0, -133, 40, 0, 0, 55], [-334, 0, 41, 0, 0, -2, -66, 0], [0, -108, 0, -64, -9, 0, 0, -20], [0, 22, 0, 5, 10, 0, 0, 90], [-138, 0, -12, 0, 0, -3, 47, 0], [0, 14, 0, -58, -68, 0, 0, 64], [-202, 0, 78, 0, 0, -77, -59, 0], [550, 0, -216, 0, 0, -50, -27, 0], [0, 24, 0, 5, 106, 0, 0, -3], [0, 100, 0, -82, -36, 0, 0, -20], [670, 0, 58, 0, 0, -65, 5, 0], [30, 0, -8, 0, 0, 32, -52, 0], [0, 28, 0, 95, -78, 0, 0, -5], [0, -106, 0, 106, -18, 0, 0, 40], [0, -12, 0, 125, 94, 0, 0, -95], [0, 0, 0, 2, -44, 0, 0, -92], [0, -76, 0, -76, 103, 0, 0, 12], [1230, 0, 8, 0, 0, 10, 17, 0], [242, 0, 96, 0, 0, -48, -8, 0], [42, 0, 127, 0, 0, -44, -2, 0], [0, -94, 0, 3, 114, 0, 0, -80], [-386, 0, 168, 0, 0, 74, -43, 0], [0, -52, 0, 52, 21, 0, 0, -60], [0, 81, 0, -80, 99, 0, 0, 43], [302, 0, 56, 0, 0, 42, 113, 0], [0, 6, 0, 66, -106, 0, 0, -40], [0, -30, 0, 46, 13, 0, 0, -136], [350, 0, -54, 0, 0, 55, -7, 0], [-558, 0, 88, 0, 0, -48, -18, 0], [-982, 0, 163, 0, 0, 8, -18, 0], [0, 40, 0, 19, 52, 0, 0, -74], [-394, 0, 46, 0, 0, -53, -71, 0], [0, 38, 0, -108, -22, 0, 0, 44], [530, 0, -48, 0, 0, 0, 80, 0], [0, 118, 0, 38, -19, 0, 0, 56], [-570, 0, -48, 0, 0, 60, 68, 0], [0, -88, 0, 37, -96, 0, 0, -80], [-126, 0, -137, 0, 0, -96, 72, 0], [0, -28, 0, 104, 25, 0, 0, 130], [-306, 0, 6, 0, 0, -29, 69, 0], [0, -86, 0, 22, 93, 0, 0, 20], [0, 47, 0, -16, -75, 0, 0, 105], [214, 0, -56, 0, 0, -31, 11, 0], [758, 0, -192, 0, 0, -12, -44, 0], [0, -56, 0, 118, -16, 0, 0, 120], [-630, 0, -149, 0, 0, 8, 74, 0], [622, 0, 0, 0, 0, 97, 95, 0], [0, -16, 0, -113, 96, 0, 0, 20], [-666, 0, -20, 0, 0, 42, 20, 0], [0, -72, 0, -28, 128, 0, 0, -20], [-254, 0, -192, 0, 0, 16, 36, 0], [0, -42, 0, -175, -2, 0, 0, -60], [0, 24, 0, -53, -48, 0, 0, -24], [-246, 0, 123, 0, 0, -88, 22, 0], [0, 120, 0, 153, 24, 0, 0, 80], [-954, 0, -56, 0, 0, -84, 88, 0], [-442, 0, 292, 0, 0, 34, -2, 0], [334, 0, -32, 0, 0, -71, -29, 0], [-938, 0, -64, 0, 0, -68, -64, 0], [0, 10, 0, -6, -10, 0, 0, -20], [306, 0, 23, 0, 0, -96, -28, 0], [0, 58, 0, -78, 63, 0, 0, 34], [-106, 0, -188, 0, 0, 109, 19, 0], [702, 0, 156, 0, 0, 94, -86, 0], [0, 143, 0, 110, -23, 0, 0, 27], [-182, 0, -111, 0, 0, -22, 72, 0], [0, 230, 0, -46, -53, 0, 0, 40], [0, -35, 0, -162, 51, 0, 0, 85], [-90, 0, 280, 0, 0, -72, 10, 0], [0, -76, 0, 205, 6, 0, 0, -23], [398, 0, 84, 0, 0, -22, -62, 0], [226, 0, -189, 0, 0, 12, -96, 0], [0, -8, 0, -60, 21, 0, 0, -250], [0, 94, 0, 5, 22, 0, 0, 60], [0, -42, 0, -78, 68, 0, 0, -64], [146, 0, -216, 0, 0, 84, -64, 0], [0, 134, 0, -46, -83, 0, 0, 10], [-138, 0, -96, 0, 0, -78, -109, 0], [0, 42, 0, -32, -150, 0, 0, -20], [-414, 0, 127, 0, 0, 56, 12, 0], [470, 0, 188, 0, 0, -11, 27, 0], [0, 56, 0, 68, 153, 0, 0, 20], [-586, 0, -24, 0, 0, 134, -11, 0], [0, -174, 0, -62, 139, 0, 0, -80], [0, -53, 0, 56, -71, 0, 0, 45], [-186, 0, -58, 0, 0, 19, 73, 0], [110, 0, -176, 0, 0, -20, -10, 0], [0, 124, 0, 40, 166, 0, 0, 44], [0, -76, 0, 0, 33, 0, 0, -240], [0, -64, 0, -9, 132, 0, 0, -166], [-266, 0, 244, 0, 0, -91, 19, 0], [0, -8, 0, -13, 74, 0, 0, -111], [0, -2, 0, 22, -37, 0, 0, 74], [998, 0, -32, 0, 0, -60, 72, 0], [0, 132, 0, -59, 38, 0, 0, -175], [0, 72, 0, -8, 170, 0, 0, 100], [-558, 0, 104, 0, 0, 24, -34, 0], [-686, 0, 31, 0, 0, 64, 12, 0], [0, -7, 0, 44, 59, 0, 0, 59], [0, 114, 0, 148, -100, 0, 0, 68], [0, -71, 0, -122, -73, 0, 0, 105], [0, 10, 0, -241, -18, 0, 0, 2], [0, -8, 0, -24, -29, 0, 0, 50], [-458, 0, -114, 0, 0, -109, 109, 0], [118, 0, -88, 0, 0, 108, -68, 0], [0, -54, 0, 114, -24, 0, 0, 76], [0, 98, 0, -94, 97, 0, 0, -160], [0, 46, 0, 91, 122, 0, 0, 22], [0, 114, 0, -141, -42, 0, 0, -38], [0, -60, 0, 28, -208, 0, 0, 100], [-634, 0, -60, 0, 0, 21, 55, 0], [-2026, 0, 24, 0, 0, -56, 6, 0], [0, 2, 0, -8, 20, 0, 0, -260], [594, 0, 256, 0, 0, -32, -116, 0], [-726, 0, 117, 0, 0, -66, 12, 0], [0, -106, 0, 135, 66, 0, 0, 30], [-634, 0, -236, 0, 0, -19, 71, 0], [0, -12, 0, 49, 0, 0, 0, -170], [122, 0, -7, 0, 0, 122, 16, 0], [-1038, 0, 56, 0, 0, 12, 136, 0], [0, -18, 0, 18, 55, 0, 0, 170], [0, -110, 0, -121, 18, 0, 0, 40], [-1306, 0, 28, 0, 0, -3, 107, 0], [1126, 0, 78, 0, 0, -109, -59, 0], [1766, 0, 208, 0, 0, -114, 25, 0], [0, 18, 0, -95, 62, 0, 0, 150], [-702, 0, 139, 0, 0, 124, -4, 0], [-650, 0, -184, 0, 0, 48, 154, 0], [0, -194, 0, 99, 12, 0, 0, -85], [354, 0, 103, 0, 0, 64, 120, 0], [1886, 0, -6, 0, 0, 111, -39, 0], [0, 12, 0, -17, -16, 0, 0, -110], [-174, 0, -192, 0, 0, -24, 136, 0], [0, -134, 0, -210, 9, 0, 0, -66], [306, 0, -192, 0, 0, -64, -112, 0], [0, -136, 0, 172, -95, 0, 0, 100], [0, -89, 0, 218, 89, 0, 0, 15], [-490, 0, 242, 0, 0, 47, 133, 0], [822, 0, -332, 0, 0, -30, 122, 0], [0, 62, 0, 15, -182, 0, 0, -18], [0, -68, 0, 8, -26, 0, 0, 140], [0, 76, 0, -204, -23, 0, 0, 6], [0, -2, 0, -200, 20, 0, 0, -40], [214, 0, -22, 0, 0, 15, 117, 0], [0, -154, 0, -89, -180, 0, 0, 95], [0, 76, 0, -28, -40, 0, 0, 96], [518, 0, -388, 0, 0, 18, -200, 0], [1346, 0, -157, 0, 0, -44, 48, 0], [0, 12, 0, 4, -45, 0, 0, -284], [438, 0, 400, 0, 0, 16, -90, 0], [-182, 0, -5, 0, 0, 40, -150, 0], [0, 38, 0, -257, -82, 0, 0, 60], [-1122, 0, -368, 0, 0, -12, 46, 0], [0, 98, 0, -83, 58, 0, 0, 244], [190, 0, 102, 0, 0, -5, -107, 0], [1466, 0, -321, 0, 0, -20, 106, 0], [0, -90, 0, 14, -133, 0, 0, 160], [-190, 0, -67, 0, 0, 50, 66, 0], [0, 63, 0, -166, -71, 0, 0, 123], [0, -130, 0, -155, 52, 0, 0, -65], [238, 0, -172, 0, 0, 58, 234, 0], [-454, 0, -269, 0, 0, 16, -82, 0], [0, 118, 0, 254, 109, 0, 0, 4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1152_g_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1152_3_g_f();". // 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_1152_3_g_f(:prec:=8) chi := MakeCharacter_1152_g(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); 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_1152_3_g_f();". // 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_1152_3_g_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1152_g(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<5,R![-128, 128, -16, -8, 1]>,<13,R![-35696, 6432, -136, -24, 1]>],Snew); return Vf; end function;