// Make newform 1728.4.d.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1728_d();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_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_1728_4_d_g();". // 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_1728_4_d_g();". // 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 := [256, 0, -144, 0, 65, 0, -9, 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, 256, 0, -377, 0, 65, 0, -9], [2328, 0, -1755, 0, 195, 0, -27, 0], [0, 1632, 0, -435, 0, 75, 0, -3], [-1782, 0, 0, 0, 0, 0, -12, 0], [216, 0, -147, 0, 27, 0, -3, 0], [0, -15456, 0, 3939, 0, -507, 0, 51], [0, -768, 0, 1515, 0, -195, 0, 27]]; Rf_basisdens := [1, 832, 260, 160, 65, 8, 832, 160]; 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_1728_d();" function MakeCharacter_1728_d() N := 1728; order := 2; char_gens := [703, 325, 1217]; 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_1728_d_Hecke();" function MakeCharacter_1728_d_Hecke(Kf) N := 1728; order := 2; char_gens := [703, 325, 1217]; 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 := 4; raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 5, 0, 0, 0], [0, -7, 0, 0, 0, 0, 0, 0], [0, 0, 0, -11, 0, 0, 0, 0], [0, 0, -26, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 10, 0], [0, 0, 0, 0, 0, 11, 0, 0], [0, 0, 0, 0, -16, 0, 0, 0], [0, -92, 0, 0, 0, 0, 0, 0], [0, 0, 0, -11, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7], [0, 0, 0, 0, 0, 21, 0, 0], [0, -353, 0, 0, 0, 0, 0, 0], [0, 0, 0, -76, 0, 0, 0, 0], [425, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -61, 0], [0, 0, 0, 0, 0, 0, 0, -24], [0, 0, 0, 0, 3, 0, 0, 0], [799, 0, 0, 0, 0, 0, 0, 0], [0, 0, -22, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 51, 0], [0, 0, 0, 0, 0, 0, 0, -31], [0, 0, 0, 0, 0, -32, 0, 0], [0, 0, 0, 0, 21, 0, 0, 0], [0, 0, 0, 0, 0, 0, -94, 0], [0, 0, 0, 0, 0, 0, 0, 102], [0, 0, 0, 0, -1, 0, 0, 0], [0, -2185, 0, 0, 0, 0, 0, 0], [0, 0, 312, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 9, 0], [0, 0, 0, 0, 0, -64, 0, 0], [0, -1321, 0, 0, 0, 0, 0, 0], [0, 0, 0, -55, 0, 0, 0, 0], [0, 0, -332, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -60], [0, 0, 0, 0, 0, 91, 0, 0], [0, 0, 0, 197, 0, 0, 0, 0], [2459, 0, 0, 0, 0, 0, 0, 0], [0, 0, 381, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 139, 0], [0, -4763, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 86, 0], [0, 0, 0, 0, 0, 0, 0, -104], [0, 0, 0, 0, 0, 188, 0, 0], [0, 0, 0, 0, 142, 0, 0, 0], [0, 0, 0, 218, 0, 0, 0, 0], [4709, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 284], [0, 0, 0, 0, -170, 0, 0, 0], [0, 0, 0, -510, 0, 0, 0, 0], [0, 0, -429, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 21, 0], [0, 0, 0, 0, 0, -328, 0, 0], [0, 0, 0, 0, -258, 0, 0, 0], [0, -8116, 0, 0, 0, 0, 0, 0], [0, 0, -367, 0, 0, 0, 0, 0], [0, 628, 0, 0, 0, 0, 0, 0], [0, 0, 0, -511, 0, 0, 0, 0], [745, 0, 0, 0, 0, 0, 0, 0], [0, 0, 24, 0, 0, 0, 0, 0], [0, -4057, 0, 0, 0, 0, 0, 0], [2023, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 254], [0, 0, 0, 0, 0, -373, 0, 0], [0, 0, 0, 0, 490, 0, 0, 0], [0, 0, 0, 971, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 135, 0], [0, 0, 0, 0, 0, 211, 0, 0], [0, 4759, 0, 0, 0, 0, 0, 0], [0, 0, 0, -106, 0, 0, 0, 0], [0, 0, 119, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -40, 0, 0], [0, 0, 0, 0, 326, 0, 0, 0], [-10555, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -69], [0, 0, 0, 0, 0, -451, 0, 0], [0, 0, 0, 227, 0, 0, 0, 0], [16450, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -270, 0], [0, 0, 0, 0, 0, 0, 0, -470], [0, 0, 0, 0, -593, 0, 0, 0], [-9574, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1377, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 845, 0], [0, 0, 0, 0, 0, 0, 0, 277], [0, 0, 0, -1266, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 173, 0], [0, 0, 0, 0, 0, 0, 0, 769], [0, -3944, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2011, 0, 0, 0, 0], [0, 0, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 401, 0, 0, 0], [0, 20923, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -931, 0, 0], [0, 875, 0, 0, 0, 0, 0, 0], [0, 0, -45, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -692], [0, 0, 0, 0, 691, 0, 0, 0], [0, -12823, 0, 0, 0, 0, 0, 0], [17773, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -163], [0, 0, 0, 0, 732, 0, 0, 0], [0, 0, 0, -2210, 0, 0, 0, 0], [-21562, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 97, 0], [0, 0, 0, 0, 0, 1209, 0, 0], [0, 0, 0, 0, 199, 0, 0, 0], [0, 12917, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -823, 0], [0, 0, 0, 0, -20, 0, 0, 0], [0, -1564, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1972, 0, 0, 0, 0], [0, 0, 1884, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 606], [0, 0, 0, 0, 0, -1263, 0, 0], [-5231, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1889, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1126], [0, 4556, 0, 0, 0, 0, 0, 0], [0, 0, 1863, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 481, 0, 0], [0, 0, 0, -900, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -138, 0], [0, 0, 0, 0, 0, -1472, 0, 0], [0, 33892, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1645, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1457, 0], [0, 0, 0, 0, 0, -1119, 0, 0], [0, 0, 0, 0, -207, 0, 0, 0], [-15649, 0, 0, 0, 0, 0, 0, 0], [0, 0, -832, 0, 0, 0, 0, 0], [0, 10217, 0, 0, 0, 0, 0, 0], [0, 0, -3086, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1084, 0, 0, 0], [0, 45556, 0, 0, 0, 0, 0, 0], [0, 0, -2381, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1453, 0], [0, 0, 0, 0, 0, 0, 0, -945], [0, 0, 0, 0, 0, 239, 0, 0], [0, 0, 0, -2714, 0, 0, 0, 0], [0, 0, 0, 0, 0, 615, 0, 0], [0, 0, 0, 0, 1008, 0, 0, 0], [0, -25031, 0, 0, 0, 0, 0, 0], [0, 0, 0, -4471, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1415, 0, 0], [0, 0, 0, 0, 1137, 0, 0, 0], [0, 34931, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2796, 0, 0, 0, 0], [0, 30283, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3172, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1254, 0], [0, 0, 0, 0, 2106, 0, 0, 0], [-13367, 0, 0, 0, 0, 0, 0, 0], [0, 0, -419, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1553], [0, 0, 0, 0, -1439, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1629, 0], [0, 0, 0, 0, 0, 0, 0, 9], [0, 0, 0, 0, -592, 0, 0, 0], [0, 0, 0, 3657, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2145, 0], [0, 0, 0, 0, 0, -1512, 0, 0], [35179, 0, 0, 0, 0, 0, 0, 0], [0, 0, 549, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1420], [0, 0, 0, 0, 0, 1008, 0, 0], [0, 0, 0, -3233, 0, 0, 0, 0], [-23717, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2825, 0], [0, 0, 0, 0, -1608, 0, 0, 0], [0, -14380, 0, 0, 0, 0, 0, 0], [0, 0, -2335, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1631, 0], [0, 0, 0, 0, 0, -1804, 0, 0], [0, 0, 0, 0, 0, 0, -797, 0], [0, 0, 0, 0, 0, 0, 0, -952], [0, 0, 0, 0, 0, 1704, 0, 0], [0, 0, 0, 0, 2357, 0, 0, 0], [0, 0, 0, 1062, 0, 0, 0, 0], [0, 0, 2166, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1053, 0, 0], [0, -31043, 0, 0, 0, 0, 0, 0], [-35561, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2087, 0, 0, 0, 0], [45677, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2090], [0, -13223, 0, 0, 0, 0, 0, 0], [0, 0, 3979, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 575], [0, 0, 0, 0, -89, 0, 0, 0], [47005, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1609, 0, 0], [0, 0, 0, 0, -961, 0, 0, 0], [0, 0, 0, -226, 0, 0, 0, 0], [0, 0, -2279, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -635, 0], [0, 0, 0, 0, 0, -1345, 0, 0], [39463, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1138], [0, 0, -7822, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 241, 0], [0, 0, 0, 0, 0, 0, 0, 2683], [0, 0, 0, 0, 371, 0, 0, 0], [0, -58025, 0, 0, 0, 0, 0, 0], [54911, 0, 0, 0, 0, 0, 0, 0], [0, 0, -6439, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1389, 0], [0, 0, 0, 0, 0, 0, 0, 2871], [0, 0, 0, 5190, 0, 0, 0, 0], [4331, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2138, 0], [0, 0, 0, 0, 1331, 0, 0, 0], [0, 0, 0, -2904, 0, 0, 0, 0], [0, 0, -9752, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3997, 0, 0], [0, 0, 0, 0, 0, 0, 786, 0], [0, 0, 0, 0, 2470, 0, 0, 0], [0, 0, 0, 0, 0, 0, 814, 0], [0, 0, 0, 0, 0, 0, 0, -31], [0, 0, 0, 0, 0, 2933, 0, 0], [0, 0, 0, 0, -3507, 0, 0, 0], [0, 0, 0, 7929, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2881, 0], [0, 0, 0, 0, 0, 0, 0, -1701], [0, 0, 0, 0, 0, -199, 0, 0], [0, -17440, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3650, 0], [0, 0, 0, 0, -481, 0, 0, 0], [0, 20365, 0, 0, 0, 0, 0, 0], [0, 0, 0, -4555, 0, 0, 0, 0], [53827, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1205, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2506], [0, 0, 0, -3732, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1390], [0, -101815, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1809, 0], [0, 0, 0, 0, 0, -2925, 0, 0], [0, 0, 0, 0, 3293, 0, 0, 0], [0, 0, 0, 13, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 47, 0], [0, 0, 0, 0, 0, 0, 0, -2098], [0, -1312, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5166, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1008, 0, 0], [0, 0, 0, 0, 1421, 0, 0, 0], [0, 0, 0, -973, 0, 0, 0, 0], [28951, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2235, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1195], [0, 0, 0, 0, 0, -645, 0, 0], [0, 24212, 0, 0, 0, 0, 0, 0], [0, 0, -8536, 0, 0, 0, 0, 0], [-47986, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1951, 0], [0, 0, 0, 0, 0, 0, 0, 407], [0, 0, 0, 0, 0, -159, 0, 0], [0, 0, 0, 0, 0, 1957, 0, 0], [0, 0, 0, 0, -5040, 0, 0, 0], [0, -89084, 0, 0, 0, 0, 0, 0], [0, 0, 6107, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2020, 0, 0, 0], [0, -108488, 0, 0, 0, 0, 0, 0], [0, 0, -481, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -6251, 0, 0], [0, -38099, 0, 0, 0, 0, 0, 0], [-12998, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3225, 0], [11234, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5541, 0], [0, 0, 0, 0, 0, 0, 0, 4725], [0, 0, 0, 0, 0, 1688, 0, 0], [-17098, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4628], [0, 0, 0, 8235, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3466, 0], [0, 0, 0, -7902, 0, 0, 0, 0], [0, 0, 0, 0, 0, -669, 0, 0], [0, 74545, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2241, 0, 0, 0, 0], [-109267, 0, 0, 0, 0, 0, 0, 0], [0, 0, -5494, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1701, 0], [0, 0, 0, 0, 3347, 0, 0, 0], [0, 0, -9193, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5452], [0, 0, 0, 0, 4526, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1856], [0, 0, 0, 0, 0, -6564, 0, 0], [0, 0, -9816, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2165, 0], [0, 0, -9537, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1278], [0, -13291, 0, 0, 0, 0, 0, 0], [-3323, 0, 0, 0, 0, 0, 0, 0], [0, 0, 6290, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2642, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1728_d_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1728_4_d_g();". // 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_1728_4_d_g(:prec:=8) chi := MakeCharacter_1728_d(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); 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_1728_4_d_g();". // 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_1728_4_d_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1728_d(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<5,R![108, 0, 1]>,<7,R![-459, 0, 1]>,<17,R![-15300, 0, 1]>,<23,R![-13068, 0, 1]>],Snew); return Vf; end function;