// Make newform 384.4.f.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_384_f();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_384_f_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_384_4_f_e();". // 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_384_4_f_e();". // 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 := [169, 0, 0, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 0, 2, 0], [0, 13, -1, -1], [0, 26, 0, 2]]; Rf_basisdens := [1, 13, 13, 13]; 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_384_f();" function MakeCharacter_384_f() N := 384; order := 2; char_gens := [127, 133, 257]; v := [1, 1, 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_384_f_Hecke();" function MakeCharacter_384_f_Hecke(Kf) N := 384; order := 2; char_gens := [127, 133, 257]; char_values := [[-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 := 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, -1, -1, 0], [0, 1, 2, 0], [0, 0, 0, 1], [0, -23, 0, 0], [0, 22, 0, 0], [0, 0, 0, 2], [0, 1, 2, 0], [-88, 0, 0, 0], [0, 25, 50, 0], [0, 0, 0, 21], [0, -166, 0, 0], [0, 0, 0, 48], [0, 23, 46, 0], [384, 0, 0, 0], [0, 45, 90, 0], [0, 315, 0, 0], [0, 118, 0, 0], [0, -5, -10, 0], [680, 0, 0, 0], [422, 0, 0, 0], [0, 0, 0, 73], [0, 93, 0, 0], [0, 0, 0, 94], [-1062, 0, 0, 0], [0, 1, 2, 0], [0, 0, 0, -195], [0, -117, 0, 0], [0, -362, 0, 0], [0, 0, 0, -100], [0, 0, 0, 221], [0, 871, 0, 0], [0, 0, 0, -236], [0, -75, -150, 0], [0, -47, -94, 0], [0, 0, 0, -147], [0, 974, 0, 0], [0, 333, 666, 0], [2008, 0, 0, 0], [0, -119, -238, 0], [0, 1191, 0, 0], [0, -1326, 0, 0], [4784, 0, 0, 0], [-3074, 0, 0, 0], [0, -475, -950, 0], [0, 0, 0, 189], [0, -337, -674, 0], [0, 0, 0, 121], [0, -635, 0, 0], [0, -2278, 0, 0], [0, 0, 0, 86], [-1920, 0, 0, 0], [-1618, 0, 0, 0], [0, 793, 0, 0], [0, 0, 0, 388], [-4344, 0, 0, 0], [0, 269, 538, 0], [0, 0, 0, -319], [0, -2478, 0, 0], [0, 0, 0, 194], [0, -783, -1566, 0], [0, 181, 362, 0], [0, -813, -1626, 0], [10104, 0, 0, 0], [1198, 0, 0, 0], [0, 493, 986, 0], [0, 601, 1202, 0], [390, 0, 0, 0], [0, -2183, 0, 0], [0, -2746, 0, 0], [0, 0, 0, -8], [2216, 0, 0, 0], [0, 0, 0, -699], [0, -1014, 0, 0], [0, -129, -258, 0], [2128, 0, 0, 0], [0, 189, 378, 0], [0, 3110, 0, 0], [0, 0, 0, 190], [-1270, 0, 0, 0], [0, -4563, 0, 0], [0, -1718, 0, 0], [-1920, 0, 0, 0], [-15766, 0, 0, 0], [0, 0, 0, 717], [0, -4095, 0, 0], [0, 0, 0, -1350], [10682, 0, 0, 0], [0, -303, -606, 0], [0, 0, 0, 261], [0, -6123, 0, 0], [-15008, 0, 0, 0], [0, 0, 0, 561], [0, -2509, 0, 0], [0, -1577, -3154, 0], [-10632, 0, 0, 0], [0, -351, -702, 0], [0, 0, 0, 156], [0, 403, 806, 0], [0, 4742, 0, 0], [0, 537, 1074, 0], [0, -107, -214, 0], [0, 8949, 0, 0], [0, 0, 0, -2068], [0, -327, -654, 0], [-5066, 0, 0, 0], [0, -10021, 0, 0], [0, 0, 0, -2584], [-4344, 0, 0, 0], [-18250, 0, 0, 0], [0, 0, 0, 517], [0, 5874, 0, 0], [0, 0, 0, -382], [0, 705, 1410, 0], [0, 0, 0, -2631], [0, 0, 0, 2318], [0, 1593, 3186, 0], [7384, 0, 0, 0], [0, -1479, -2958, 0], [0, -10217, 0, 0], [0, 6074, 0, 0], [12730, 0, 0, 0], [0, 1929, 3858, 0], [0, -3279, 0, 0], [0, -2987, -5974, 0], [0, 3085, 6170, 0], [0, 12698, 0, 0], [7888, 0, 0, 0], [0, 0, 0, -635], [0, -7386, 0, 0], [0, -2193, -4386, 0], [-19080, 0, 0, 0], [0, 0, 0, 1993], [0, -2094, 0, 0], [0, 0, 0, 3016], [-10370, 0, 0, 0], [0, 1057, 2114, 0], [0, 2293, 4586, 0], [0, 317, 634, 0], [0, 0, 0, 3320], [0, -1029, -2058, 0], [0, 1481, 2962, 0], [0, 0, 0, 1761], [0, 3299, 0, 0], [0, -20514, 0, 0], [-17816, 0, 0, 0], [0, 14154, 0, 0], [0, 0, 0, 3308], [0, -33, -66, 0], [48208, 0, 0, 0], [0, 21422, 0, 0], [0, 0, 0, -684], [0, -471, -942, 0], [37336, 0, 0, 0], [0, -1617, -3234, 0], [-35360, 0, 0, 0], [0, 0, 0, -431], [0, 0, 0, -178], [34034, 0, 0, 0], [0, 3625, 7250, 0], [0, 12431, 0, 0], [0, 0, 0, -2236], [0, 0, 0, 3929], [0, 21761, 0, 0], [0, 0, 0, -1734], [42360, 0, 0, 0], [0, 0, 0, -2471], [0, -2678, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_384_f_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_384_4_f_e();". // 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_384_4_f_e(:prec:=4) chi := MakeCharacter_384_f(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 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_384_4_f_e();". // 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_384_4_f_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_384_f(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<5,R![-104, 0, 1]>,<23,R![88, 1]>],Snew); return Vf; end function;