// Make newform 768.3.e.k in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_768_e();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_768_e_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_768_3_e_k();". // 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_768_3_e_k();". // 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]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 0, 2, 0], [0, 2, -1, -2], [0, 4, 0, 4]]; Rf_basisdens := [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_768_e();" function MakeCharacter_768_e() N := 768; order := 2; char_gens := [511, 517, 257]; v := [2, 2, 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_768_e_Hecke();" function MakeCharacter_768_e_Hecke(Kf) N := 768; order := 2; char_gens := [511, 517, 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 := 3; raw_aps := [[0, 0, 0, 0], [0, 1, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -7, 0, 0], [0, 0, 0, 0], [0, 0, 0, -6], [0, 3, 6, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 12], [0, 15, 30, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -41, 0, 0], [0, 0, 0, 0], [0, 21, 42, 0], [0, 0, 0, 0], [-142, 0, 0, 0], [0, 0, 0, 0], [0, 79, 0, 0], [0, 0, 0, -18], [-94, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -89, 0, 0], [0, 0, 0, 0], [0, 0, 0, 36], [0, 0, 0, 0], [0, -31, 0, 0], [0, 0, 0, -24], [0, 33, 66, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -9, -18, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 17, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-98, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -63, -126, 0], [0, 0, 0, 0], [0, 223, 0, 0], [0, 0, 0, 0], [0, 0, 0, 30], [0, 0, 0, 0], [-194, 0, 0, 0], [0, 233, 0, 0], [0, 0, 0, -60], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 90], [0, -99, -198, 0], [0, 0, 0, 0], [0, -51, -102, 0], [0, 0, 0, 0], [-526, 0, 0, 0], [0, 0, 0, 0], [0, 117, 234, 0], [478, 0, 0, 0], [0, 329, 0, 0], [0, 0, 0, 0], [0, 0, 0, 120], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -57, -114, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -42], [334, 0, 0, 0], [0, -257, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [578, 0, 0, 0], [0, 0, 0, 0], [0, -439, 0, 0], [0, 0, 0, 42], [238, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -17, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 391, 0, 0], [0, 105, 210, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 48], [0, 171, 342, 0], [0, 0, 0, 0], [0, -69, -138, 0], [0, 0, 0, 0], [0, -113, 0, 0], [0, 0, 0, 168], [0, 165, 330, 0], [-2, 0, 0, 0], [0, -569, 0, 0], [0, 0, 0, 144], [0, 0, 0, 0], [914, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 210], [0, -195, -390, 0], [0, 0, 0, 0], [0, 0, 0, -210], [0, 75, 150, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 497, 0, 0], [0, 0, 0, 0], [-1246, 0, 0, 0], [0, 0, 0, 0], [0, -199, 0, 0], [0, 207, 414, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -255, -510, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 108], [1054, 0, 0, 0], [0, 0, 0, 0], [0, -225, -450, 0], [0, 0, 0, 0], [0, 0, 0, -60], [0, -285, -570, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 631, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -216], [0, 87, 174, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -180], [0, -21, -42, 0], [0, 0, 0, 0], [0, -105, -210, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 270], [718, 0, 0, 0], [0, 0, 0, 0], [0, 497, 0, 0], [0, 0, 0, -336], [0, 0, 0, 0], [0, -487, 0, 0], [0, 0, 0, 66], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_768_e_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_768_3_e_k();". // 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_768_3_e_k(:prec:=4) chi := MakeCharacter_768_e(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); 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_768_3_e_k();". // 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_768_3_e_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_768_e(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<5,R![0, 1]>,<7,R![0, 1]>,<11,R![196, 0, 1]>,<19,R![-288, 0, 1]>,<37,R![0, 1]>],Snew); return Vf; end function;