// Make newform 2004.1.g.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2004_g();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2004_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_2004_1_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 ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then Kf := CyclotomicField(22); end if; return [ #coeff eq Kf!0 select 0 else &+[ elt[1]*Kf.1^elt[2] : elt in coeff] : coeff in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_2004_g();" function MakeCharacter_2004_g() N := 2004; order := 2; char_gens := [1003, 1337, 673]; 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_2004_g_Hecke();" function MakeCharacter_2004_g_Hecke(Kf) N := 2004; order := 2; char_gens := [1003, 1337, 673]; char_values := [[[-1, 0]], [[-1, 0]], [[-1, 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 := 1; raw_aps := [[[-1, 6]], [[1, 9]], [], [[1, 3], [1, 8]], [[-1, 1], [1, 10]], [], [], [[-1, 4], [-1, 7]], [], [[-1, 3], [-1, 8]], [[-1, 5], [-1, 6]], [], [], [], [[1, 4], [-1, 7]], [], [], [[-1, 2], [1, 9]], [], [], [], [], [], [[1, 2], [1, 9]], [[-1, 1], [1, 10]], [], [], [[-1, 3], [1, 8]], [], [], [[-1, 1], [-1, 10]], [], [[1, 1], [1, 10]], [], [], [], [[-1, 5], [1, 6]], [], [[-1, 0]], [[1, 1], [1, 10]], [[-1, 4], [1, 7]], [[-1, 4], [1, 7]], [[-1, 5], [1, 6]], [], [], [[1, 1], [1, 10]], [[-1, 5], [-1, 6]], [[-1, 2], [-1, 9]], [], [[1, 3], [-1, 8]], [[-1, 5], [-1, 6]], [[1, 2], [-1, 9]], [], [[1, 3], [-1, 8]], [], [[-1, 2], [1, 9]], [], [], [], [[1, 2], [1, 9]], [[1, 1], [1, 10]], [[-1, 5], [-1, 6]], [], [[-1, 0], [-1, 0]], [], [[-1, 2], [-1, 9]], [], [[1, 4], [-1, 7]], [], [], [[1, 3], [1, 8]], [[1, 5], [-1, 6]], [[1, 4], [1, 7]], [], [], [[-1, 3], [1, 8]], [], [[-1, 2], [1, 9]], [], [[1, 1], [-1, 10]], [[-1, 5], [1, 6]], [[-1, 3], [1, 8]], [[-1, 2], [1, 9]], [[-1, 4], [1, 7]], [], [], [[-1, 4], [-1, 7]], [], [[1, 4], [1, 7]], [], [[1, 3], [-1, 8]], [], [], [[1, 0], [1, 0]], [], [[1, 2], [-1, 9]], [[-1, 1], [-1, 10]], [], [[1, 5], [1, 6]], [], [], [[-1, 1], [-1, 10]], [[-1, 4], [1, 7]], [], [], [[1, 5], [-1, 6]], [], [], [[1, 3], [-1, 8]], [[-1, 3], [1, 8]], [], [[1, 5], [-1, 6]], [[-1, 3], [-1, 8]], [], [[-1, 2], [-1, 9]], [], [], [], [[1, 5], [1, 6]], [], [], [], [], [], [], [[-1, 2], [-1, 9]], [], [], [], [[-1, 4], [1, 7]], [], [[-1, 0], [-1, 0]], [], [[-1, 1], [1, 10]], [[1, 2], [1, 9]], [], [], [], [], [[-1, 1], [-1, 10]], [], [], [], [], [], [[1, 2], [-1, 9]], [[1, 1], [-1, 10]], [[-1, 4], [-1, 7]], [[1, 3], [1, 8]], [[1, 1], [-1, 10]], [[1, 4], [-1, 7]], [], [[1, 3], [1, 8]], [], [[-1, 2], [-1, 9]], [[-1, 4], [1, 7]], [[-1, 4], [-1, 7]], [[-1, 5], [-1, 6]], [], [], [[1, 2], [-1, 9]], [], [[-1, 5], [-1, 6]], [], [], [], [], [[1, 3], [-1, 8]], [[1, 1], [-1, 10]], [[1, 3], [1, 8]], [], [[-1, 1], [1, 10]], [[1, 1], [-1, 10]], [[1, 5], [-1, 6]], [], [[1, 5], [1, 6]], [[1, 5], [1, 6]], [], [[-1, 1], [-1, 10]], [], [[-1, 4], [-1, 7]], [[1, 5], [-1, 6]], [], [], [], [], [[1, 5], [-1, 6]], [[-1, 3], [-1, 8]], [[1, 3], [-1, 8]], [], [], [], [[-1, 2], [-1, 9]], [[-1, 3], [-1, 8]], [[1, 4], [-1, 7]], [[1, 4], [1, 7]], [[-1, 4], [1, 7]], [[1, 5], [-1, 6]], [[1, 4], [1, 7]], [[-1, 5], [1, 6]], [], [[1, 2], [1, 9]], [], [], [], [[1, 1], [1, 10]], [], [[1, 4], [-1, 7]], [], [[-1, 3], [-1, 8]], [[1, 1], [-1, 10]], [[-1, 1], [-1, 10]], [], [], [[-1, 3], [1, 8]], [[1, 3], [-1, 8]], [], [[1, 1], [1, 10]], [[1, 1], [-1, 10]], [], [], [[-1, 5], [-1, 6]], [], [[-1, 4], [-1, 7]], [], [[1, 5], [-1, 6]], [[-1, 4], [-1, 7]], [], [], [[-1, 2], [1, 9]], [], [], [], [], [[1, 1], [1, 10]], [], [], [[-1, 4], [-1, 7]], [], [[-1, 1], [1, 10]], [], [[1, 4], [1, 7]], [], [], [[-1, 2], [-1, 9]], [[-1, 3], [1, 8]], [], [], [[1, 3], [1, 8]], [], [[-1, 2], [1, 9]], [[-1, 4], [-1, 7]], [], [], [], [[1, 5], [-1, 6]], [], [[1, 2], [1, 9]], [], [[1, 4], [-1, 7]], [], [], [], [], [[-1, 5], [-1, 6]], [], [], [], [], [[1, 2], [1, 9]], [], [[1, 5], [1, 6]], [], [], [[-1, 2], [1, 9]], [], [], [], [], [[-1, 5], [1, 6]], [], [], [], [[-1, 4], [1, 7]], [], [], [[-1, 2], [1, 9]], [], [[-1, 1], [-1, 10]], [], [[-1, 5], [-1, 6]], [], [[-1, 3], [-1, 8]], [[1, 0], [1, 0]], [[1, 3], [-1, 8]], [[-1, 4], [-1, 7]], [[-1, 1], [-1, 10]], [], [], [], [], [], [[-1, 2], [-1, 9]]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2004_g_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2004_1_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_2004_1_g_f(:prec:=10) chi := MakeCharacter_2004_g(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 1)); 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;