// Make newform 800.2.f.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_800_f();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_800_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_800_2_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_800_2_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 := [1, 0, -1, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 2, 0, -1], [0, 0, 0, 2], [-1, 0, 2, -1]]; 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_800_f();" function MakeCharacter_800_f() N := 800; order := 2; char_gens := [351, 101, 577]; v := [2, 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_800_f_Hecke();" function MakeCharacter_800_f_Hecke(Kf) N := 800; order := 2; char_gens := [351, 101, 577]; 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 := 2; raw_aps := [[0, 0, 0, 0], [1, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, -1], [0, 0, -1, 0], [0, 2, 0, 0], [0, 0, 1, 2], [0, 0, -1, 2], [0, 0, -2, -3], [0, 0, 2, 4], [2, 2, 0, 0], [-2, 0, 0, 0], [-2, 2, 0, 0], [-7, 1, 0, 0], [0, 0, -2, 1], [-8, -2, 0, 0], [0, 0, 3, 2], [0, 0, 3, 4], [-9, -1, 0, 0], [-2, -2, 0, 0], [0, 0, -3, -2], [-8, 4, 0, 0], [3, -1, 0, 0], [-2, -4, 0, 0], [0, 0, -5, -6], [0, 0, 0, -4], [0, 0, 6, 5], [-1, -1, 0, 0], [0, 0, 7, 4], [0, 0, 5, 4], [0, 0, -8, -1], [0, 0, -7, -8], [0, 0, 1, 4], [0, 0, 1, -2], [0, 0, -1, -8], [2, 6, 0, 0], [-10, 4, 0, 0], [5, -3, 0, 0], [0, 0, 6, -3], [-2, 0, 0, 0], [0, 0, 5, -2], [0, 0, -8, 0], [2, -10, 0, 0], [0, 0, -1, 2], [-16, -2, 0, 0], [12, -8, 0, 0], [0, 0, 9, 12], [0, 0, -4, 3], [9, -11, 0, 0], [0, 0, -2, 0], [0, 0, -1, 10], [-20, 0, 0, 0], [6, 6, 0, 0], [0, 0, 11, 4], [0, 0, -1, 0], [0, 0, -4, -5], [0, 0, -3, -4], [2, 10, 0, 0], [2, 0, 0, 0], [14, -2, 0, 0], [-1, -5, 0, 0], [2, 8, 0, 0], [-1, 15, 0, 0], [-14, -10, 0, 0], [0, 0, 5, -8], [12, -2, 0, 0], [0, 0, -7, 0], [0, 0, -7, -8], [-7, 5, 0, 0], [0, 0, -14, 0], [0, 0, -5, -4], [12, 4, 0, 0], [0, 0, -4, 7], [-2, 16, 0, 0], [0, 0, 13, 14], [0, 0, 8, 7], [0, 0, -1, 12], [8, 14, 0, 0], [6, -8, 0, 0], [-6, 10, 0, 0], [0, 0, -7, -6], [0, 0, -3, 8], [18, 2, 0, 0], [0, 0, 9, 2], [20, 12, 0, 0], [-5, 15, 0, 0], [6, 10, 0, 0], [0, 0, -9, -12], [0, 0, 4, 4], [0, 0, -6, 1], [17, 5, 0, 0], [-8, 8, 0, 0], [0, 0, 0, -9], [0, 0, -7, -4], [0, 0, -15, -2], [0, 0, -2, 5], [0, 0, 2, -8], [-30, -8, 0, 0], [-17, 3, 0, 0], [0, 0, 12, 4], [-35, 1, 0, 0], [-6, 12, 0, 0], [-3, -11, 0, 0], [10, 2, 0, 0], [0, 0, -1, 12], [0, 0, 11, 24], [9, -3, 0, 0], [0, 0, -9, -20], [0, -20, 0, 0], [22, -2, 0, 0], [0, 0, -10, -15], [-16, -6, 0, 0], [0, 0, -11, 2], [0, 0, 3, -10], [6, 10, 0, 0], [-10, 6, 0, 0], [27, 7, 0, 0], [0, 0, -8, -1], [12, -14, 0, 0], [0, 0, 5, 10], [0, 0, 7, -8], [0, 0, -7, 2], [28, -6, 0, 0], [9, -15, 0, 0], [0, 0, 9, 0], [0, 0, -11, 4], [0, 0, -2, -12], [12, -8, 0, 0], [0, 0, 8, 11], [42, -4, 0, 0], [0, 0, -11, -10], [0, 0, 0, 15], [-2, -14, 0, 0], [-26, 4, 0, 0], [22, -16, 0, 0], [-6, -4, 0, 0], [12, 6, 0, 0], [-1, -9, 0, 0], [12, 22, 0, 0], [-10, -8, 0, 0], [0, 0, -13, -4], [0, 0, -13, -12], [0, 0, -2, -7], [-11, -7, 0, 0], [0, 0, 13, 4], [4, 12, 0, 0], [32, -6, 0, 0], [0, 0, -7, -8], [0, 0, -9, -14], [0, 0, 22, 5], [26, -16, 0, 0], [6, 2, 0, 0], [5, -19, 0, 0], [0, 0, 0, 19], [-17, 7, 0, 0], [18, 18, 0, 0], [-32, -4, 0, 0], [14, 2, 0, 0], [0, 0, -3, 14], [0, 0, 16, 0], [-9, 7, 0, 0], [0, 0, 13, -12], [0, 0, -10, 5], [0, 0, 17, 4], [0, 0, -13, 2], [0, 0, -24, -1], [-38, -2, 0, 0], [20, -18, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_800_f_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_800_2_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_800_2_f_e(:prec:=4) chi := MakeCharacter_800_f(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); 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_800_2_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_800_2_f_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_800_f(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-2, -2, 1]>],Snew); return Vf; end function;