// Make newform 570.2.i.e in Magma, downloaded from the LMFDB on 19 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_570_i();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_570_i_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_570_2_i_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_570_2_i_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, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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_570_i();" function MakeCharacter_570_i() N := 570; order := 3; char_gens := [191, 457, 211]; v := [3, 3, 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_570_i_Hecke();" function MakeCharacter_570_i_Hecke(Kf) N := 570; order := 3; char_gens := [191, 457, 211]; char_values := [[1, 0], [1, 0], [0, -1]]; 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 := [[1, -1], [1, -1], [-1, 1], [-5, 0], [1, 0], [0, -6], [-4, 4], [-3, 5], [0, -7], [0, -6], [0, 0], [7, 0], [5, -5], [-6, 6], [0, 8], [0, -11], [8, -8], [0, 4], [0, -12], [-2, 2], [2, -2], [-10, 10], [-10, 0], [0, -13], [-2, 2], [0, -2], [15, 0], [-4, 0], [16, -16], [4, 0], [0, 5], [3, -3], [0, 12], [0, 16], [4, -4], [-20, 0], [-9, 9], [-10, 0], [0, -21], [9, -9], [7, 0], [0, -18], [-14, 0], [4, -4], [1, 0], [0, 4], [-13, 13], [21, -21], [2, 0], [10, 0], [18, -18], [0, 0], [0, 26], [0, -12], [0, -18], [-21, 21], [-32, 32], [-2, 2], [-14, 0], [0, 11], [-22, 22], [5, 0], [-18, 18], [-20, 0], [0, 0], [0, -3], [15, 0], [-2, 2], [2, -2], [16, 0], [36, 0], [18, -18], [0, 4], [5, 0], [-4, 0], [-32, 32], [0, 10], [33, -33], [18, -18], [0, 3], [-3, 0], [8, -8], [0, 4], [0, -6], [32, -32], [0, -24], [-39, 0], [8, 0], [6, -6], [-9, 0], [38, 0], [0, -42], [-13, 0], [33, -33], [7, -7], [0, -1], [0, -2], [-14, 0], [0, 14], [0, -8], [0, -14], [0, -1], [24, 0], [33, 0], [20, 0], [-32, 0], [20, -20], [0, -22], [0, -8], [-3, 0], [-1, 0], [-35, 35], [0, -48], [43, 0], [0, -38], [26, -26], [44, -44], [3, 0], [9, 0], [0, 11], [0, 20], [-40, 0], [21, 0], [0, 0], [1, 0], [-34, 34], [0, 4], [44, -44], [-32, 32], [19, 0], [47, -47], [-15, 15], [0, -40], [13, -13], [-31, 0], [0, -6], [0, 21], [-26, 0], [47, 0], [-18, 0], [0, -43], [0, -42], [0, 23], [0, 30], [-28, 0], [18, -18], [-18, 18], [36, -36], [0, 17], [-1, 0], [7, -7], [33, 0], [0, 36], [0, 0], [-22, 22], [-18, 0], [-26, 0], [15, -15], [0, 22], [0, -46], [32, -32], [36, -36], [-24, 24], [-12, 12], [-6, 0], [-3, 3], [-10, 10], [0, 31]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_570_i_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_570_2_i_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_570_2_i_e(:prec:=2) chi := MakeCharacter_570_i(); 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_570_2_i_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_570_2_i_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_570_i(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![5, 1]>,<11,R![-1, 1]>],Snew); return Vf; end function;