// Make newform 4032.2.a.bw in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4032_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_4032_2_a_bw();". // 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_4032_2_a_bw();". // 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; Rf_num := [[1, 0], [-1, 2]]; Rf_basisdens := [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_4032_a();" function MakeCharacter_4032_a() N := 4032; order := 1; char_gens := [127, 3781, 1793, 577]; v := [1, 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; function MakeCharacter_4032_a_Hecke(Kf) return MakeCharacter_4032_a(); 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], [1, 0], [-2, -2], [-3, 1], [0, -2], [-1, 1], [-4, 0], [0, -2], [2, -2], [0, 2], [4, 2], [-2, -2], [-6, -2], [-10, 0], [-7, -1], [-9, -1], [-4, 0], [-4, 4], [6, -4], [4, 4], [-7, -1], [6, 0], [8, 2], [5, -3], [-10, 2], [0, 4], [-4, -2], [8, 2], [0, 4], [5, 3], [-6, -4], [1, -1], [-6, 4], [0, -4], [-3, 1], [-10, 6], [-10, -6], [-1, 3], [-16, 4], [1, 5], [4, -4], [-4, 2], [2, 4], [2, 6], [12, 0], [4, 4], [-15, -1], [1, -11], [-2, 8], [-4, 8], [8, 2], [11, -3], [14, 0], [-20, 4], [7, -5], [-24, 0], [10, -8], [-26, 0], [-5, 5], [-15, 1], [-13, 5], [-8, 0], [-4, -2], [18, -4], [-18, -2], [-12, 10], [2, 2], [-11, -7], [-2, 8], [8, 4], [4, -12], [6, -4], [18, -6], [2, 6], [0, -2], [13, 1], [28, -2], [-8, -6], [11, 5], [2, 0], [28, 0], [-4, 6], [-32, 0], [4, -8], [-10, 8], [-12, 2], [15, 3], [0, -16], [-3, -13], [22, -6], [20, 0], [20, 8], [-4, -8], [-4, 4], [-21, -9], [20, -6], [11, -3], [14, -4], [18, -6], [18, -4], [-29, 5], [-4, -10], [10, 2], [-6, 0], [3, -3], [-6, -12], [0, -8], [-6, 16], [-8, 8], [-16, -6], [-12, -10], [19, 13], [-36, -4], [-28, 6], [37, 3], [-2, 10], [4, -6], [6, 14], [7, 7], [6, -4], [31, -5], [-12, 16], [3, 5], [-28, -6], [-24, 10], [-2, -22], [-10, 10], [7, -1], [-10, 6], [-16, -4], [36, 0], [-8, -14], [8, -2], [4, -10], [29, 5], [19, -3], [-41, -5], [4, 6], [15, 17], [22, -8], [16, -16], [24, -4], [-9, -1], [26, -2], [-7, -3], [32, -2], [45, -5], [12, 4], [0, -6], [2, -12], [-32, 4], [2, 14], [8, -4], [8, 12], [0, -16], [16, 14], [-18, 4], [-11, -11], [-18, 6], [-34, -8], [12, 8], [25, -1], [-14, 12], [10, -2], [-24, 0], [11, -13], [28, 2], [7, 19], [20, 8], [-7, -11], [-4, -12], [34, -8], [18, 6], [34, -12], [24, 4], [-32, -10], [-36, -4], [-29, 3], [36, 0], [3, 5], [-36, -10], [-18, 16], [-8, -4], [11, -1], [10, 8], [7, 9], [38, 4], [-46, -2], [34, -8], [4, -16], [-24, 12], [3, -1], [16, -4], [58, -4], [-12, -6], [-32, -14], [-30, -12], [-48, 0], [6, 24], [14, -14], [19, 11], [46, -4], [-11, 3], [45, -3], [-2, 18], [-14, -6], [0, 2], [-15, -25], [-20, -6], [-7, -7], [16, 12], [-3, 19], [40, -8], [-24, 2], [16, 16], [20, 18], [-16, 8], [-44, 10], [12, -2], [40, -8], [-10, 8], [-28, 8], [9, 23], [-28, -10], [-26, -16], [4, -16], [-12, -20], [-18, -2], [4, -18], [-35, 3], [-8, 24], [14, 8], [-9, -23], [16, 24], [24, -6], [-6, -28], [36, -8], [-12, -12], [-34, -2], [-53, -3], [20, -4], [42, 8], [-6, 12], [14, -6], [-16, 0], [-3, 11], [22, -2], [24, 0], [6, 20], [30, -8], [-68, -4], [-6, 0], [17, 9], [32, 4], [-20, 14], [-3, 3], [17, 1], [-48, -14], [-12, -8], [34, -6], [-17, -1], [-25, 15], [-16, -26], [1, 23], [-22, -4], [14, 8], [-24, 4], [-12, 10], [1, -3], [-38, 10], [-2, 12], [-24, -4], [-22, 24], [-4, 12], [-38, 2], [38, 12], [16, 14], [-39, 15], [36, 20], [24, 8], [-14, 30], [39, 7], [-41, -7], [-72, -4], [-38, -8], [26, -20], [-50, 2], [30, 24], [14, -32], [17, -9], [36, 6], [-55, -9], [-46, -8], [59, 7], [40, 0], [11, 15], [11, -19], [1, 15], [-56, -6], [20, 18], [-8, 8], [-46, -6], [-42, -18], [-42, -4], [-6, 26], [-45, -13], [12, 24], [10, 24], [-12, 4], [8, 22], [-14, -20], [54, -2], [20, -28], [10, -16], [-21, -19], [-32, 0], [-2, -36], [6, -8], [-11, -5], [-50, -4], [-35, -3], [4, -12], [44, -18], [20, -2], [26, 2], [-25, -23], [-32, -12], [-28, 2], [-26, -4], [12, -14], [46, 10], [7, 1], [40, 20], [-43, 3], [-20, -18], [14, -8], [26, 24], [-10, -22], [-2, -36], [-12, 30], [55, -13], [-20, -4], [28, 2], [38, 6], [-63, -11], [-22, -30], [4, 20], [37, -3], [-15, -17], [-30, 8], [30, 16], [-18, 34], [22, 20], [-30, 4], [30, 18], [25, -1], [-18, 16], [4, -24], [-18, 4], [-46, -4], [-40, 4], [-16, 12], [55, -7], [36, 2], [-25, -5], [-76, 0], [20, -14], [8, -12], [-39, -9], [-24, 0], [2, 12], [0, -8], [-2, -4], [-31, 15], [60, -8], [-26, -12], [16, -10], [64, 2], [19, -1], [-44, -10], [32, 16], [-10, -24], [-11, -13], [-64, 16], [-28, 20], [23, -25], [-12, 0], [-22, 6], [52, 10], [-79, 1], [-32, 4], [-43, -5], [-24, -4], [6, -12], [-6, -26], [4, 26], [-8, 4], [0, 38], [-37, 27], [22, 24], [-64, 0], [-64, -2], [-17, 11], [22, -14], [34, 24], [36, -18], [45, 19], [67, 5], [46, -4], [-4, 42], [-42, 14], [12, -8], [-6, -32], [-57, -5], [-28, -12], [-18, -14], [16, 6], [92, 4], [-52, 10], [27, -29], [-8, -16], [23, -15], [4, -10], [5, -43], [-50, -2], [-34, -24], [-7, -17], [30, 18]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4032_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4032_2_a_bw();". // 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_4032_2_a_bw(:prec:=2) chi := MakeCharacter_4032_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(2999) - 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_4032_2_a_bw();". // 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_4032_2_a_bw( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4032_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-4, -2, 1]>,<11,R![-16, 4, 1]>,<13,R![4, 6, 1]>,<17,R![-20, 0, 1]>,<19,R![-4, 2, 1]>],Snew); return Vf; end function;