// Make newform 4140.2.a.s in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4140_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_4140_2_a_s();". // 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_4140_2_a_s();". // 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 := [-8, -16, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-10, -1, 1]]; Rf_basisdens := [1, 1, 2]; 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_4140_a();" function MakeCharacter_4140_a() N := 4140; order := 1; char_gens := [2071, 461, 1657, 3961]; 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_4140_a_Hecke(Kf) return MakeCharacter_4140_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, 0, 0], [1, 0, 0], [1, -1, 0], [-1, 0, -1], [1, 0, -1], [0, -1, 1], [3, 0, 1], [-1, 0, 0], [0, 1, -1], [3, 1, 0], [1, 1, -2], [-2, -3, 1], [6, -2, 0], [1, 2, -1], [-2, -3, 1], [0, -1, 1], [3, 2, -1], [3, 1, 0], [2, 3, -1], [5, 0, 3], [4, 2, 0], [-4, 1, 1], [-2, 2, 2], [-6, -4, 2], [0, 3, -3], [8, -2, 2], [8, -1, 3], [-1, 0, -3], [-2, -3, 1], [-11, 4, 1], [8, 2, 0], [2, -2, -2], [1, -3, 2], [-5, 0, 1], [-6, -2, 0], [-5, 5, 0], [6, 2, -4], [11, -2, 1], [0, 0, 0], [6, 4, -4], [10, -2, -2], [7, -2, -3], [6, 6, -2], [8, -2, -2], [-8, 0, 2], [3, -1, 2], [-6, 2, -4], [8, 0, -4], [6, 2, 2], [4, 4, 0], [10, -3, 1], [17, 0, 3], [6, 2, 4], [3, -2, 5], [4, -3, 1], [-2, -1, -1], [-9, 1, 0], [-14, 0, -4], [9, -2, 5], [9, -3, -2], [-2, 1, -3], [-11, 2, 3], [28, -2, 0], [3, -5, 6], [3, 2, 1], [-3, -3, -2], [-18, -2, 0], [-10, 4, -2], [-1, -1, 4], [3, -2, 5], [1, 2, -1], [-9, -5, 6], [-8, 8, 0], [12, -2, 0], [10, -5, -3], [-12, 0, -6], [-16, 2, 4], [0, 0, 6], [-3, 5, -4], [9, 0, -3], [15, 4, -3], [14, -8, -2], [-7, 3, 0], [16, -6, 2], [-9, 6, -3], [-8, -3, 7], [-9, -1, 2], [8, 4, -2], [-9, 2, -7], [-16, -1, 3], [3, -2, 5], [-19, 0, -3], [0, 3, -3], [7, -1, -6], [-6, 3, -3], [-18, 2, -2], [3, 6, -9], [12, -4, 8], [14, 2, -2], [-28, 0, 0], [-6, -3, -3], [-8, -7, 5], [-12, 4, 2], [11, 2, -5], [24, 2, -4], [-22, 2, 0], [-15, 6, 3], [2, 2, 6], [3, 9, -2], [11, 4, -1], [-22, 2, 4], [-18, 5, 1], [8, -10, 4], [-1, -6, 3], [-5, -2, 3], [21, -3, -2], [33, 0, -3], [1, 0, 1], [3, 4, -1], [12, -4, 2], [-13, 6, -9], [-18, -1, 7], [5, 2, 3], [2, 2, 4], [-27, 4, -1], [-45, 2, -1], [-34, 3, -1], [-7, 9, -6], [19, 1, 4], [-11, 5, 6], [-24, 4, -4], [11, 0, -3], [-23, -3, 2], [-26, -5, 3], [7, -6, 11], [-16, 0, 2], [1, 1, -2], [-6, -1, 7], [-6, 11, -5], [11, 3, -6], [2, -2, -2], [-16, -8, 8], [-14, -3, -5], [-29, 3, -4], [6, -2, -4], [-4, 12, -6], [-6, -2, 8], [27, 3, -2], [-24, 10, 2], [-16, 4, 2], [-9, 0, -3], [5, -4, 13], [-4, -2, 10], [-1, 9, 0], [8, 4, -8], [4, -6, -4], [6, 5, -11], [-18, 4, -6], [-23, 8, -1], [-2, 0, -2], [26, 4, -8], [9, 0, 1], [-6, 8, -2], [-4, 1, -5], [12, -7, 13], [19, 3, 2], [-10, 6, -6], [-4, -10, 4], [11, -14, 1], [-12, 8, 4], [7, 9, -10], [10, 14, -4], [-45, 3, -2], [0, -8, 0], [-21, -8, 5], [4, 2, 6], [10, -3, 1], [-11, 3, -4], [5, 8, -5], [-27, -6, 7], [1, 10, -9], [24, -8, -6], [3, -4, 7], [8, -3, 5], [45, 6, -3], [5, -4, 7], [-23, -1, 0], [-5, 11, 0], [30, -1, 7], [-15, -8, 3], [-18, 2, 4], [17, -8, -1], [-16, 5, 3], [-30, -6, 0], [30, 3, 3], [41, 8, -5], [-34, -12, 6], [-10, 12, -4], [-10, 2, 0], [7, -4, 5], [-14, 10, 4], [-1, 4, 5], [-9, -4, 5], [11, 10, 1], [-32, 2, 2], [9, -6, 1], [25, 8, -7], [26, 5, -3], [-6, -2, 12], [28, -4, 6], [24, 0, 6], [-4, -2, 2], [-30, -6, 0], [10, -9, -5], [-49, -6, 3], [-24, -2, 6], [30, -3, -3], [-8, 9, 1], [23, -8, 7], [-13, -9, 12], [-28, -16, 10], [-22, -3, 5], [-15, 7, -12], [-27, 4, -7], [-45, 3, -8], [0, -4, 10], [-2, 9, -11], [-4, 0, 0], [-20, 7, 9], [1, -8, -5], [-23, 0, -13], [52, 4, -2], [52, 9, -5], [9, -11, 6], [18, 18, -12], [-3, 6, 1], [18, 0, 0], [30, -9, 3], [58, -3, 1], [-6, -7, 7], [-17, 17, 0], [36, -6, -2], [-6, -10, 2], [-12, 4, -16], [22, -2, -6], [-31, 4, 5], [37, -12, 1], [20, 4, 2], [6, -3, 3], [-6, -16, 14], [-22, -16, 12], [26, 11, -3], [-12, -4, 14], [-13, -8, -5], [0, -9, -3], [59, -2, -1], [8, 0, 0], [3, 4, -13], [30, -4, -4], [-13, -9, 6], [6, -11, -1], [13, -3, -4], [37, 1, -2], [-18, 8, -14], [-1, 6, 3], [-39, 4, -1], [-2, 4, 0], [5, 5, -2], [8, 0, -4], [17, -1, -14], [5, 1, -10], [-7, -3, 6], [14, 0, -12], [21, 14, -7], [25, 2, 3], [19, -2, 9], [-27, -9, 10], [9, -10, 17], [-3, 18, -9], [-2, -14, 12], [-15, -6, 1], [-6, 12, 6], [-12, -10, 8], [23, 2, -11], [-20, 9, -5], [-25, 11, -14], [-54, 4, -4], [-20, 10, -2], [-12, 0, 6], [-16, -12, 2], [-12, 3, 3], [34, 4, -6], [1, -10, -1], [1, -2, -11], [-36, 6, -12], [-15, 8, -13], [-7, 0, 5], [-30, 17, 1], [34, 6, -4], [1, 5, 6], [12, 11, -11], [2, 10, 2], [-11, 14, -1], [-24, 10, 0], [51, 10, -9], [45, -8, -7], [2, 0, -12], [21, -2, -1], [-31, -2, -7], [-12, 8, -20], [-11, 2, 5], [-29, -4, 5], [-28, 0, 0], [24, -1, 7], [-5, -2, -5], [34, 7, 3], [6, 9, 9], [17, -9, 24], [-30, 2, -2], [21, -18, 1], [-29, -5, -2], [14, 0, 20], [2, -14, 20], [-2, 9, 1], [33, 10, -3], [-27, 18, 1], [-39, 6, -5], [12, -4, 16], [18, 1, -7], [7, -1, 0], [-23, 12, -11], [53, -9, 0], [-11, 4, 9], [-8, -2, 4], [-56, -4, -4], [-16, -8, 14], [-8, 4, 18], [2, 18, -18], [-27, 5, 2], [-28, 4, 2], [54, -4, 4], [-24, -20, 14], [43, -6, -1], [8, -17, 1], [38, -5, 7], [-9, 7, 0], [49, -10, 3], [-54, 2, 4], [0, 17, -17], [69, -1, -4], [27, 6, 7], [14, -9, -13], [-3, 23, -4], [-9, -14, 3], [22, 2, -4], [-9, 20, -5], [18, -18, 12], [35, 20, -9], [-36, 14, -2], [-1, 13, -16], [-10, 9, -7], [16, 14, 2], [-18, -4, 10], [-24, -10, 2], [20, 12, -6], [-11, 14, -13], [-71, 1, -2], [28, -22, 0], [-14, -7, 17], [9, -7, -10], [-38, 2, 8], [66, -6, 6], [12, -2, 12], [-25, 12, 9], [26, 15, -13], [12, -14, 2], [21, -5, 12], [-42, -10, 10], [8, -12, 6], [24, -12, -6], [28, -4, 14], [36, 12, -8], [-17, 8, -1], [-6, 6, -2], [30, 6, 0], [21, -17, 12], [31, -14, -5], [15, -5, -6], [14, -11, 19], [-31, -12, 3], [32, 2, 6], [-6, 3, -21], [-7, -4, 1], [-17, 0, 7], [57, -1, 8], [0, -12, 10], [42, 6, 6], [-28, 14, 4], [18, -7, 1], [29, 3, -6], [46, 14, -4], [39, 1, 0], [-12, 13, -13], [-18, -17, -1], [9, 3, -8], [33, -5, 0], [-48, 4, -10], [-42, 14, -4], [12, -18, 0], [-21, -6, -5], [67, -4, 11], [12, -16, -8], [10, 6, 2], [-21, -3, 4], [14, -8, -8], [-18, 2, 10], [-41, -4, 3], [30, 7, 11], [-42, 12, -6], [-3, 6, -3], [-14, -20, 22], [39, 0, 3], [24, -6, 24], [47, -2, -1], [0, 0, 12], [-22, -4, -6], [8, -13, -3], [-4, -10, -2], [24, -7, 7]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4140_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4140_2_a_s();". // 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_4140_2_a_s(:prec:=3) chi := MakeCharacter_4140_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_4140_2_a_s();". // 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_4140_2_a_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4140_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![24, -15, -2, 1]>,<11,R![-48, -14, 4, 1]>,<13,R![-12, -18, -2, 1]>,<17,R![-18, -21, 0, 1]>],Snew); return Vf; end function;