// Make newform 5586.2.a.bu in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5586_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_5586_2_a_bu();". // 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_5586_2_a_bu();". // 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, -5, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-3, -1, 1]]; Rf_basisdens := [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_5586_a();" function MakeCharacter_5586_a() N := 5586; order := 1; char_gens := [3725, 4903, 4999]; 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; function MakeCharacter_5586_a_Hecke(Kf) return MakeCharacter_5586_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 := [[1, 0, 0], [-1, 0, 0], [2, -1, 0], [0, 0, 0], [1, -1, -1], [2, 2, -2], [5, -1, 0], [1, 0, 0], [-1, 0, -3], [-1, 3, -1], [3, 2, 0], [-1, -3, 2], [5, -2, 1], [1, -1, 4], [-2, -2, 4], [-3, -2, 2], [2, 2, -1], [-4, 2, 0], [-4, 2, -2], [-3, -1, 4], [0, -1, 3], [-7, 3, 3], [9, -6, -2], [5, -2, 1], [10, 2, -1], [8, 0, 2], [-8, -3, 3], [-4, -5, 6], [-1, 7, 0], [-1, 4, -5], [1, 6, -2], [-3, 2, -2], [6, -6, 4], [11, -1, -4], [8, -2, 4], [-9, 4, 0], [-8, -6, 4], [-2, 8, -6], [-11, 3, 0], [8, 3, -1], [0, -2, 2], [6, 2, 6], [-6, -2, -4], [-4, 3, -2], [8, -1, 5], [-12, -2, 2], [7, 0, 9], [7, 3, 1], [-2, 1, 0], [-4, 12, -2], [2, 4, 4], [-4, 6, -2], [6, 2, 3], [-11, 7, 1], [10, 8, -4], [-15, 4, -3], [7, -13, 1], [-4, -7, 0], [-1, 0, -7], [-6, -2, 6], [-5, -9, 6], [1, 5, -7], [13, -2, 7], [17, 8, -3], [3, -7, 3], [11, 4, 2], [1, 4, 5], [14, -8, -7], [-22, 1, 1], [9, -6, -3], [9, 5, -2], [-6, -4, -4], [-8, 7, 0], [-18, 8, 2], [-4, -6, 10], [-18, 0, -2], [-12, -8, 2], [-16, 6, 0], [0, 12, -2], [2, 4, -7], [12, -9, 1], [8, -4, -2], [13, -1, -4], [-4, 1, -5], [7, 3, -1], [-7, 1, -7], [7, -16, 3], [1, -5, -9], [-6, 12, -8], [12, 6, 2], [-4, 9, 7], [0, 8, 4], [17, 0, 4], [13, -10, -8], [2, -2, -6], [-7, -6, 1], [-3, 9, 9], [-4, -10, 14], [3, 2, 3], [17, -16, -3], [-11, 6, 1], [0, -4, 5], [-12, -2, 1], [18, 2, 6], [1, -11, 4], [17, 6, -10], [5, -17, 3], [-30, 8, 2], [-1, -15, 12], [14, 12, -7], [-25, -5, 5], [21, -4, 5], [15, -11, -6], [-12, -6, 0], [-6, 9, -14], [-17, -2, -5], [23, 1, -6], [5, 4, -17], [-6, -8, -9], [8, 1, 1], [-11, 5, -6], [-14, 15, -4], [-5, 8, 2], [14, 21, -14], [19, -9, 6], [-6, 0, 5], [5, 14, -9], [0, 18, -12], [-30, -1, 4], [-17, 2, -7], [-11, 15, -8], [-6, -2, -12], [-17, -9, 3], [21, 4, 5], [2, -12, 0], [37, -7, 1], [-14, -1, -3], [-10, 8, -16], [-29, 15, -1], [-13, 5, 2], [-7, -4, -1], [6, 6, -1], [26, 4, -10], [-20, -2, -9], [-8, 4, -12], [11, 7, -8], [-1, 6, 9], [-12, 0, 0], [0, 8, -20], [-29, -1, 0], [13, -1, -10], [-6, 0, 6], [11, -5, -12], [-21, 21, -2], [13, 6, 9], [16, -12, 2], [16, -1, -5], [3, -17, 0], [-3, 4, -8], [5, 14, -2], [2, 7, -1], [-42, 2, 0], [-4, 8, -7], [-38, 14, 5], [18, -18, -6], [-24, 2, -2], [-5, -4, -12], [6, 2, 8], [-2, 15, -2], [-2, 14, 7], [2, -8, 17], [11, -19, 4], [-46, 10, 2], [-5, 13, -11], [19, -1, -7], [-34, 0, -8], [-2, -8, 14], [2, 2, 5], [-10, 5, -19], [-1, -10, 13], [8, 14, -5], [14, -5, 6], [-14, 0, -14], [5, -18, 1], [2, 0, 18], [22, 1, -6], [2, -6, 10], [-13, -18, 11], [-44, 0, 1], [4, -14, 0], [-16, 1, -2], [-17, -7, -3], [-42, 10, -4], [18, -15, -1], [-17, 10, -10], [19, -4, 3], [16, -3, -3], [12, -2, 10], [20, 0, 8], [-9, 18, -11], [1, -1, -1], [-4, -27, 15], [3, -13, 16], [0, -10, 8], [4, 1, -1], [-14, 25, -4], [-14, -4, -10], [-42, -6, 7], [11, 19, -6], [39, 18, -13], [34, 19, -11], [44, -12, -1], [-22, -14, 5], [16, 4, -11], [-6, -18, 2], [-4, 21, -13], [32, -1, 10], [-4, -2, 4], [12, -6, 10], [-48, -6, 4], [23, 25, -20], [24, -5, 13], [2, -8, 0], [-12, -6, 7], [10, 2, -7], [23, -4, -7], [-27, 6, 9], [-19, 17, 0], [13, -11, -5], [4, -13, 17], [-6, 18, 12], [6, 12, -12], [-17, -18, 14], [17, -8, -13], [16, 16, -10], [10, -4, 10], [1, 16, -4], [-12, -1, 0], [8, 13, -1], [-19, 1, 4], [8, -8, 4], [-24, 6, 8], [44, -15, -6], [36, 12, 0], [-35, -6, -9], [7, -12, -3], [-38, 19, 4], [36, 8, -4], [23, -22, -5], [-18, -12, -10], [-15, 26, -11], [34, -14, -14], [33, 12, -1], [10, -2, 6], [-14, -6, 9], [-57, 2, 2], [7, 18, -15], [7, -6, 11], [3, -18, 26], [-1, 1, -13], [-8, 16, 8], [-12, 10, -1], [35, -24, -3], [35, 5, 6], [27, 9, -16], [1, 12, -15], [23, -19, -1], [-23, 7, 2], [13, -12, 15], [6, -12, 15], [-4, 10, 14], [46, -14, 0], [27, 3, -3], [38, -15, 6], [-12, -21, 4], [-14, -10, -16], [-41, 8, -6], [5, -1, 0], [32, -7, 11], [-14, 1, -19], [28, 8, 8], [30, 14, 1], [5, -28, 5], [11, -21, -10], [52, 0, 8], [32, 16, -12], [9, 14, 4], [-43, 10, -4], [-6, 4, -29], [39, 20, -5], [-30, 26, 2], [25, -9, -9], [-15, 16, -21], [-12, -5, -1], [22, -16, -20], [42, 6, -5], [-10, 32, -6], [0, -10, 4], [-20, 10, -2], [-33, -21, 6], [6, 24, -24], [-43, -2, -4], [30, -2, 19], [-26, -3, -16], [-35, 1, -20], [-32, 12, 6], [26, 8, -1], [45, 1, 16], [9, -2, -7], [17, -18, 29], [36, -4, 6], [-24, 6, -8], [-18, 14, -18], [12, 2, 16], [7, -28, 19], [-8, 10, 8], [49, 14, -14], [0, -32, 6], [27, 0, 12], [-44, 8, -12], [11, 8, -19], [-6, 19, -15], [37, 11, -17], [38, 1, 17], [-53, 5, 4], [12, 2, 21], [47, 6, 3], [17, -2, 7], [14, -18, -10], [-36, 22, 7], [-2, -28, 28], [-23, -2, -16], [20, -10, 14], [22, 12, 14], [-1, 23, -12], [36, 36, -24], [2, 16, -14], [10, -12, 16], [29, -5, 7], [34, -7, 4], [0, 14, -26], [-2, -26, 24], [11, 37, -13], [15, 7, -5], [23, -1, 11], [46, -4, 1], [27, -20, -11], [14, -4, 6], [-38, -14, -2], [16, -4, -7], [12, -18, 32], [-40, 5, 6], [8, -5, -7], [-67, -5, 11], [23, -1, -18], [9, -2, -1], [22, -16, 6], [-1, 16, 14], [23, 1, 2], [18, -6, 22], [-27, -12, 17], [7, -5, 24], [-27, -9, 16], [12, -2, 6], [22, 16, -12], [-24, 2, -17], [-8, -2, -20], [-14, -19, 5], [46, -11, 2], [67, -2, 8], [26, -6, 10], [24, 12, -12], [17, 8, 12], [4, 0, -12], [28, -6, -4], [-4, 10, -14], [-16, 26, -4], [-9, 6, 0], [35, -11, 20], [-42, 32, -4], [-24, 3, -13], [-68, -14, 10], [-40, -15, 9], [4, 2, 18], [-55, 0, 13], [-21, -9, 24], [-50, -28, 14], [-42, -32, 14], [2, -10, 28], [31, 1, -12], [-56, -4, -1], [4, 10, -16], [-17, 9, 3], [-7, -1, -6], [55, -9, 18], [38, -33, -2], [-31, -2, 24], [80, -2, 0], [14, 24, -20], [-30, 27, 6], [66, -24, -4], [-31, -13, 4], [22, 15, -17], [-11, 10, 7], [31, -5, -7], [30, 18, -15], [5, 2, -5], [40, -16, 0], [-3, -26, 9], [-3, 13, 23], [2, 23, -24], [-38, 7, 18], [-14, 17, 18], [-26, -2, -14], [-36, -17, -1], [12, 30, -21], [26, 16, -8], [-32, -15, 15], [31, 7, -12], [-35, -11, -8], [-25, 8, -4], [33, -33, 12], [-18, 2, -10], [46, -7, 26], [4, 14, 16], [-18, 7, 5], [18, -4, 9], [66, 12, -16], [-11, 21, 16], [5, 16, -23]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5586_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5586_2_a_bu();". // 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_5586_2_a_bu(:prec:=3) chi := MakeCharacter_5586_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_5586_2_a_bu();". // 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_5586_2_a_bu( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5586_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![7, 3, -5, 1]>,<11,R![21, -17, -1, 1]>,<13,R![88, -20, -6, 1]>,<17,R![-74, 60, -14, 1]>],Snew); return Vf; end function;