// Make newform 6240.2.a.bv in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6240_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_6240_2_a_bv();". // 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_6240_2_a_bv();". // 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_6240_a();" function MakeCharacter_6240_a() N := 6240; order := 1; char_gens := [1951, 2341, 2081, 2497, 5761]; v := [1, 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_6240_a_Hecke(Kf) return MakeCharacter_6240_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], [1, 0], [1, 0], [3, 1], [2, -2], [-1, 0], [3, -1], [3, 1], [5, -1], [7, -1], [0, -4], [-4, -2], [-8, 2], [-4, 4], [4, 4], [-4, -2], [-2, 2], [2, 4], [-8, 0], [-4, 0], [3, 3], [-4, 4], [8, 0], [-6, -4], [-1, 7], [9, -3], [4, 0], [-12, 0], [-1, -5], [9, 1], [4, 0], [-5, -7], [4, -2], [2, 2], [0, -2], [8, 4], [0, 6], [-6, -2], [-10, -6], [0, 2], [-19, -1], [16, -2], [-12, 4], [9, 5], [4, 6], [8, 8], [-8, -4], [1, -5], [-4, 8], [-5, -1], [-13, -1], [-20, -4], [10, 0], [-7, 3], [13, -3], [3, -7], [9, 5], [4, -8], [-8, -2], [-10, -8], [-16, 0], [-8, 2], [-12, -4], [12, -4], [8, -6], [-2, 0], [7, -3], [28, -2], [2, -6], [-17, 3], [0, 2], [16, -4], [-8, 12], [8, 6], [-35, -1], [-6, 14], [5, 1], [0, 2], [-8, -6], [12, -2], [-5, 9], [21, -3], [-8, 8], [-20, 6], [-4, -8], [-26, -2], [2, 12], [27, 3], [30, 0], [1, 3], [0, 4], [-4, 12], [15, 5], [-25, -3], [13, -1], [-9, -11], [14, -8], [4, -10], [-16, -12], [9, 9], [12, -8], [14, -8], [6, 14], [-4, -10], [6, 14], [17, -11], [-8, -4], [-6, 0], [-16, 0], [20, 10], [8, 4], [-16, -6], [4, 6], [-19, 7], [10, -6], [26, 0], [-8, 0], [23, -3], [26, 0], [-33, -3], [-7, 9], [14, 4], [-30, 0], [-32, 8], [15, -3], [21, 1], [-29, 7], [30, 2], [-10, -2], [-12, -2], [9, -5], [-34, -6], [20, 0], [-18, 4], [-16, -6], [6, -12], [-24, 10], [10, -10], [-32, -6], [-2, 20], [-1, -3], [20, -14], [12, -16], [12, 4], [-26, 8], [0, 8], [-4, 6], [1, 9], [8, -4], [14, -6], [-38, 0], [20, 6], [4, -8], [-15, -13], [24, 0], [50, -2], [-20, -16], [30, -8], [-54, 0], [20, 2], [-16, -8], [31, -5], [3, 1], [-27, 15], [-22, -16], [6, -6], [-36, -4], [-12, 2], [-30, 0], [-20, 6], [-40, 0], [-17, -5], [-26, -14], [13, -15], [40, 8], [16, -6], [43, 1], [0, 22], [34, 2], [-24, -2], [-25, -3], [-41, 5], [-34, 4], [-26, -4], [18, 14], [27, -13], [42, -8], [-40, 0], [-24, -14], [-24, 0], [-30, 8], [-4, -16], [12, 8], [-4, -22], [34, 2], [9, -23], [-12, -2], [24, -2], [-50, 4], [39, 5], [-46, -4], [-12, -16], [-22, 8], [4, -6], [-22, -10], [-2, 20], [20, 0], [26, -6], [32, 10], [40, -12], [0, -22], [-13, -5], [-18, -18], [0, -8], [8, -16], [14, 4], [-18, -2], [-32, -6], [-18, 18], [-26, -16], [20, -22], [34, 10], [-4, -26], [-29, 1], [22, -2], [24, 6], [-29, 15], [2, 6], [18, 2], [-22, -18], [16, 22], [32, 12], [28, -16], [-12, 6], [-36, 12], [-42, 10], [-12, -18], [-38, -12], [41, -5], [38, -6], [-4, 12], [16, -4], [10, -6], [9, -15], [-14, 24], [0, 0], [-43, 7], [-16, 0], [-25, -11], [39, -3], [-28, -10], [42, 4], [-24, 0], [-22, 16], [24, -6], [-16, 24], [-30, -20], [-46, -2], [-36, -10], [-13, -21], [30, -2], [40, -12], [15, -5], [2, 24], [-30, 0], [-2, 6], [38, 0], [-38, 12], [-52, -4], [54, -4], [60, -6], [-18, 26], [-15, 13], [-8, -20], [-30, -8], [-1, 29], [-40, 4], [25, 17], [40, 18], [-23, -13], [-1, -19], [-14, 2], [31, -3], [47, 3], [-32, 0], [-6, 6], [28, -2], [0, 26], [-10, 30], [-42, -4], [47, -9], [-20, -16], [58, -8], [4, -4], [-50, -4], [49, 13], [-28, 20], [50, 0], [51, 9], [10, 6], [-56, -6], [-12, 14], [4, -12], [-10, -10], [8, 4], [-11, -7], [-52, -16], [44, 2], [-36, -20], [2, -8], [9, 11], [30, -8], [-52, 14], [-4, -8], [-32, -8], [50, 8], [18, -2], [-24, 20], [15, -1], [-6, 8], [14, -18], [-61, 11], [49, -11], [-15, -5], [46, 12], [-36, 18], [-19, -25], [-24, 16], [11, -7], [4, -2], [21, 5], [6, -20], [48, 4], [16, 28], [35, 9], [-4, -16], [-51, -11], [-16, 10], [-12, -10], [8, 4], [-6, 16], [53, 5], [28, -14], [60, 4], [32, 26], [-17, 21], [10, 12], [-76, 4], [-48, 4], [26, 0], [61, 15], [17, -11], [12, 10], [-30, 26], [16, 14], [39, -21], [60, 4], [-2, 18], [43, 23], [36, -4], [-30, 8], [-14, 8], [-45, 17], [32, 8], [-8, 8], [16, 2], [-56, 2], [-27, 7], [34, -24], [3, -7], [10, 2], [46, -14], [-5, 11], [8, 32], [-2, -36], [-40, -8], [-8, -32], [37, 1], [-24, 26], [56, 2], [-36, 26], [-28, 14], [-35, -9], [-2, -12], [9, -5], [-28, -4], [10, 10], [-14, 16], [-4, -12], [-13, -15], [-10, -20], [30, 16], [34, 14], [-12, -16], [44, -8], [-28, 14], [-12, -24], [14, -28], [16, 28], [38, -8], [-39, -23], [49, 1], [-3, -33], [58, 0], [46, -8], [16, 20], [40, 10], [-22, 20], [38, 2], [-16, -32], [36, -10], [12, 22], [-42, -2], [-6, -30], [-18, 28], [39, 23], [-4, -4], [-44, -8], [-38, 0], [-13, 9], [-45, 11], [-6, 16], [0, -8], [-23, 19], [31, -17], [22, 16], [20, 0], [-6, -12], [-15, -5], [-60, 4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6240_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6240_2_a_bv();". // 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_6240_2_a_bv(:prec:=2) chi := MakeCharacter_6240_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_6240_2_a_bv();". // 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_6240_2_a_bv( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6240_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![4, -6, 1]>,<11,R![-16, -4, 1]>,<17,R![4, -6, 1]>,<19,R![4, -6, 1]>],Snew); return Vf; end function;