// Make newform 3840.2.a.bp in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3840_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_3840_2_a_bp();". // 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_3840_2_a_bp();". // 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 := [2, -4, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [-1, 2, 0], [-6, 0, 2]]; 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_3840_a();" function MakeCharacter_3840_a() N := 3840; order := 1; char_gens := [511, 2821, 2561, 1537]; 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_3840_a_Hecke(Kf) return MakeCharacter_3840_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], [-1, 0, 0], [1, 0, 0], [-1, -1, 0], [-1, 1, -1], [0, 0, -1], [2, 0, -1], [-1, 1, 0], [1, -1, 0], [2, 0, 0], [-2, 0, -1], [-2, -2, 1], [4, 2, 0], [-2, -2, 0], [1, -1, 2], [2, 0, 0], [-3, -1, 1], [2, -2, 2], [4, 0, 0], [2, 2, -2], [6, 0, 0], [6, 0, -1], [-6, -2, 0], [4, -2, 0], [6, 0, -2], [2, 0, 0], [1, 1, 2], [-6, 2, 2], [-2, -2, 0], [10, 4, -1], [-7, 1, 2], [1, 3, -1], [10, 0, 1], [9, -1, -2], [-2, 0, 0], [-4, 2, 1], [-10, -2, -1], [6, -2, 0], [-5, 1, -2], [-6, -4, 2], [-5, 1, 1], [14, -2, 0], [-8, 0, 0], [6, -4, 2], [-14, 0, -2], [12, -2, -1], [13, 3, -2], [-13, -1, 2], [0, 0, 2], [-6, 2, 2], [14, 0, 1], [10, -2, 0], [6, 4, -2], [-5, -3, -3], [6, 0, 3], [13, -1, 2], [-10, -4, 0], [20, -2, 3], [-2, 2, 3], [6, 4, -2], [20, 0, 0], [2, 0, 4], [14, -2, 4], [-10, -6, 4], [-6, 0, -2], [-14, -8, 2], [11, -3, 4], [2, -4, 4], [4, -4, 2], [-6, 6, -2], [2, 4, 1], [2, 2, -2], [1, 5, -4], [10, 2, -1], [5, 3, 2], [3, 1, 0], [-10, 0, -4], [0, 4, -1], [-4, 2, -2], [6, 0, 4], [-1, 1, -3], [22, -2, -2], [-2, -2, 2], [-2, 0, -4], [0, 2, -1], [-4, 4, 2], [-18, 0, -4], [-14, -4, -2], [10, -4, 0], [19, -1, 0], [-18, 2, -4], [-14, 2, 2], [11, 3, -2], [-13, 1, -1], [7, 1, -4], [13, 3, -2], [-6, 8, 4], [-10, 0, 0], [4, 8, -4], [10, 2, 4], [-2, 6, -4], [-14, 4, -2], [12, 4, 2], [-16, -2, 2], [15, 1, 0], [2, -4, 2], [-10, -2, -2], [-2, 0, -1], [-14, 2, -6], [6, 8, -4], [-15, 1, -4], [4, 0, -1], [-18, 0, -3], [17, -1, -2], [18, 4, -1], [-6, 4, -6], [2, 2, 4], [-5, -11, 6], [-2, 8, -2], [-13, -7, 3], [18, 6, -2], [-18, 0, 0], [-10, -8, 6], [-26, 2, 0], [5, -5, 0], [14, -4, 4], [-18, -2, 2], [16, 4, -2], [-9, -9, 6], [-14, -10, 3], [15, -7, 8], [21, -1, -4], [14, 0, -5], [-22, -2, -1], [-6, -4, 8], [2, 0, 6], [2, 0, 2], [8, -4, 8], [-10, 0, 0], [-14, 4, 6], [-13, -3, -6], [22, -4, -4], [1, -3, 6], [-20, 0, 0], [-34, -2, 0], [6, -10, 2], [-12, -8, 7], [-10, -4, 3], [-11, -5, -2], [3, -7, 6], [6, 6, -1], [-4, -2, 8], [6, -2, -8], [15, 5, -6], [16, 4, 4], [6, 2, -8], [-30, -4, 3], [-14, -4, 4], [-2, 12, 0], [18, 4, 0], [4, 0, -4], [-26, -4, -3], [19, 7, 2], [-13, 5, 5], [-10, -4, -1], [9, 11, 0], [-8, 6, 1], [-4, 0, -7], [-18, 8, -2], [-10, 8, 0], [-11, -1, -7], [-6, 6, -12], [-8, 8, 8], [14, 4, -2], [26, -4, -1], [16, -6, 0], [-1, -7, -2], [34, -4, 0], [-11, -3, 0], [34, -2, 4], [17, 5, 2], [9, 7, -3], [-16, -8, 5], [6, 0, 7], [47, 5, 0], [-22, 4, 4], [-38, 6, 1], [-6, -6, 4], [-2, -4, -8], [-4, 4, 0], [2, 8, -6], [-24, -12, 4], [17, 7, -2], [38, 0, 0], [-10, -10, 2], [2, 4, -5], [14, -8, 0], [-20, 4, 7], [22, -8, 3], [45, -1, 4], [2, -12, 8], [22, 12, -3], [4, 4, -5], [2, -12, 2], [21, -5, -1], [-14, 8, -12], [2, 8, -11], [-4, -12, 10], [0, 10, -8], [7, 9, 0], [14, 4, 6], [-2, 8, 4], [-55, 1, 2], [24, 4, 0], [-16, 4, 6], [2, 12, 2], [1, -3, 4], [0, 6, 2], [-5, 9, 0], [14, -8, -8], [10, 10, 2], [20, -2, -1], [-4, 2, 8], [-5, -9, 8], [18, 6, 4], [-10, 10, -4], [-2, -16, -1], [-28, 8, -2], [11, -1, 0], [-13, -7, 5], [44, -4, -1], [-25, -7, 2], [12, -2, 7], [2, 8, -6], [-48, 4, -8], [1, 7, -4], [-14, 4, -10], [-18, -8, -2], [-3, 11, 3], [-14, 10, 6], [8, -8, -2], [31, 9, -8], [-27, 1, -10], [14, 10, -2], [30, -8, 1], [-44, 0, -6], [1, -3, -4], [37, -1, 1], [29, 3, -4], [23, 5, -6], [-24, 8, 9], [-6, 0, -4], [-5, -3, -8], [14, -12, 10], [-2, 4, -2], [-45, 1, 3], [34, 10, 0], [-6, 2, 4], [-18, 12, -2], [42, 12, -4], [-5, -17, 10], [12, 8, 0], [-30, 2, 2], [-50, -2, -3], [46, 4, -3], [31, 1, -4], [-14, 4, 4], [40, -6, 2], [0, -4, 8], [30, 0, 4], [-6, -2, 8], [-22, 10, -8], [14, 4, -2], [22, 8, -1], [34, 8, -2], [1, 9, 6], [16, 0, 2], [-14, -2, 2], [-6, -4, -6], [21, -9, -5], [11, 5, -4], [12, -10, 11], [23, 1, 6], [-30, 6, 0], [-36, 0, 8], [-28, -12, 4], [14, 4, 8], [-26, 0, 2], [-4, 2, -3], [24, 2, 2], [-26, 8, 4], [10, 10, 2], [6, 0, 13], [19, 1, -11], [26, 18, -3], [-10, 0, 12], [6, -4, 5], [46, 12, -6], [-63, -1, 3], [-50, -2, 4], [18, 8, -10], [-10, 0, -12], [20, 2, -3], [42, 2, 2], [-17, -15, 4], [-10, 12, 4], [14, -6, 12], [34, 10, -2], [32, 20, -6], [-46, -6, -1], [47, -3, 6], [18, 0, 0], [4, -14, 14], [-8, -4, 8], [-33, 5, -4], [42, -8, 2], [11, -7, 13], [-12, 16, 6], [34, -4, 0], [8, 10, 2], [5, 11, 0], [2, 8, -10], [22, 12, -8], [7, 15, -2], [2, -16, 7], [-10, -8, 12], [-17, 9, 6], [10, 2, 4], [-43, -1, -4], [6, -8, 8], [34, 2, -6], [26, 4, -6], [38, -4, 9], [26, 10, -2], [-29, -19, 6], [2, -2, -8], [-30, -18, 4], [10, -12, -7], [6, 0, 0], [-51, -3, -2], [-6, 10, -1], [-2, 12, -9], [6, -12, -4], [-8, -6, -5], [-2, -4, 10], [-21, -19, 3], [-2, -18, 14], [4, -8, 0], [-52, 8, -6], [42, 8, -8], [-19, -13, 10], [26, 4, -10], [-10, 4, -12], [-45, 7, -10], [46, -6, 6], [22, 0, 5], [44, 4, 4], [-49, -19, 5], [-70, -8, 1], [57, -5, -2], [-8, 12, 5], [-4, -2, 10], [-15, -5, 8], [-63, -1, -1], [-20, 0, 4], [6, 0, 14], [10, -16, 8], [-3, -15, 16], [-30, -12, 0], [29, 11, -5], [49, -9, 2], [-25, 5, 4], [30, 20, -4], [60, 10, -7], [44, -4, 1], [31, 9, -9], [-50, 2, -4], [10, -8, 12], [34, -12, 8], [34, -8, 4], [-14, 12, 4], [-50, 4, -7], [-1, -1, 8], [30, 0, -3], [-9, -7, -10], [9, -5, 10], [-34, 0, 3], [14, -14, -1], [18, 18, -8], [9, -9, -12], [-22, -16, 2], [-30, 4, -14], [17, 19, -3], [-20, -8, 0], [-56, -8, 8], [10, -12, 6], [-24, -14, 3], [-6, 4, -10], [-57, 1, 4], [-34, -16, 0], [-6, -2, 4], [-54, -8, 5], [-51, -3, 2], [-46, -8, 9], [-18, -4, 12], [-18, 8, -9], [24, -8, 13], [0, 14, -2], [18, 10, 0], [35, -3, 9], [-54, 4, -6], [46, 0, -4], [22, -14, 20], [-19, 3, -16], [-34, 8, 12], [-46, -8, 0], [24, -16, -8], [23, -1, -8], [-14, 8, 9], [57, 7, -4], [22, 8, -12], [-4, 8, 1], [-59, -9, 4], [-11, -21, 13], [34, -8, 6], [26, -16, -4], [38, -6, -12], [6, -12, -8], [-9, 9, 4], [74, 2, 2]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3840_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3840_2_a_bp();". // 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_3840_2_a_bp(:prec:=3) chi := MakeCharacter_3840_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_3840_2_a_bp();". // 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_3840_2_a_bp( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3840_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-16, -16, 2, 1]>,<11,R![-64, -24, 4, 1]>,<13,R![-16, -28, 0, 1]>,<17,R![32, -16, -6, 1]>,<19,R![-16, -12, 4, 1]>],Snew); return Vf; end function;