// Make newform 1728.2.a.bd in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1728_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_1728_2_a_bd();". // 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_1728_2_a_bd();". // 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 := [-3, -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_1728_a();" function MakeCharacter_1728_a() N := 1728; order := 1; char_gens := [703, 325, 1217]; 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_1728_a_Hecke(Kf) return MakeCharacter_1728_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, -1], [0, -1], [1, 0], [-4, 0], [0, 0], [0, -2], [6, 0], [0, 2], [0, 1], [-10, 0], [0, -2], [0, 2], [10, 0], [0, 1], [-4, 0], [0, 0], [0, 2], [-8, 0], [-3, 0], [0, 4], [9, 0], [0, 2], [7, 0], [0, -3], [0, 0], [17, 0], [-2, 0], [0, -2], [0, 1], [1, 0], [0, -6], [0, -4], [0, 3], [0, -3], [4, 0], [0, 4], [-2, 0], [0, 3], [15, 0], [-8, 0], [12, 0], [-15, 0], [0, 5], [0, 1], [0, -2], [0, -4], [12, 0], [10, 0], [0, 6], [-6, 0], [22, 0], [-24, 0], [0, 4], [-18, 0], [0, -6], [0, 3], [8, 0], [0, 0], [0, 2], [0, 6], [0, 0], [-18, 0], [1, 0], [0, -3], [0, 2], [26, 0], [-1, 0], [2, 0], [0, 2], [-10, 0], [0, -3], [16, 0], [0, 4], [-8, 0], [0, 3], [-4, 0], [0, -4], [7, 0], [4, 0], [-32, 0], [-18, 0], [-35, 0], [0, 5], [12, 0], [0, 2], [23, 0], [0, 3], [0, 7], [-23, 0], [42, 0], [0, -4], [7, 0], [0, -6], [42, 0], [0, -3], [0, 12], [0, 0], [-4, 0], [0, 0], [0, 11], [39, 0], [0, 4], [0, 4], [6, 0], [-3, 0], [0, 6], [-14, 0], [-17, 0], [0, 4], [18, 0], [0, -6], [0, 12], [0, 7], [0, -10], [0, -12], [0, 0], [0, -5], [27, 0], [-14, 0], [-19, 0], [0, -2], [-12, 0], [0, 4], [0, -3], [-4, 0], [4, 0], [0, -9], [46, 0], [0, -8], [36, 0], [0, 13], [-42, 0], [0, 0], [-11, 0], [0, -6], [0, 0], [0, 3], [0, 14], [0, -2], [0, 2], [0, 5], [16, 0], [46, 0], [40, 0], [42, 0], [0, 8], [0, 8], [28, 0], [-34, 0], [0, 12], [0, 14], [36, 0], [0, -2], [-30, 0], [0, -15], [0, -12], [35, 0], [0, 7], [27, 0], [0, -12], [0, 3], [25, 0], [0, 10], [-6, 0], [0, -17], [-36, 0], [43, 0], [0, -2], [-33, 0], [18, 0], [2, 0], [-38, 0], [0, 5], [0, 14], [0, -10], [0, 3], [0, -13], [8, 0], [0, 13], [13, 0], [-22, 0], [0, 4], [-36, 0], [0, 7], [18, 0], [0, -16], [23, 0], [-18, 0], [-14, 0], [31, 0], [0, 14], [0, -5], [44, 0], [0, -12], [43, 0], [26, 0], [0, 4], [4, 0], [0, -9], [0, 1], [16, 0], [9, 0], [67, 0], [0, -6], [0, 8], [-55, 0], [0, 14], [0, 8], [-19, 0], [0, -6], [0, -9], [51, 0], [-32, 0], [33, 0], [0, -20], [0, 14], [62, 0], [0, -18], [34, 0], [0, 16], [0, 0], [0, 8], [-27, 0], [-32, 0], [0, 0], [-42, 0], [0, -4], [4, 0], [-46, 0], [0, -16], [0, -4], [0, -12], [0, 2], [-28, 0], [-34, 0], [0, 6], [-21, 0], [-52, 0], [36, 0], [0, -6], [0, -9], [24, 0], [0, -16], [68, 0], [0, -7], [-63, 0], [0, -14], [-20, 0], [36, 0], [0, -8], [42, 0], [-63, 0], [0, 17], [57, 0], [-36, 0], [0, 16], [0, -15], [1, 0], [0, -13], [3, 0], [58, 0], [56, 0], [0, -12], [0, -20], [0, -14], [0, 10], [0, -4], [0, -18], [-8, 0], [0, -8], [53, 0], [0, 3], [-23, 0], [0, -15], [73, 0], [-56, 0], [-53, 0], [-76, 0], [-36, 0], [0, -11], [22, 0], [54, 0], [0, -2], [6, 0], [-17, 0], [0, -1], [0, -4], [0, -16], [0, -18], [-27, 0], [0, 12], [-61, 0], [66, 0], [0, 6], [0, 17], [0, 9], [-3, 0], [0, -10], [-86, 0], [0, -1], [0, 7], [-35, 0], [0, 22], [67, 0], [84, 0], [-54, 0], [-24, 0], [-32, 0], [-84, 0], [0, 13], [0, -2], [0, 4], [48, 0], [10, 0], [57, 0], [-28, 0], [34, 0], [0, -6], [0, -12], [-7, 0], [0, -19], [0, -13], [0, 0], [-65, 0], [0, 6], [0, 24], [80, 0], [0, 19], [-10, 0], [0, 14], [0, 1], [8, 0], [0, -8], [37, 0], [-66, 0], [0, 14], [5, 0], [0, 15], [46, 0], [0, 8], [0, -11], [0, -4], [0, -7], [84, 0], [-66, 0], [0, -22], [-50, 0], [0, 2], [0, -16], [11, 0], [0, -2], [0, 9], [-6, 0], [0, 6], [-30, 0], [-33, 0], [0, -2], [-58, 0], [-64, 0], [0, -24], [44, 0], [-5, 0], [0, -2], [94, 0], [0, -6], [0, 17], [-47, 0], [33, 0], [0, 4], [42, 0], [0, -3], [0, -19], [-68, 0], [72, 0], [26, 0], [59, 0], [0, -18], [30, 0], [0, 15], [0, 10], [0, 20], [0, -24], [0, 20], [50, 0], [0, 0], [-68, 0], [0, -10], [20, 0], [89, 0], [0, 22], [-84, 0], [0, -10], [44, 0], [-66, 0], [0, 4], [0, 6], [0, -2], [0, -13], [44, 0], [0, 18], [0, -20], [0, -24], [0, 1], [0, 17], [70, 0], [0, 4], [0, -12], [-49, 0], [91, 0], [0, -13], [-27, 0], [0, 12], [37, 0], [0, 1], [38, 0], [0, 20], [0, -10], [-2, 0], [0, 2], [46, 0], [28, 0], [-51, 0], [89, 0], [0, -19], [-3, 0], [0, 10], [0, 8], [12, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1728_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1728_2_a_bd();". // 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_1728_2_a_bd(:prec:=2) chi := MakeCharacter_1728_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_1728_2_a_bd();". // 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_1728_2_a_bd( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1728_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-13, 0, 1]>,<7,R![-13, 0, 1]>,<11,R![-1, 1]>],Snew); return Vf; end function;