// Make newform 4730.2.a.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4730_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_4730_2_a_e();". // 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_4730_2_a_e();". // 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 Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_4730_a();" function MakeCharacter_4730_a() N := 4730; order := 1; char_gens := [947, 431, 1981]; 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_4730_a_Hecke(Kf) return MakeCharacter_4730_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], [-1], [-1], [-2], [1], [4], [0], [-2], [-2], [3], [4], [-4], [8], [-1], [-2], [-9], [-8], [-13], [14], [-4], [-13], [-13], [9], [-8], [3], [-18], [2], [-3], [16], [-8], [-13], [0], [4], [-11], [-5], [6], [-22], [5], [17], [-24], [-9], [11], [11], [-4], [-12], [-20], [-12], [-1], [-14], [-5], [3], [15], [2], [-2], [22], [-8], [-3], [-20], [-10], [0], [-3], [24], [29], [10], [28], [3], [-20], [2], [4], [10], [-9], [1], [-10], [-17], [14], [29], [-30], [25], [35], [-29], [15], [40], [12], [-24], [0], [-2], [-22], [-42], [12], [32], [-20], [-25], [-22], [-18], [-13], [28], [-34], [-14], [-24], [-10], [7], [-6], [20], [-38], [10], [28], [-36], [37], [-44], [-26], [26], [-26], [-2], [20], [5], [10], [14], [35], [34], [-15], [51], [17], [-15], [40], [-19], [22], [-19], [-10], [13], [-17], [24], [-6], [-35], [-30], [27], [18], [-48], [32], [34], [-36], [2], [-48], [-18], [19], [-28], [17], [50], [-8], [40], [27], [12], [11], [-20], [48], [10], [1], [53], [-28], [25], [-24], [2], [-41], [-27], [36], [51], [56], [21], [38], [-19], [15], [50], [-22], [44], [40], [-21], [-14], [20], [-2], [-56], [49], [-19], [8], [-22], [61], [-12], [30], [-57], [-4], [-4], [-18], [49], [4], [-14], [-28], [0], [-12], [14], [-3], [0], [56], [38], [-32], [-48], [-18], [36], [-54], [-63], [-5], [-1], [-42], [-51], [49], [-32], [64], [-32], [19], [18], [-37], [67], [14], [-55], [45], [-26], [-38], [68], [-8], [1], [-51], [-2], [0], [18], [-25], [-64], [-47], [54], [-16], [-34], [-58], [-72], [-14], [21], [50], [14], [-11], [-27], [24], [-61], [36], [-44], [-57], [-42], [-34], [32], [56], [-15], [-18], [22], [20], [48], [8], [-14], [-26], [-24], [-10], [-48], [-15], [-10], [21], [-36], [-12], [37], [-36], [36], [-4], [-18], [-64], [42], [13], [-10], [38], [-60], [-10], [-27], [-42], [-52], [-27], [-39], [-43], [-40], [-24], [-30], [-18], [66], [-58], [-20], [36], [-62], [-54], [29], [33], [-50], [63], [45], [78], [12], [-82], [42], [30], [45], [75], [72], [-54], [36], [-51], [64], [74], [-38], [19], [39], [78], [65], [-2], [30], [-54], [-58], [-23], [76], [20], [-61], [58], [-2], [-2], [84], [10], [47], [22], [-20], [42], [-58], [58], [-7], [-42], [-30], [-93], [94], [73], [28], [-72], [-2], [57], [29], [48], [9], [-74], [38], [6], [-9], [74], [6], [38], [-85], [38], [-72], [-20], [-10], [-16], [-87], [-64], [29], [81], [51], [32], [-6], [63], [48], [69], [-53], [52], [-18], [34], [-18], [-9], [24], [37], [-4], [-42], [-32], [-81], [41], [30], [39], [-18], [-45], [34], [-49], [56], [-25], [-26], [31], [42], [-70], [27], [-1], [70], [81], [-38], [72], [88], [-47], [-62], [-64], [-33], [-27], [28], [85], [-63], [0], [-14], [-13], [-26], [10], [-70], [-27], [-35], [-58], [-30], [4], [-90], [-14], [56]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4730_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4730_2_a_e();". // 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_4730_2_a_e(:prec:=1) chi := MakeCharacter_4730_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_4730_2_a_e();". // 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_4730_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4730_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, 1]>,<7,R![2, 1]>,<13,R![-4, 1]>],Snew); return Vf; end function;