// Make newform 6034.2.a.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6034_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_6034_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_6034_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_6034_a();" function MakeCharacter_6034_a() N := 6034; order := 1; char_gens := [1725, 869]; v := [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_6034_a_Hecke(Kf) return MakeCharacter_6034_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], [-3], [-1], [3], [-2], [8], [-5], [5], [3], [-6], [-8], [-2], [4], [-2], [9], [9], [-6], [-8], [10], [-4], [4], [-6], [-6], [-7], [14], [-14], [-8], [6], [-8], [4], [-12], [-2], [9], [-23], [-15], [-1], [-25], [-12], [-15], [-24], [-20], [8], [16], [15], [14], [-18], [-15], [-17], [-1], [28], [20], [-24], [12], [-4], [-23], [-18], [28], [19], [-12], [-23], [24], [-20], [8], [-10], [18], [-8], [-18], [-12], [-2], [15], [20], [22], [-34], [36], [24], [15], [5], [-38], [-20], [-10], [10], [1], [-5], [-21], [-9], [31], [-16], [-9], [-31], [-28], [0], [26], [28], [6], [33], [8], [-38], [16], [17], [-3], [-8], [-4], [9], [-40], [42], [-16], [-3], [0], [10], [20], [-22], [30], [-4], [5], [-28], [12], [-13], [16], [33], [-1], [-26], [15], [-24], [-42], [-22], [-1], [24], [-26], [-43], [26], [-4], [37], [14], [14], [5], [-51], [16], [-14], [36], [-29], [-24], [57], [-35], [-14], [0], [4], [18], [-4], [-40], [31], [-25], [56], [43], [-27], [-41], [29], [-14], [-9], [16], [24], [43], [-18], [12], [51], [9], [-28], [-19], [-33], [-14], [-15], [-47], [-23], [-29], [-21], [-38], [-42], [-42], [32], [22], [53], [64], [-48], [-18], [-36], [14], [-44], [29], [-24], [32], [-37], [11], [-14], [-51], [44], [-36], [-24], [-62], [-48], [-16], [-52], [-33], [13], [-16], [55], [54], [56], [22], [-30], [30], [43], [-13], [-1], [-38], [-24], [50], [22], [-30], [66], [-6], [-7], [67], [-66], [58], [-6], [-22], [42], [-60], [46], [-46], [-50], [2], [-56], [60], [-40], [63], [49], [31], [-4], [-11], [-44], [71], [63], [39], [46], [-42], [-16], [31], [-72], [37], [-53], [-8], [48], [-32], [26], [72], [-39], [18], [12], [-35], [-26], [66], [26], [10], [54], [-2], [-10], [6], [-32], [26], [-12], [-63], [-2], [54], [49], [-40], [-40], [2], [-28], [-12], [-9], [0], [-36], [-26], [12], [-19], [7], [-36], [-20], [3], [42], [63], [-60], [-33], [57], [-66], [64], [36], [6], [-35], [-86], [18], [-36], [-18], [18], [-44], [44], [29], [14], [41], [-30], [9], [66], [14], [-3], [-78], [-29], [2], [58], [7], [0], [70], [-15], [-20], [-48], [-25], [51], [-5], [8], [9], [-66], [39], [30], [7], [-12], [78], [-41], [-45], [20], [-52], [38], [36], [66], [12], [8], [9], [28], [-1], [3], [-50], [-4], [-66], [-66], [-11], [-66], [45], [59], [15], [72], [-54], [-80], [50], [32], [-36], [-84], [71], [0], [-11], [-25], [-24], [-44], [25], [41], [-73], [-12], [-11], [3], [-42], [27], [88], [68], [40], [-97], [30], [88], [-62], [28], [-42], [-15], [82], [-80], [30], [54], [21], [57], [-20], [2], [-4], [32], [-94], [-30], [64], [-102], [68], [62], [63], [102], [-98], [59], [-56], [-40], [-20], [96], [43], [-70], [-87], [-46], [-8], [18], [-48], [-98], [-68], [-84], [-45], [54], [6], [30], [76], [-42], [96]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6034_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6034_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_6034_2_a_e(:prec:=1) chi := MakeCharacter_6034_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_6034_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_6034_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6034_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, 1]>,<5,R![3, 1]>,<11,R![-3, 1]>],Snew); return Vf; end function;