// Make newform 1050.2.a.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1050_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_1050_2_a_l();". // 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_1050_2_a_l();". // 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_1050_a();" function MakeCharacter_1050_a() N := 1050; order := 1; char_gens := [701, 127, 451]; 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_1050_a_Hecke(Kf) return MakeCharacter_1050_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], [0], [-1], [6], [1], [-3], [-4], [3], [3], [5], [10], [9], [1], [0], [-9], [9], [11], [4], [-12], [10], [-10], [-9], [-6], [-14], [-12], [-17], [-18], [8], [6], [10], [0], [0], [-16], [-9], [2], [10], [-5], [-18], [12], [18], [2], [-15], [22], [15], [8], [-25], [19], [-3], [-22], [-12], [0], [2], [-27], [-15], [-9], [-18], [20], [10], [-24], [4], [0], [-2], [-18], [10], [27], [-19], [13], [-18], [17], [30], [-3], [37], [-8], [35], [24], [-6], [-29], [-12], [38], [-21], [-34], [-15], [-38], [41], [-24], [24], [25], [12], [-32], [3], [24], [-2], [42], [-19], [-6], [-24], [-21], [-20], [20], [-17], [18], [-21], [-6], [-13], [34], [-33], [18], [9], [-4], [28], [-2], [18], [26], [-22], [-18], [-14], [-6], [-18], [42], [38], [37], [12], [-18], [-34], [3], [-40], [30], [7], [-29], [-37], [9], [8], [-14], [18], [-4], [48], [-26], [6], [0], [-34], [-42], [34], [-6], [29], [36], [19], [-18], [-40], [48], [-32], [33], [-5], [-54], [-53], [15], [20], [-27], [-2], [24], [24], [-24], [28], [-12], [-42], [-24], [-16], [-26], [-25], [-60], [18], [-43], [-39], [1], [-28], [-3], [44], [33], [55], [35], [10], [21], [22], [15], [24], [30], [-20], [-26], [41], [12], [4], [12], [56], [-42], [-60], [-57], [5], [10], [30], [36], [-6], [-37], [31], [-10], [-27], [48], [17], [-42], [-54], [62], [10], [30], [-26], [39], [6], [56], [28], [-18], [21], [-33], [32], [-55], [-30], [16], [-60], [-22], [-30], [-24], [28], [30], [40], [20], [-4], [66], [4], [12], [-22], [6], [-6], [-30], [-12], [2], [61], [2], [-15], [36], [7], [-12], [-4], [-72], [-32], [21], [21], [-10], [-12], [30], [2], [58], [-42], [34], [-56], [36], [-19], [34], [-63], [-22], [75], [-63], [13], [-69], [41], [7], [4], [-4], [76], [76], [48], [-46], [-25], [-51], [0], [-16], [12], [-37], [-32], [-39], [13], [-42], [11], [-30], [-30], [-60], [-36], [-24], [34], [-24], [35], [30], [33], [-26], [-68], [63], [14], [-24], [-52], [-53], [18], [65], [63], [-38], [-18], [-33], [54], [-56], [15], [38], [12], [69], [-8], [-12], [14], [58], [54], [16], [-30], [-28], [-25], [34], [0], [27], [-46], [-69], [35], [-39], [11], [-12], [68], [69], [14], [-29], [10], [-66], [72], [8], [-15], [48], [-94], [-29], [-48], [-18], [32], [-11], [-15], [13], [-58], [-69], [6], [63], [-66], [63], [4], [-42], [48], [-84], [70], [-17], [-6], [16], [17], [72], [-16], [3], [-15], [5], [-92], [45], [24], [64], [27], [4], [30], [24], [-20], [0], [68], [-96], [2], [-86], [52], [-66], [-73], [-36], [-6], [58], [-15], [-26], [23], [51], [92], [30], [-13], [90], [-32], [39], [-48], [-13], [-50], [-102], [-86], [-24], [-8], [90], [84], [71], [13], [36], [57], [25], [105], [-78], [-99], [-11], [9], [-21], [16], [0], [30], [78], [8], [-18]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1050_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1050_2_a_l();". // 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_1050_2_a_l(:prec:=1) chi := MakeCharacter_1050_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_1050_2_a_l();". // 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_1050_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1050_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-6, 1]>,<13,R![-1, 1]>,<17,R![3, 1]>,<19,R![4, 1]>],Snew); return Vf; end function;