// Make newform 8036.2.a.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8036_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_8036_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_8036_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_8036_a();" function MakeCharacter_8036_a() N := 8036; order := 1; char_gens := [4019, 493, 785]; 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_8036_a_Hecke(Kf) return MakeCharacter_8036_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], [1], [1], [0], [-5], [2], [3], [3], [-1], [-10], [11], [7], [1], [8], [7], [-11], [7], [1], [7], [-12], [5], [9], [0], [3], [-2], [11], [1], [-13], [17], [6], [4], [-3], [9], [20], [-7], [-5], [3], [-17], [-16], [21], [7], [22], [1], [17], [26], [21], [-12], [-8], [9], [27], [13], [-16], [-19], [-24], [7], [19], [-11], [25], [23], [-14], [-17], [18], [-12], [1], [11], [17], [-3], [-14], [-9], [2], [9], [-29], [15], [19], [-20], [27], [3], [-29], [-13], [-35], [-12], [-18], [-35], [-14], [7], [39], [22], [5], [-2], [8], [33], [17], [-35], [-20], [-3], [-40], [-5], [27], [-39], [11], [-4], [33], [25], [15], [15], [39], [-36], [-5], [9], [22], [-35], [27], [-6], [11], [0], [-15], [36], [19], [9], [-4], [-51], [34], [-15], [31], [-9], [50], [-19], [11], [-32], [-3], [-15], [-48], [-7], [-38], [49], [10], [19], [47], [-2], [-31], [12], [-17], [-5], [-12], [-51], [-8], [14], [13], [-11], [-45], [11], [50], [-32], [43], [17], [-28], [5], [-17], [-6], [-27], [15], [30], [4], [11], [-7], [11], [-13], [-37], [-14], [39], [-33], [-34], [-3], [-21], [41], [-2], [12], [43], [8], [27], [-25], [16], [-42], [45], [-27], [-47], [23], [-17], [-43], [21], [25], [20], [-53], [9], [-15], [25], [13], [21], [18], [3], [-9], [56], [23], [3], [36], [-61], [9], [-11], [30], [71], [-39], [42], [-8], [33], [-39], [-55], [19], [33], [-23], [-6], [-71], [-48], [21], [-59], [-52], [22], [9], [-67], [71], [15], [-41], [-55], [-48], [-45], [44], [-47], [-49], [-11], [20], [-8], [39], [-37], [-31], [-13], [2], [13], [16], [-7], [-15], [0], [-22], [-7], [-55], [-54], [51], [-17], [21], [77], [-54], [-13], [49], [-48], [-37], [10], [-11], [-3], [54], [-22], [-20], [-51], [-5], [17], [-15], [31], [62], [37], [-39], [5], [-23], [75], [37], [-65], [24], [-2], [-61], [-47], [-11], [-6], [45], [-46], [-79], [3], [-51], [28], [14], [83], [-61], [78], [-21], [-20], [-39], [9], [-7], [24], [-49], [-42], [-63], [22], [-27], [-19], [63], [-3], [63], [-39], [-44], [-79], [84], [45], [-46], [-18], [-45], [-69], [-10], [72], [23], [63], [1], [-21], [69], [50], [5], [-9], [16], [-55], [-3], [8], [-30], [69], [38], [-31], [41], [14], [6], [-80], [-13], [-20], [17], [15], [56], [81], [-77], [59], [30], [-65], [-23], [30], [21], [-49], [11], [16], [-26], [-69], [-81], [3], [-75], [49], [-30], [-19], [-10], [79], [19], [67], [-66], [67], [-43], [19], [-84], [-55], [35], [38], [-9], [-42], [-32], [-9], [-4], [-9], [71], [55], [99], [-24], [78], [-69], [99], [-3], [67], [57], [45], [42], [100], [-27], [45], [69], [-37], [-93], [-35], [35], [39], [-62], [-7], [-63], [29], [-9], [8], [85], [102], [-27], [-81], [25], [-34], [-103], [-33], [-77], [48], [0], [58], [-83], [81], [-46], [89], [-39]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8036_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8036_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_8036_2_a_e(:prec:=1) chi := MakeCharacter_8036_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_8036_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_8036_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8036_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-1, 1]>,<5,R![-1, 1]>,<11,R![5, 1]>],Snew); return Vf; end function;