// Make newform 4650.2.a.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4650_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_4650_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_4650_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_4650_a();" function MakeCharacter_4650_a() N := 4650; order := 1; char_gens := [3101, 2977, 1801]; 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_4650_a_Hecke(Kf) return MakeCharacter_4650_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], [-2], [2], [-4], [-1], [4], [-2], [3], [-1], [4], [-8], [13], [1], [6], [0], [-12], [-2], [5], [-2], [-3], [-12], [15], [1], [10], [-4], [1], [11], [-2], [-11], [-11], [-17], [-5], [16], [-17], [7], [-16], [8], [18], [6], [-16], [-5], [-10], [13], [-8], [-8], [-15], [-5], [-4], [14], [30], [4], [-22], [-6], [8], [1], [-21], [0], [-22], [-6], [-10], [-20], [-25], [20], [-14], [4], [18], [-14], [-5], [-10], [8], [3], [7], [-28], [-26], [6], [-14], [-30], [-20], [-9], [17], [-16], [-38], [-30], [17], [35], [0], [-18], [-35], [-3], [23], [11], [0], [-35], [9], [-18], [-16], [-15], [35], [-24], [11], [-13], [-6], [12], [-3], [-18], [-22], [-15], [30], [12], [24], [12], [11], [-25], [-18], [-35], [24], [-24], [-36], [33], [42], [-25], [-9], [-2], [42], [-30], [-6], [30], [-5], [-21], [-2], [36], [2], [11], [1], [51], [44], [-22], [9], [-12], [42], [-4], [6], [-16], [7], [21], [-32], [-21], [-6], [-47], [-5], [-8], [7], [-24], [40], [38], [18], [22], [21], [48], [14], [40], [27], [-62], [52], [37], [-6], [6], [21], [-20], [14], [0], [17], [10], [50], [-52], [-16], [-38], [6], [2], [12], [17], [-6], [3], [31], [-25], [56], [46], [43], [32], [8], [-52], [54], [-4], [31], [-36], [19], [-24], [-44], [18], [-16], [18], [55], [5], [-53], [0], [48], [-30], [-28], [-10], [-45], [6], [12], [14], [-1], [-28], [-6], [60], [-52], [-46], [-38], [28], [16], [28], [-46], [-40], [-36], [59], [-1], [-10], [-68], [66], [-66], [47], [-52], [71], [-66], [-12], [-8], [39], [-23], [-16], [-10], [61], [-24], [63], [60], [11], [-8], [34], [56], [-58], [-49], [6], [5], [51], [-31], [60], [-65], [-74], [52], [-28], [-2], [-34], [16], [-20], [62], [16], [0], [31], [-16], [-14], [-77], [-7], [-16], [1], [26], [72], [-78], [7], [-50], [-72], [-63], [8], [-48], [-2], [37], [-46], [9], [0], [-15], [-42], [-16], [16], [53], [13], [-58], [34], [37], [-39], [-3], [16], [4], [-29], [-3], [72], [35], [-62], [73], [-34], [-75], [-22], [-47], [-48], [40], [40], [-34], [26], [67], [3], [38], [30], [-16], [64], [56], [26], [-86], [60], [-42], [-29], [-54], [0], [-70], [-70], [20], [-20], [-72], [-77], [-43], [-64], [21], [-15], [-3], [24], [-45], [18], [-17], [66], [-90], [4], [-72], [-6], [-71], [74], [-39], [-84], [-32], [41], [-9], [-22], [-78], [38], [29], [38], [4], [92], [48], [78], [-96], [-10], [23], [0], [-10], [-26], [-72], [72], [-86], [30], [70], [43], [71], [-23], [70], [28], [-93], [90], [-70], [-17], [-54], [20], [-89], [-9], [86], [-45], [26], [-16], [82], [52], [19], [20], [-32], [-8], [28], [63], [14], [-25], [-19], [4], [-8], [-26], [-68], [56], [57], [-57], [-104], [92], [8], [28], [0], [-90], [92], [-12], [-33], [101], [-42], [99], [9], [-9], [30]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4650_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4650_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_4650_2_a_e(:prec:=1) chi := MakeCharacter_4650_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_4650_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_4650_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4650_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![2, 1]>,<11,R![-2, 1]>,<13,R![4, 1]>,<19,R![-4, 1]>],Snew); return Vf; end function;