// Make newform 2160.2.a.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2160_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_2160_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_2160_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_2160_a();" function MakeCharacter_2160_a() N := 2160; order := 1; char_gens := [271, 1621, 2081, 1297]; v := [1, 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_2160_a_Hecke(Kf) return MakeCharacter_2160_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], [0], [-1], [-2], [4], [-2], [-5], [5], [1], [2], [-7], [-6], [0], [-4], [4], [-9], [14], [-11], [-14], [0], [-12], [3], [-1], [0], [16], [-12], [-4], [-12], [-19], [-6], [-6], [-18], [17], [-12], [16], [-8], [16], [14], [-3], [13], [-4], [-19], [14], [10], [5], [16], [19], [10], [3], [-29], [-6], [-6], [11], [-24], [-27], [-24], [10], [9], [-4], [14], [-8], [-7], [-22], [-22], [8], [27], [-12], [22], [20], [-3], [-26], [18], [22], [-32], [23], [-27], [36], [-14], [30], [25], [36], [3], [14], [26], [17], [39], [-6], [28], [-42], [18], [-13], [28], [-40], [-6], [25], [-15], [-28], [-40], [-10], [22], [28], [10], [4], [10], [5], [-16], [-15], [9], [18], [-11], [36], [0], [-33], [-32], [-13], [0], [-6], [33], [-41], [18], [-42], [30], [2], [-9], [19], [10], [26], [46], [-32], [0], [15], [0], [3], [2], [30], [-35], [15], [40], [-3], [24], [-44], [-46], [20], [-39], [-2], [-16], [-28], [-21], [-41], [33], [-32], [54], [22], [-23], [-12], [-2], [-32], [8], [-56], [-34], [-53], [-6], [30], [14], [-2], [-3], [55], [-48], [35], [25], [-4], [-17], [-58], [-4], [-17], [-30], [-55], [-26], [-16], [-14], [28], [0], [50], [38], [-1], [-20], [-58], [-32], [-31], [34], [-34], [-39], [-5], [-56], [31], [-11], [-3], [24], [-14], [56], [48], [-28], [14], [-21], [6], [46], [44], [31], [-36], [4], [-50], [28], [-56], [36], [28], [45], [8], [-56], [57], [23], [50], [-55], [-34], [48], [3], [31], [-35], [-40], [34], [20], [4], [-3], [-32], [-10], [-18], [27], [9], [-3], [-50], [12], [4], [41], [-70], [61], [-2], [-10], [8], [74], [-52], [71], [66], [-22], [-33], [41], [-29], [0], [-7], [-6], [3], [-30], [-16], [5], [3], [44], [-59], [-28], [8], [78], [-54], [26], [23], [2], [24], [1], [-4], [34], [-11], [-19], [17], [-24], [72], [-3], [-43], [19], [-76], [10], [2], [54], [15], [-24], [8], [21], [-18], [66], [16], [-60], [1], [-54], [-56], [-42], [24], [15], [16], [-1], [87], [-32], [-12], [-5], [36], [-16], [55], [60], [0], [-58], [11], [-53], [40], [-72], [-2], [38], [-17], [32], [-60], [32], [83], [54], [71], [88], [-16], [81], [50], [62], [32], [52], [-64], [-36], [47], [75], [-26], [12], [56], [-57], [86], [79], [26], [-32], [7], [-52], [12], [-39], [20], [30], [-70], [16], [-70], [35], [-90], [10], [74], [-19], [-10], [98], [-40], [-90], [-60], [-2], [63], [-74], [-46], [-54], [-11], [-37], [36], [-63], [8], [-18], [-10], [-28], [-30], [-10], [28], [-19], [-18], [-33], [-60], [-44], [35], [78], [12], [40], [34], [3], [6], [-22], [74], [-16], [79], [18], [19], [-36], [18], [-30], [-46], [-6], [-102], [-27], [-26], [42], [-28], [78], [-74], [-27], [89], [-67], [-38], [-18], [22], [32], [-75], [99], [-60], [58], [43], [-14], [6], [33], [21], [18], [-17], [80]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2160_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2160_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_2160_2_a_e(:prec:=1) chi := MakeCharacter_2160_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_2160_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_2160_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2160_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![2, 1]>,<11,R![-4, 1]>,<13,R![2, 1]>,<17,R![5, 1]>],Snew); return Vf; end function;