// Make newform 4410.2.a.u in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4410_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_4410_2_a_u();". // 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_4410_2_a_u();". // 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_4410_a();" function MakeCharacter_4410_a() N := 4410; order := 1; char_gens := [3431, 2647, 1081]; 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_4410_a_Hecke(Kf) return MakeCharacter_4410_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], [0], [1], [0], [5], [5], [-4], [7], [-1], [0], [2], [1], [5], [12], [-11], [9], [4], [-4], [-12], [-2], [-10], [-12], [-12], [14], [8], [0], [-8], [2], [-2], [14], [9], [-9], [2], [4], [-12], [14], [-11], [-24], [11], [-13], [23], [-20], [-14], [10], [3], [4], [17], [12], [-12], [-10], [14], [22], [-15], [1], [-16], [0], [-24], [0], [14], [-7], [10], [9], [-8], [8], [-16], [22], [31], [-16], [18], [4], [24], [20], [-7], [-22], [1], [21], [-18], [14], [21], [-10], [15], [10], [-12], [-24], [40], [28], [29], [-14], [-4], [-19], [-20], [18], [0], [36], [40], [-20], [-10], [-33], [22], [2], [28], [-37], [-18], [-39], [40], [-26], [-30], [36], [10], [10], [27], [-43], [8], [-25], [6], [-21], [-34], [-23], [19], [20], [40], [-4], [33], [4], [-28], [-30], [28], [-6], [3], [-9], [-15], [-15], [-50], [-34], [-37], [5], [-5], [-2], [-54], [-25], [-21], [-10], [40], [6], [52], [12], [11], [2], [4], [-45], [-37], [3], [-50], [-44], [-18], [-30], [28], [19], [0], [-6], [42], [-12], [4], [15], [30], [-25], [-38], [22], [-37], [63], [-4], [34], [-14], [44], [40], [-2], [5], [-30], [19], [54], [-17], [-5], [-11], [38], [47], [-4], [-5], [16], [-7], [14], [-26], [-26], [-17], [-14], [-48], [-12], [-43], [10], [18], [-36], [-48], [-30], [-26], [-21], [-40], [-47], [-48], [-48], [17], [-3], [4], [-30], [16], [18], [24], [-55], [17], [11], [37], [-37], [-52], [6], [39], [1], [8], [10], [26], [-36], [-55], [-60], [-41], [7], [30], [-15], [-74], [-16], [62], [-35], [47], [-22], [12], [-43], [48], [-34], [26], [-14], [31], [51], [4], [3], [-2], [69], [-45], [-71], [30], [27], [2], [-30], [-33], [-44], [25], [2], [-46], [2], [16], [-3], [60], [9], [2], [51], [8], [22], [-30], [-72], [30], [-47], [-4], [-28], [1], [-23], [-9], [-4], [65], [8], [46], [84], [56], [-41], [-28], [-51], [56], [58], [-20], [-57], [45], [-80], [22], [-42], [-59], [62], [28], [-25], [-34], [-74], [68], [2], [-12], [66], [66], [-37], [69], [-2], [-18], [54], [33], [27], [-69], [-76], [56], [78], [68], [30], [-66], [91], [40], [53], [-11], [42], [-48], [30], [-42], [90], [-50], [52], [23], [18], [-38], [-48], [67], [-1], [-14], [-22], [74], [34], [62], [-52], [72], [34], [8], [-59], [-43], [-2], [-26], [-55], [30], [-10], [-12], [80], [-36], [68], [39], [-22], [-48], [-11], [18], [34], [-73], [3], [-77], [-43], [21], [4], [-50], [-38], [2], [-11], [-40], [28], [-14], [-54], [22], [36], [-49], [-8], [-35], [-25], [44], [83], [88], [-24], [-50], [-22], [-44], [20], [-10], [-74], [-84], [29], [-67], [30], [-28], [10], [-23], [12], [-66], [20], [93], [-46], [-48], [-5], [36], [-82], [72], [67], [12], [-14], [-36], [47], [36], [31], [-34], [9], [48], [-48], [40], [-15], [28], [-93], [-7], [66]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4410_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4410_2_a_u();". // 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_4410_2_a_u(:prec:=1) chi := MakeCharacter_4410_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_4410_2_a_u();". // 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_4410_2_a_u( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4410_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-5, 1]>,<13,R![-5, 1]>,<17,R![4, 1]>,<19,R![-7, 1]>,<29,R![0, 1]>,<31,R![-2, 1]>],Snew); return Vf; end function;