// Make newform 4410.2.a.ba 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_ba();". // 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_ba();". // 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], [1], [-1], [0], [3], [-7], [8], [2], [11], [-11], [8], [-5], [11], [4], [0], [0], [6], [6], [-8], [8], [-10], [16], [0], [16], [10], [6], [-6], [-17], [-5], [18], [-20], [0], [6], [7], [16], [-3], [-15], [19], [24], [6], [22], [1], [24], [5], [-12], [8], [-14], [-18], [18], [-7], [13], [-20], [-24], [20], [-32], [-22], [1], [14], [27], [-20], [12], [12], [18], [-13], [-12], [-14], [-12], [24], [-4], [-25], [6], [-19], [35], [-10], [-14], [5], [-2], [-5], [22], [0], [32], [-32], [-16], [-11], [-18], [-12], [-13], [0], [22], [-16], [-20], [-32], [36], [-34], [-33], [-2], [-10], [16], [33], [-38], [9], [-32], [14], [-18], [16], [-2], [26], [37], [-41], [-28], [-29], [-26], [3], [10], [-17], [-7], [-36], [24], [28], [-13], [-4], [12], [18], [44], [2], [-11], [-11], [-19], [-49], [26], [2], [27], [5], [33], [-22], [-18], [39], [-9], [-42], [-24], [10], [8], [-40], [41], [-14], [-12], [53], [17], [-53], [-42], [44], [-10], [2], [-36], [-21], [-12], [54], [-22], [20], [-20], [-45], [-54], [49], [22], [-14], [11], [45], [36], [2], [10], [-44], [-20], [14], [-7], [-10], [-11], [-22], [57], [23], [-9], [-10], [9], [-24], [-23], [-24], [25], [-42], [30], [6], [35], [22], [-20], [-16], [29], [46], [14], [4], [-32], [10], [-62], [-13], [-32], [27], [20], [-44], [-15], [1], [16], [34], [32], [18], [-40], [-7], [-17], [-69], [3], [-31], [44], [-38], [-25], [-49], [48], [46], [38], [16], [-17], [4], [-19], [-37], [10], [-55], [58], [-56], [22], [39], [43], [26], [-24], [49], [8], [-46], [-18], [18], [-55], [-25], [60], [-3], [-14], [-27], [-27], [17], [66], [55], [38], [2], [5], [-8], [15], [18], [34], [54], [16], [41], [72], [25], [14], [65], [56], [-66], [-14], [-48], [66], [39], [-72], [20], [-31], [-27], [1], [-76], [-33], [-64], [22], [-60], [-16], [69], [-76], [61], [68], [6], [36], [35], [7], [0], [26], [-6], [-15], [42], [-28], [-3], [-46], [42], [-44], [38], [44], [10], [-6], [-31], [-45], [10], [-18], [-2], [-57], [91], [-89], [12], [80], [-74], [-60], [-22], [-6], [37], [24], [-59], [9], [74], [-48], [-54], [54], [38], [-30], [12], [3], [74], [-74], [36], [11], [17], [14], [-30], [90], [70], [58], [-12], [-76], [-58], [-12], [47], [-7], [14], [-42], [-17], [66], [-54], [-36], [24], [-24], [-20], [29], [-22], [-48], [25], [42], [22], [37], [45], [-93], [57], [-63], [-36], [90], [10], [-34], [-15], [44], [48], [-14], [34], [6], [12], [33], [12], [89], [9], [20], [-63], [32], [-56], [22], [54], [-36], [20], [46], [94], [8], [69], [81], [42], [-40], [-6], [15], [-84], [90], [-36], [103], [66], [84], [-81], [-48], [-14], [-84], [-49], [36], [10], [80], [9], [-92], [1], [-106], [-77], [24], [-72], [-64], [-77], [-32], [-69], [-19], [90]]; 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_ba();". // 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_ba(: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_ba();". // 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_ba( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4410_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-1, 1]>,<13,R![1, 1]>,<17,R![0, 1]>,<19,R![-3, 1]>,<29,R![-8, 1]>,<31,R![-2, 1]>],Snew); return Vf; end function;