// Make newform 4400.2.a.u in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4400_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_4400_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_4400_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_4400_a();" function MakeCharacter_4400_a() N := 4400; order := 1; char_gens := [2751, 3301, 177, 1201]; 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_4400_a_Hecke(Kf) return MakeCharacter_4400_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], [0], [-1], [1], [0], [1], [-1], [0], [-1], [1], [1], [0], [-6], [-8], [9], [-4], [-7], [4], [-5], [14], [-4], [-16], [-7], [-16], [-10], [-8], [-8], [10], [2], [16], [-21], [12], [-12], [17], [-10], [13], [-19], [3], [-22], [10], [-22], [-24], [13], [18], [-17], [-17], [10], [-12], [12], [1], [16], [0], [-6], [24], [21], [8], [6], [-8], [-22], [20], [16], [16], [13], [-18], [-1], [34], [-13], [18], [10], [-14], [-20], [0], [4], [-2], [-18], [6], [-14], [-29], [-32], [-8], [24], [0], [26], [-26], [-20], [-2], [-1], [9], [36], [5], [34], [-18], [27], [-28], [24], [8], [-2], [28], [3], [12], [8], [12], [24], [-7], [4], [-33], [-18], [5], [14], [-3], [-46], [-8], [0], [-21], [-1], [1], [-48], [27], [33], [28], [-35], [36], [15], [-40], [33], [18], [41], [18], [-6], [16], [-21], [29], [10], [-20], [-22], [45], [-30], [-54], [34], [17], [42], [-2], [14], [4], [-24], [-16], [53], [-2], [-52], [-38], [-10], [-41], [-48], [-7], [27], [26], [15], [10], [27], [35], [39], [25], [48], [0], [-42], [-24], [10], [0], [9], [43], [16], [62], [-3], [-12], [-5], [44], [-30], [-11], [1], [-14], [-37], [63], [-11], [-58], [-24], [20], [52], [0], [40], [-10], [50], [-52], [-2], [-38], [48], [-12], [34], [42], [33], [11], [40], [35], [-70], [-54], [25], [8], [-44], [-36], [58], [-17], [42], [16], [-49], [-34], [15], [-23], [42], [8], [31], [58], [-66], [33], [-52], [24], [-15], [6], [3], [24], [40], [-58], [37], [46], [-10], [4], [-35], [-29], [-14], [8], [-51], [15], [-52], [-62], [20], [30], [30], [-24], [-4], [-45], [33], [-62], [60], [48], [-5], [-36], [51], [-54], [10], [26], [7], [44], [-6], [17], [44], [-18], [-50], [-36], [5], [-28], [54], [-30], [83], [46], [-76], [-83], [-8], [12], [62], [-32], [-63], [67], [-76], [-44], [-63], [8], [-39], [-50], [-64], [35], [-48], [-24], [-60], [-75], [85], [-62], [-30], [-32], [-69], [76], [50], [55], [-40], [52], [20], [-78], [8], [12], [-64], [21], [-4], [-8], [36], [-72], [21], [-57], [-56], [-72], [-6], [-52], [20], [5], [56], [-42], [-50], [-49], [-65], [12], [28], [-41], [-86], [31], [30], [-76], [36], [-9], [41], [28], [-7], [-19], [-25], [-54], [-48], [-9], [-53], [45], [-11], [-46], [-41], [82], [-1], [63], [-4], [24], [-19], [-47], [51], [-15], [-27], [14], [64], [-88], [30], [72], [29], [-67], [-12], [-12], [-33], [-26], [-46], [-28], [-56], [-75], [36], [46], [-66], [42], [77], [48], [-89], [-26], [14], [88], [-57], [-100], [-38], [64], [3], [-74], [-14], [-11], [39], [-36], [-52], [-60], [6], [-74], [51], [-2], [-63], [-57], [-50], [24], [-38], [-22], [-66], [43], [68], [-21], [30], [-6], [48], [8], [8], [94], [-42], [18], [-70], [0], [-69], [-86], [58], [-42], [59], [16], [31], [79], [70], [14], [-18]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4400_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4400_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_4400_2_a_u(:prec:=1) chi := MakeCharacter_4400_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_4400_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_4400_2_a_u( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4400_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-1, 1]>,<7,R![1, 1]>,<13,R![0, 1]>],Snew); return Vf; end function;