// Make newform 8450.2.a.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8450_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_8450_2_a_l();". // 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_8450_2_a_l();". // 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_8450_a();" function MakeCharacter_8450_a() N := 8450; order := 1; char_gens := [677, 3551]; v := [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_8450_a_Hecke(Kf) return MakeCharacter_8450_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], [3], [0], [0], [3], [0], [-7], [-1], [-4], [4], [10], [-12], [5], [12], [4], [6], [4], [4], [5], [0], [11], [4], [-15], [11], [2], [12], [14], [-9], [8], [19], [10], [4], [9], [-5], [0], [2], [12], [1], [14], [-18], [-5], [16], [-18], [-15], [-16], [-6], [-11], [14], [4], [16], [6], [-20], [-23], [-19], [-10], [8], [24], [4], [-2], [-2], [-11], [-30], [-25], [12], [6], [-10], [-25], [27], [-25], [10], [6], [-4], [0], [24], [21], [-26], [-12], [-18], [33], [-23], [-9], [-16], [24], [-23], [30], [-39], [9], [-31], [10], [-14], [20], [10], [32], [-4], [-20], [-16], [-24], [-7], [-5], [26], [-13], [-14], [0], [13], [32], [19], [5], [-29], [-10], [-25], [-42], [-2], [22], [-36], [0], [18], [44], [-46], [-2], [-15], [-22], [10], [-44], [35], [31], [32], [10], [-50], [-10], [-14], [24], [8], [-22], [28], [-27], [5], [38], [4], [8], [-22], [-12], [-36], [-4], [21], [-10], [-10], [20], [21], [-13], [50], [40], [14], [55], [44], [44], [-6], [8], [-22], [-13], [-18], [-32], [-7], [36], [-13], [9], [24], [14], [-56], [45], [36], [5], [26], [30], [-17], [-16], [-9], [-55], [-26], [-22], [-6], [32], [-45], [8], [15], [-54], [28], [14], [-29], [41], [-34], [-9], [-27], [-51], [-6], [-25], [-2], [43], [4], [-41], [-24], [66], [20], [16], [-1], [36], [-34], [-46], [36], [10], [4], [-38], [-18], [-24], [11], [14], [69], [10], [-27], [50], [-26], [-22], [-2], [-59], [50], [-25], [66], [-21], [24], [-70], [-23], [-22], [37], [8], [-21], [4], [-22], [49], [-16], [13], [2], [-45], [13], [52], [50], [-57], [-46], [-52], [0], [13], [-54], [52], [-45], [-32], [-21], [42], [-27], [2], [9], [60], [-3], [40], [9], [8], [16], [-27], [11], [-82], [6], [23], [-48], [36], [-23], [-1], [-52], [38], [64], [-45], [-56], [-42], [23], [-4], [60], [48], [28], [-79], [40], [26], [-52], [64], [-18], [66], [36], [-81], [-59], [-4], [42], [26], [-16], [-48], [-43], [49], [52], [-22], [-1], [-35], [-7], [36], [16], [-60], [16], [18], [-50], [21], [28], [-36], [-26], [-29], [72], [-33], [15], [76], [53], [42], [-74], [10], [-14], [-43], [-4], [-10], [0], [-14], [48], [-92], [-81], [15], [36], [-38], [34], [47], [32], [34], [45], [-90], [-40], [24], [-44], [60], [5], [90], [-12], [-71], [-38], [-22], [16], [-56], [-81], [-42], [23], [-54], [90], [-70], [-45], [-30], [-36], [17], [51], [-82], [-14], [-59], [-72], [11], [-46], [-96], [-6], [-88], [63], [-2], [-21], [-39], [-46], [-28], [-27], [18], [39], [20], [30], [-86], [-62], [-4], [66], [79], [38], [21], [-23], [40], [25], [76], [61], [55], [-26], [-64], [-43], [-54], [25], [10], [46], [24], [42], [32], [77], [-1], [4], [13], [76], [55], [18], [-66], [-54], [77], [90], [28], [-38], [-24], [-27], [-91], [84], [12], [-65], [67], [-44]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8450_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8450_2_a_l();". // 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_8450_2_a_l(:prec:=1) chi := MakeCharacter_8450_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_8450_2_a_l();". // 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_8450_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8450_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-3, 1]>,<7,R![0, 1]>,<11,R![-3, 1]>,<17,R![7, 1]>,<31,R![-10, 1]>],Snew); return Vf; end function;