// Make newform 5184.2.a.bf in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5184_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_5184_2_a_bf();". // 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_5184_2_a_bf();". // 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_5184_a();" function MakeCharacter_5184_a() N := 5184; order := 1; char_gens := [2431, 325, 1217]; 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_5184_a_Hecke(Kf) return MakeCharacter_5184_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], [4], [2], [5], [2], [3], [1], [-6], [-2], [4], [8], [-1], [-7], [2], [-4], [-5], [0], [-13], [8], [3], [-8], [-12], [10], [-11], [12], [-6], [-5], [-8], [2], [-2], [-16], [3], [-13], [-18], [18], [-20], [-20], [-16], [18], [-12], [-2], [-6], [-3], [4], [-2], [28], [14], [-3], [-14], [21], [6], [-23], [21], [5], [6], [0], [12], [14], [18], [-28], [30], [-9], [18], [13], [18], [-4], [-25], [1], [-16], [-35], [-14], [0], [22], [1], [20], [-6], [32], [-5], [25], [-4], [-8], [30], [1], [8], [27], [31], [-7], [-18], [16], [-7], [18], [-22], [-7], [21], [24], [24], [3], [-12], [32], [33], [10], [-15], [11], [13], [3], [-9], [-6], [-40], [19], [-20], [-12], [45], [21], [28], [-5], [-15], [6], [-22], [12], [-32], [2], [-28], [15], [-8], [42], [20], [32], [30], [34], [49], [-6], [52], [-18], [30], [10], [-42], [-36], [24], [-17], [55], [-50], [-28], [-28], [-14], [-4], [-30], [10], [5], [-28], [-40], [-30], [-13], [36], [7], [-24], [-54], [6], [-58], [-28], [27], [39], [42], [20], [11], [36], [16], [-60], [-38], [-40], [-27], [-60], [-44], [19], [-34], [-14], [-55], [30], [32], [-52], [46], [-1], [14], [-19], [18], [32], [0], [-40], [-43], [48], [31], [-55], [-43], [-28], [-47], [-51], [31], [-52], [41], [-10], [18], [-62], [-8], [3], [5], [-6], [44], [37], [-35], [-49], [29], [18], [30], [12], [56], [9], [-44], [43], [-62], [-48], [46], [4], [-21], [62], [3], [22], [54], [-72], [62], [-40], [14], [20], [26], [57], [-16], [-2], [26], [-66], [39], [-2], [51], [-36], [72], [6], [49], [-8], [-28], [45], [28], [2], [-42], [-43], [12], [-6], [-14], [-15], [72], [4], [36], [13], [56], [57], [4], [2], [-30], [55], [-28], [5], [23], [18], [-20], [-44], [-61], [-6], [-17], [-12], [68], [58], [55], [-59], [42], [-14], [-10], [42], [-23], [30], [-38], [-44], [-28], [49], [36], [-33], [-27], [67], [-36], [30], [14], [6], [-84], [-4], [1], [58], [28], [44], [40], [-38], [12], [-60], [78], [52], [-24], [-52], [86], [-32], [72], [43], [9], [40], [-29], [18], [-27], [53], [-8], [-22], [6], [-47], [81], [-39], [-20], [38], [-28], [-8], [-92], [13], [25], [-28], [-30], [-71], [23], [-48], [52], [-38], [56], [-52], [34], [63], [-84], [23], [-52], [22], [-4], [65], [2], [60], [0], [-54], [-84], [84], [-58], [46], [38], [-39], [-68], [29], [-53], [-17], [-84], [38], [-11], [36], [85], [-36], [12], [62], [-26], [87], [-2], [14], [27], [42], [30], [-1], [20], [21], [23], [-92], [-90], [52], [-16], [-32], [-25], [-28], [-15], [-79], [-86], [54], [16], [51], [-17], [52], [-82], [-33], [-8], [54], [-10], [20], [-14], [-13], [45], [76], [103], [58], [39], [-57], [-2], [2], [46], [2], [67], [-52], [16], [28], [-16], [51], [2], [4], [-87], [38], [20], [60]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5184_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5184_2_a_bf();". // 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_5184_2_a_bf(:prec:=1) chi := MakeCharacter_5184_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_5184_2_a_bf();". // 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_5184_2_a_bf( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5184_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-4, 1]>,<7,R![-2, 1]>,<11,R![-5, 1]>],Snew); return Vf; end function;