// Make newform 5184.2.a.w in Magma, downloaded from the LMFDB on 28 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_w();". // 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_w();". // 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], [1], [2], [2], [-1], [-3], [-2], [-6], [1], [-8], [-1], [2], [-10], [-4], [-10], [4], [-9], [14], [-10], [-9], [10], [12], [-11], [-2], [-6], [-12], [16], [-5], [17], [-2], [14], [9], [20], [21], [12], [-17], [4], [2], [21], [0], [-14], [-18], [-9], [-11], [16], [-2], [2], [6], [7], [-15], [-12], [-5], [-18], [29], [12], [-15], [-6], [2], [9], [-16], [21], [0], [0], [7], [-3], [14], [14], [10], [-34], [-2], [34], [-24], [10], [-8], [14], [-18], [-13], [25], [-29], [-40], [7], [-12], [19], [-28], [12], [-14], [-37], [30], [28], [-28], [42], [-16], [-34], [-6], [0], [-18], [18], [30], [-13], [12], [-11], [36], [5], [-20], [27], [-30], [-15], [32], [-17], [22], [-6], [-15], [48], [-20], [-35], [-36], [-6], [-34], [-30], [31], [23], [-34], [-36], [10], [45], [-1], [14], [6], [-14], [-8], [24], [22], [42], [-39], [31], [9], [-18], [-39], [37], [16], [-23], [8], [-22], [-38], [-22], [42], [13], [44], [-46], [35], [42], [20], [-54], [-8], [30], [42], [-27], [-25], [-43], [54], [21], [-18], [38], [2], [-24], [-38], [15], [43], [14], [-24], [-21], [4], [-5], [-58], [-38], [-58], [45], [-52], [59], [28], [38], [59], [26], [-36], [5], [-45], [14], [26], [-30], [-5], [56], [-22], [-55], [22], [-42], [55], [-22], [-7], [-34], [-63], [22], [22], [15], [14], [54], [50], [28], [10], [26], [14], [-15], [42], [-6], [-64], [-42], [-38], [-23], [-44], [-15], [-5], [-50], [57], [-4], [24], [-5], [18], [66], [44], [-4], [47], [-46], [-10], [-15], [-64], [-68], [-13], [-54], [-24], [-38], [30], [6], [36], [42], [-14], [34], [2], [54], [28], [-16], [-57], [-7], [12], [15], [73], [-18], [-9], [58], [57], [-2], [32], [0], [-41], [-34], [-18], [16], [-19], [17], [-16], [-27], [31], [58], [-58], [-24], [-65], [-36], [-16], [-17], [-17], [4], [-60], [40], [56], [-33], [4], [-24], [31], [22], [-46], [13], [54], [42], [-15], [-8], [87], [21], [-64], [57], [-18], [-28], [43], [-62], [34], [-46], [-56], [-26], [-18], [-30], [24], [19], [-6], [-58], [-31], [-74], [12], [-50], [-60], [16], [-29], [66], [84], [-58], [49], [-88], [66], [-29], [-78], [42], [22], [-34], [26], [37], [16], [34], [34], [-28], [15], [37], [-58], [-12], [-5], [-38], [-19], [32], [-71], [-78], [-93], [56], [80], [-23], [-94], [-25], [53], [-42], [15], [-75], [36], [-18], [-22], [-2], [-85], [-75], [34], [-28], [-92], [79], [78], [-88], [-38], [-30], [-38], [-48], [21], [68], [-65], [30], [76], [23], [-18], [-81], [-87], [41], [-28], [9], [14], [-62], [72], [19], [74], [64], [-85], [53], [-78], [2], [-92], [39], [94], [-27], [-38], [-23], [-1], [-102], [70], [93], [-31], [38], [91], [-79], [-6], [-32], [7], [25], [78], [36], [-5], [-43], [-32], [-64], [1], [-52], [-62], [-26], [68], [-48], [23], [37], [48], [-43], [-16], [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_w();". // 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_w(: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_w();". // 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_w( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5184_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-1, 1]>,<7,R![-2, 1]>,<11,R![-2, 1]>],Snew); return Vf; end function;