// Make newform 552.2.a.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_552_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_552_2_a_f();". // 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_552_2_a_f();". // 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 poly := [-1, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0], [-1, 2]]; Rf_basisdens := [1, 1]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_552_a();" function MakeCharacter_552_a() N := 552; order := 1; char_gens := [415, 277, 185, 97]; 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_552_a_Hecke(Kf) return MakeCharacter_552_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, 0], [1, 1], [2, 0], [1, 1], [0, -2], [-2, -2], [1, -1], [1, 0], [-4, -2], [6, -2], [1, 3], [-2, 0], [3, 1], [0, -4], [1, 5], [-4, 0], [-5, -3], [1, 3], [0, 4], [0, 2], [14, 0], [-9, -1], [-8, -4], [0, 2], [0, -2], [0, 2], [-9, -1], [-5, 1], [-4, 0], [8, -4], [-6, -2], [-6, 6], [0, 4], [-3, 5], [6, 2], [-7, 3], [-10, -2], [-8, 4], [8, 2], [-14, -2], [3, 9], [22, 2], [6, -8], [6, -8], [-14, -4], [-6, -6], [14, 2], [1, -3], [-17, -3], [26, 0], [4, 4], [4, -2], [-5, -9], [14, 0], [12, 0], [6, 8], [12, 8], [8, 2], [4, 0], [-1, 9], [5, -7], [6, 6], [24, 0], [20, -6], [12, -10], [22, -2], [-6, 12], [-2, 10], [-14, 0], [2, 4], [2, -10], [18, 4], [-3, -1], [-23, -1], [-12, 0], [-25, 3], [-30, -4], [2, -6], [22, 0], [-21, 3], [-7, -1], [14, -10], [-16, 6], [-20, -4], [0, -12], [-10, 0], [20, -10], [18, -4], [-8, 8], [27, -1], [-12, -4], [-16, 8], [-6, -10], [-14, 2], [-2, 14], [8, 2], [-10, 10], [1, 3], [2, 8], [14, 6], [-5, -1], [-23, -3], [-34, 6], [9, 3], [-4, -14], [42, -2], [-2, 16], [0, 8], [4, 2], [2, 6], [-21, -3], [-4, -16], [-9, 17], [16, 2], [-18, 6], [27, 1], [-24, -8], [36, 2], [-1, 19], [7, 1], [8, 6], [1, 9], [-6, -10], [42, 2], [-37, -5], [1, 7], [-24, 8], [-10, -12], [29, 3], [-32, -4], [32, -4], [32, 6], [-21, 1], [-10, -8], [48, -2], [7, -17], [-5, 1], [7, -5], [-2, 0], [22, 14], [20, 10], [-10, -10], [27, -1], [18, -16], [-16, -4], [54, 0], [-6, 16], [-20, -16], [16, 8], [-24, 10], [-24, 4], [22, -10], [20, 0], [39, -7], [-2, 14], [18, 4], [-14, -4], [18, -16], [-11, -11], [14, -6], [14, -10], [-18, 14], [17, -3], [-14, 6], [-2, 14], [-2, 14], [-18, 12]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_552_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_552_2_a_f();". // 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_552_2_a_f(:prec:=2) chi := MakeCharacter_552_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(997) - 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_552_2_a_f();". // 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_552_2_a_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_552_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-4, -2, 1]>,<7,R![-2, 1]>],Snew); return Vf; end function;