// Make newform 4704.2.a.bw in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4704_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_4704_2_a_bw();". // 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_4704_2_a_bw();". // 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, 2, -3, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [1, -2, -2, 1], [-3, -4, 2, 0], [-2, 4, 2, -1]]; Rf_basisdens := [1, 1, 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_4704_a();" function MakeCharacter_4704_a() N := 4704; order := 1; char_gens := [1471, 1765, 3137, 4609]; 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_4704_a_Hecke(Kf) return MakeCharacter_4704_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, 0, 0], [-1, 0, 0, 0], [-1, 0, 0, 1], [0, 0, 0, 0], [1, 0, -1, 0], [-2, 0, -1, -1], [-1, -1, 1, 0], [-2, -1, 1, 1], [3, 0, 1, 0], [0, 1, 1, -1], [2, 3, 1, 1], [0, -4, 0, 0], [-5, 3, -1, -2], [-2, 2, 0, 2], [4, -1, -1, -3], [0, 2, 0, -2], [0, 3, -1, -3], [-4, 0, 1, -1], [-2, -4, 2, 0], [3, -4, 1, 0], [-2, -1, 0, 2], [4, -2, -2, -2], [0, 2, -2, -2], [-7, -5, -1, 0], [-10, 3, 0, 2], [-1, 2, 0, 1], [6, 3, -3, 1], [3, 0, 1, 4], [-6, 2, 0, 2], [-2, 8, 0, 0], [-2, 0, -2, -4], [-4, -6, 2, -2], [4, 5, 1, 3], [-4, -8, 0, 0], [4, -10, 0, 2], [-10, 6, 0, 2], [0, 2, 1, -5], [-2, 8, 2, 0], [4, 3, 3, 1], [-7, 10, 0, -1], [7, -8, -3, -4], [-14, -6, 3, -1], [-5, -12, 1, 0], [0, 4, -2, 4], [4, 2, 0, 6], [4, 4, 0, -4], [-12, 0, 0, -4], [-4, -8, -4, 0], [-8, -9, 3, 1], [-10, -4, -3, 1], [0, -9, -1, 1], [3, 0, 1, 0], [-10, -5, -2, 4], [-4, 5, 5, 3], [-5, -3, 1, -4], [1, 8, -1, 4], [-11, -8, 0, 3], [6, 3, 5, 1], [0, 4, 6, 4], [-8, 3, -5, -3], [-10, 3, -3, -3], [-9, -4, -2, 3], [6, -11, -1, -5], [12, -3, -3, -1], [-4, 1, 0, -4], [0, -6, -4, -2], [-12, 0, -4, 0], [4, -2, 2, 2], [5, -8, 3, 0], [0, 12, -1, -3], [5, 1, 3, 0], [1, 4, -1, 4], [4, 2, -2, 6], [2, -4, -4, 4], [-8, 12, -4, -4], [8, -8, 0, 8], [12, 1, -3, -1], [8, -2, -1, -3], [-16, -1, -1, 1], [2, 5, 6, 0], [4, 13, -3, 3], [-4, 0, -2, -8], [7, 8, 5, 0], [0, 3, -4, -4], [24, 8, 0, 0], [-1, -4, 1, -8], [-18, 0, 0, 0], [10, 0, 4, 0], [-3, -4, -6, -7], [-10, 4, -2, 4], [-12, 9, 1, 7], [0, -5, -1, 1], [-26, 6, 0, 2], [-11, 8, -1, -4], [2, 10, 0, -2], [20, 2, -2, 6], [-3, 10, -2, 5], [-7, 9, 3, 4], [-16, 10, -2, -10], [-8, 4, 2, 4], [0, -2, 10, 6], [8, 2, 4, -2], [-8, 3, -1, 5], [-4, 9, 5, 7], [-12, -2, -2, -2], [-4, -13, 0, 4], [-24, 3, -1, -3], [15, -7, -1, 2], [11, -4, 1, 8], [4, -15, 0, -4], [8, 12, 4, -4], [2, -12, 2, 8], [-16, -11, 5, -5], [-12, 14, 6, 10], [-18, 4, -2, -4], [-24, -1, -1, -7], [14, -13, -3, -3], [-4, -5, 3, 1], [8, 5, 5, 3], [13, 4, -1, -4], [-20, 6, 5, 3], [6, 4, -6, -4], [17, 2, -4, -5], [15, -4, 1, 8], [4, -6, 2, 6], [0, -17, -1, 1], [-14, 18, 0, -6], [-8, 10, 2, 6], [22, -17, 1, 5], [6, -16, -3, -7], [-6, -12, 2, 4], [3, -12, -3, -4], [0, -20, 0, 0], [-6, -14, 0, 2], [9, -17, -5, -4], [16, -9, 0, 8], [17, -6, 4, -5], [4, 14, -2, -6], [11, -10, 4, 9], [2, -4, -4, 12], [-4, -6, -6, -10], [12, 22, -4, 2], [-16, -10, 2, -2], [35, 12, -3, 0], [2, 2, 1, 1], [0, -1, 3, 5], [16, 0, 7, 5], [-15, 3, -1, 0], [-10, -3, 7, 3], [27, -12, 1, 0], [6, 20, -2, 0], [11, -3, -1, 6], [-22, -6, -8, -2], [20, -7, -7, -5], [-10, 18, -4, -2], [-5, 16, 1, 0], [14, 6, 4, 6], [-3, -11, -3, -2], [32, -5, 2, -2], [-11, -10, 2, 1], [15, 8, 1, 0], [-2, 8, 4, 0], [6, -16, -2, -12], [-8, 4, 0, 4], [-28, 1, -3, 7], [20, 4, 0, 4], [-10, 10, 8, 10], [-4, 6, -3, 11], [12, -4, -2, -4], [15, 24, 2, -1], [3, 12, -3, -8], [0, 10, 3, 1], [-21, 8, 5, 4], [2, -14, -4, -10], [-12, 4, 0, -4], [5, 15, 1, -6], [-12, -10, 10, 2], [20, -5, -1, 5], [10, -5, 1, -7], [28, 10, -3, -5], [4, 2, -10, -10], [28, 12, -4, 4], [-20, 6, 6, 6], [9, -17, 5, 2], [-9, -4, -7, 4], [1, -24, -2, 1], [14, 8, -4, -8], [-12, -12, -4, 12], [2, 6, 8, 10], [-12, -6, -2, -2], [10, 7, -2, 8], [-7, 0, 3, 4], [-28, 16, 0, -4], [15, 16, 6, -5], [15, 8, -11, -4], [15, -3, 7, 10], [18, 0, 10, 8], [-36, -2, -2, -2], [-11, 1, -7, 2], [4, 27, 3, -7], [-4, -6, -4, -10], [-32, 6, 6, 10], [-10, -6, -7, 5], [-10, 1, 8, 2], [-20, 2, -6, -10], [17, 8, 2, 5], [-20, 8, -4, 0], [-31, 0, 7, 8], [6, -4, -12, -4], [10, -13, 9, 13], [-12, 16, -4, -8], [-7, 6, -6, 5], [2, 6, 12, 10], [-4, 2, 10, 6], [-16, -9, -5, -3], [-12, -3, -6, -6], [2, -14, -4, 10], [-23, 11, 5, 10], [15, 16, 9, 4], [-20, 17, -3, -1], [-26, 10, -4, 2], [34, -3, -5, -1], [-2, 16, -12, -8], [-36, -12, -4, 8], [8, 0, 4, 8], [-32, -8, -4, 4], [-15, -19, -3, 2], [-31, 0, -5, 8], [-2, 7, 9, 13], [-9, -12, 5, 12], [10, -10, -4, -10], [22, 1, -5, -1], [-34, 10, 0, 10], [22, 12, 12, 4], [-26, 17, -5, -1], [0, -2, -10, -14], [-6, -7, 12, 2], [4, -10, 0, 2], [-19, 4, 3, -8], [-20, -8, -12, 0], [5, 24, -5, -8], [16, -8, -4, 8], [-28, -20, 0, -4], [8, 0, -2, 8], [23, 1, -7, -8], [16, 5, 1, 7], [-2, 5, -1, 7], [8, 13, 1, 3], [12, 14, 6, -2], [21, 24, 3, 4], [-10, -12, 0, -4], [-3, -5, -7, -6], [-1, -4, -3, -16], [-22, -15, -8, -2], [43, 10, 8, -3], [11, -8, 1, -4], [4, 6, 6, -2], [6, -11, -1, 11], [-29, 0, -2, -9], [12, 21, 2, -6], [-28, -10, 2, -6], [-7, -16, -1, -8], [-2, 0, -5, 11], [20, 12, 3, -3], [-37, 1, 3, -6], [16, 0, -4, -8], [20, 14, 0, 10], [-13, 27, 5, 16], [32, 10, 2, -2], [12, 7, 3, -7], [-20, 10, 5, -5], [36, 12, 8, 4], [26, 21, -4, -6], [12, 4, 4, 0], [-10, 11, 6, -12], [-10, -1, 1, 5], [15, -32, 1, -8], [-20, 12, 2, 12], [-8, -4, 6, 12], [32, -17, -5, -7], [-28, 24, -4, 0], [6, -8, -6, 8], [52, -2, 2, -6], [-28, 2, 11, 13], [8, -2, -6, 2], [-1, -28, 1, -4], [-2, -16, -4, 16], [16, 2, -4, 6], [14, 7, 1, 13], [-7, -9, 7, -8], [-4, 18, 4, 14], [4, -12, -4, -4], [-20, -15, -11, 7], [8, 11, -9, -3], [14, 36, -4, -4], [-5, 6, 4, 9], [-32, -18, 6, -6], [3, 8, -4, -7], [0, 39, 3, 1], [4, 4, -4, 4], [-8, 19, -6, -10], [36, -17, -5, -7], [34, -16, -2, 8], [-15, -8, 7, -8], [-12, -6, 2, -22], [-4, 6, -6, 18], [-31, 4, 3, -4], [-42, -8, 7, 7], [-11, 20, 3, -4], [-4, -36, -6, -4], [-20, 0, 12, 8], [-16, 7, -1, 17], [4, -1, 3, 1], [10, 6, -8, -2], [35, -28, 1, 8], [-18, -9, -8, -6], [24, -2, 2, 18], [3, 4, -7, 8], [14, -13, 0, 18], [-16, -1, -1, -15], [-2, -5, -3, -11], [-50, -16, -2, 0], [-27, -8, -2, 5], [2, 10, 12, -6], [14, -8, 4, 16], [6, 5, -10, -20], [46, 0, -6, 8], [10, -3, 3, 3], [3, 12, 5, 12], [-4, 33, 5, 7], [-14, -16, -4, 0], [-16, 13, -11, -5], [-26, -5, 5, 9], [20, 23, 7, 1], [10, 20, -2, 8], [-28, -25, -9, 1], [-22, 8, -2, -12], [9, -5, 3, -20], [-42, 5, -8, 2], [6, 3, -11, -15], [-10, -12, -4, -4], [-4, 7, -13, -7], [-13, 28, 0, 5], [40, -24, -8, -16], [-20, -29, -1, -3], [31, 8, -7, 8], [-14, 6, 1, -7], [-2, -18, -4, 6], [-8, -12, 4, 12], [-11, 2, 0, -5], [50, -21, 1, 5], [-52, 16, -2, -8], [32, -10, 4, -14], [4, -20, -8, -4], [-18, -16, 16, 0], [-27, 15, 1, 10], [-40, -24, -8, 8], [24, 2, -2, 14], [14, 4, -4, -12], [-13, 24, -7, -8], [2, 4, 4, -20], [-1, -5, 19, 14], [-5, -20, 5, -4], [-23, -16, 3, 12], [0, -4, 0, 4], [26, -36, -2, -4], [-13, -32, 0, -11], [36, 6, 2, -6], [-8, -36, 6, -4], [-3, -20, 3, 8], [-22, -33, -3, 1], [13, -12, -9, -16], [12, -10, 8, 18], [38, -21, -3, -7], [44, 0, 2, 8], [4, 6, -10, 2], [45, 16, -9, 0], [44, -3, 8, 12], [-33, 5, 3, 6], [-62, -1, -4, -6], [-67, -2, 0, -5], [22, 32, -4, 0], [8, 8, 0, 8], [-30, 24, 4, 0], [0, 6, -6, -22], [-52, -6, -2, -2], [26, 6, 0, -2], [-42, 8, 1, 5], [-8, -14, -14, 6], [-24, -1, -5, 13], [2, -2, 8, -6], [-11, -16, 12, -1], [9, 24, 11, 4], [10, -13, 1, -3], [-13, -12, -3, 4], [-40, -10, 6, -14], [-12, 12, 0, -12], [5, 23, -15, -14], [-36, -6, 6, 14], [24, 27, -5, -3], [8, -22, 9, 19], [2, -12, -8, -4], [6, 12, -2, -12], [-37, 15, -5, -14], [23, -16, 8, 9], [0, -4, 4, -12], [36, -8, 6, 0], [6, -4, -4, -4], [20, -20, -12, -4], [4, 9, 9, 23], [52, -27, -2, -2], [-12, 21, -7, -21], [21, 4, -1, -4], [-22, 22, 12, 10], [-14, -2, 8, -6], [17, 16, -4, -5], [7, 12, -3, 0], [-10, -35, 7, -1], [-13, -35, -9, -2], [-8, -35, 9, -9], [-4, -1, 3, -23], [-24, -12, 11, 1], [13, 4, 7, -24], [-52, -6, 2, -2], [16, -19, 14, 10], [-13, 24, 2, 3], [13, 4, -1, 12], [4, -7, -3, 7], [32, -18, 2, 2], [28, -7, 1, -5]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4704_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4704_2_a_bw();". // 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_4704_2_a_bw(:prec:=4) chi := MakeCharacter_4704_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_4704_2_a_bw();". // 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_4704_2_a_bw( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4704_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![28, -24, -8, 4, 1]>,<11,R![16, 48, -20, -4, 1]>,<13,R![68, -80, -4, 8, 1]>,<19,R![64, -64, -8, 8, 1]>,<31,R![-64, 128, -40, -8, 1]>],Snew); return Vf; end function;