// Make newform 4830.2.a.bw in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4830_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_4830_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_4830_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 := [2, -4, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [-1, 2, 0], [-6, 0, 2]]; Rf_basisdens := [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_4830_a();" function MakeCharacter_4830_a() N := 4830; order := 1; char_gens := [3221, 967, 2761, 1891]; 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_4830_a_Hecke(Kf) return MakeCharacter_4830_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, 0, 0], [-1, 0, 0], [-1, 0, 0], [-1, 0, 0], [-1, 1, 0], [-1, -1, 1], [-2, 0, -1], [2, 2, -1], [1, 0, 0], [-3, 1, 0], [4, 0, 1], [-6, 0, 0], [0, -2, 0], [-3, -1, 0], [-2, 2, 1], [-6, 0, 0], [-4, 0, 0], [0, 2, 0], [5, -1, 0], [1, -1, 2], [2, 0, 2], [2, -2, -2], [-2, 2, -1], [-1, -1, -1], [1, 1, -3], [3, -1, 1], [-2, 2, -2], [-6, 2, 0], [6, 0, 0], [6, 0, 2], [1, 3, 0], [-4, 0, 0], [-2, 0, -4], [4, 4, 0], [-2, -4, 0], [2, -2, -2], [8, -2, 2], [-8, 0, 0], [4, -4, 3], [2, 4, -1], [-4, 0, 2], [12, -2, 4], [-4, 0, -2], [8, 2, 2], [2, 0, 2], [-2, 2, 2], [10, 2, -2], [13, 3, -1], [8, 0, 1], [14, 4, -2], [6, 0, 2], [1, -1, 2], [16, -2, -1], [5, -1, -3], [12, -2, 2], [-14, -2, -2], [1, 1, -3], [8, -4, 1], [1, 5, -2], [7, 7, 0], [7, -3, -1], [4, -2, -2], [-2, 6, -2], [11, -3, -1], [23, 3, -1], [-16, 2, -2], [2, 2, -2], [3, 3, -4], [-8, -4, 0], [20, 2, -1], [14, 4, -2], [-12, -8, 2], [-6, 6, 2], [14, -4, 4], [14, -2, 0], [6, -2, 0], [14, 4, -2], [-9, -9, 5], [-19, 1, 0], [0, -6, 2], [15, -3, 5], [22, 0, -2], [2, 2, 2], [21, -3, -3], [14, -2, 3], [0, 4, 4], [14, 4, -2], [-7, -3, 6], [15, -5, 5], [27, 1, 2], [-6, -2, -3], [2, 6, -2], [7, 5, -2], [8, -4, 6], [-22, 2, -2], [26, 2, -2], [-1, -5, 1], [-9, 7, -5], [-9, 5, 3], [0, -2, 8], [-20, -4, 4], [30, 4, 0], [18, 6, -1], [1, 5, 4], [14, -2, 6], [-16, 2, -2], [12, 0, -4], [20, -2, -2], [-13, -3, 4], [16, 2, 0], [13, -5, 1], [10, -8, -2], [0, -2, -4], [-6, -6, 1], [-2, 2, 2], [19, 11, -4], [-1, -3, 1], [-10, -6, 7], [-4, -2, 4], [7, 9, -2], [-26, 4, -6], [30, 4, -2], [-2, -4, 0], [-4, 0, 4], [8, -8, -2], [-16, 2, 0], [10, 4, -6], [1, -1, -7], [-14, -2, -2], [2, 4, -2], [8, 4, 0], [-2, 10, -2], [-12, 4, -8], [2, 0, -2], [8, 6, -6], [10, 4, 3], [0, -6, 0], [-5, 9, -7], [-26, 4, -2], [12, -2, -6], [28, -4, 4], [-13, 3, -10], [-19, 7, 0], [2, 2, 4], [-8, -10, 1], [12, 12, -8], [-5, -5, 1], [10, 0, 2], [-12, 4, -2], [2, -10, -2], [-11, 1, -8], [-5, -5, 5], [0, -8, -4], [-14, 6, 3], [-19, -9, 2], [-18, -10, 2], [-12, -4, -4], [6, 0, 0], [-35, 5, -5], [-10, -4, 4], [-6, -6, 6], [-14, -4, 6], [-25, 5, -10], [5, -1, 5], [36, 2, -4], [2, -6, 4], [-22, 6, -2], [-25, -1, 1], [-14, 0, 4], [-18, 8, 5], [21, 3, 4], [22, 0, -5], [-4, 0, 2], [9, 13, -4], [-18, 6, -1], [-27, -7, -1], [-8, 4, 0], [-9, -1, -4], [36, -4, -4], [2, 0, 4], [-33, 5, 4], [1, 3, -3], [-7, -3, 0], [-8, -6, 8], [-20, -4, 8], [6, -4, 10], [13, 9, -2], [-15, 3, -7], [-18, -4, 0], [-19, -5, -1], [28, 14, -4], [-18, -2, 2], [-22, -6, 4], [11, -9, -1], [-4, -8, 10], [8, -2, -7], [8, -6, 10], [4, 6, -2], [0, -2, 7], [-2, 2, -5], [-14, -8, -2], [-18, 6, 5], [-23, -3, 1], [48, 6, -1], [1, 11, -1], [-30, -12, 5], [-2, 2, -2], [-10, 6, -10], [-14, 0, -4], [24, 8, -4], [-12, -2, -8], [-9, 11, -3], [8, 16, 0], [6, 10, -7], [23, -7, 7], [-26, 0, -1], [-11, 7, 6], [-4, 2, -2], [-10, 10, -10], [48, 10, -4], [2, 4, -6], [6, 2, -8], [-26, -4, 8], [28, 12, 0], [20, 0, -4], [26, -4, 6], [-20, 2, -3], [19, -3, -2], [2, 6, -6], [-32, 4, 2], [35, -5, 4], [-32, 4, 1], [-12, -12, 0], [30, 12, -2], [3, 1, 11], [22, -2, -6], [-6, 12, -1], [22, 4, 0], [4, 0, 2], [45, -5, 3], [-8, -4, -6], [-24, -8, 0], [-21, 5, -3], [4, -6, 0], [12, 6, 6], [21, -5, -9], [23, -7, 9], [23, -3, 1], [8, -12, -2], [-2, -6, 2], [2, 0, -6], [11, -5, 7], [-4, 4, 0], [18, 4, 7], [12, -6, 11], [26, -6, 6], [-34, 0, -8], [-29, 9, 5], [8, -2, 1], [-26, -12, 0], [0, -8, -8], [38, -2, 2], [-18, -8, 1], [8, 6, -8], [-10, 12, 2], [-32, -12, 5], [-24, 2, -12], [47, -1, -5], [-23, -21, 8], [-28, -2, -4], [30, -8, -1], [-17, -3, 6], [49, 9, -3], [26, -10, 6], [-38, 0, 2], [3, -3, 13], [38, 6, 2], [8, 6, -4], [8, -6, 16], [-15, -5, -9], [40, 16, -7], [-18, 10, 6], [2, -14, 4], [12, 6, 4], [-42, -2, -6], [-11, -5, 10], [-13, -13, 2], [34, 0, -2], [-6, 2, 11], [14, 8, 0], [-6, 8, 8], [32, -8, -7], [30, -8, -2], [17, -13, -3], [7, 15, -14], [38, -4, 4], [-6, -2, 8], [-14, 12, -7], [10, -6, 6], [-12, -16, 0], [23, 11, 1], [30, -4, 6], [12, -4, -8], [-38, -6, 2], [32, 4, -2], [20, 6, -8], [12, 16, 0], [-14, -16, 10], [-16, -12, 2], [51, 3, -2], [-18, -6, -1], [-44, -2, -6], [7, -17, 8], [-6, 6, 0], [42, 6, 2], [52, 2, 1], [26, -6, 14], [-9, 1, 6], [21, -3, -7], [2, -8, 4], [26, -6, 7], [-25, -9, -4], [-3, 5, -11], [17, -5, -2], [28, 18, -6], [16, 14, -1], [-48, -4, -2], [-6, -6, 2], [-12, 4, 4], [2, 8, -12], [34, -4, 0], [-16, 10, 4], [2, -14, 13], [16, 12, 4], [0, -12, -2], [-32, -12, 0], [18, -4, 0], [10, -4, 1], [44, 6, 2], [-8, 8, -12], [-17, -9, 10], [20, 10, 0], [-17, -5, -9], [-26, 10, 6], [-34, 12, 6], [0, 12, -14], [-46, -4, -3], [-10, 10, 4], [16, 16, -4], [22, -4, -12], [20, 12, -8], [-10, -12, -8], [-7, 13, -10], [4, 12, -12], [2, -12, -4], [42, 0, -2], [28, -4, 0], [-41, -11, -1], [6, 8, 2], [-28, -20, 8], [30, 4, -2], [34, -4, 12], [36, 4, -4], [-11, -5, 10], [-60, 0, 0], [18, 16, -10], [-12, 10, -3], [20, 12, -8], [4, 14, -8], [-16, 4, 2], [4, 4, -2], [12, 12, 4], [24, 2, -4], [2, -10, 2], [-13, 3, -2], [66, -6, -2], [-20, -24, 6], [11, 7, -1], [43, -5, 11], [-34, -12, 6], [-10, 4, 10], [10, -12, 18], [23, -3, 0], [50, 4, -2], [-44, 8, -3], [22, 2, -7], [-18, 10, -18], [-49, 7, 5], [25, 3, -4], [22, 6, 0], [-40, -6, 10], [8, -10, 1], [22, -2, 2], [-14, -14, 6], [-24, -20, 10], [-49, -9, 0], [-14, -2, -4], [1, -19, -5], [-30, 10, -12], [21, -7, -8], [-26, 8, -17], [30, -16, -6], [0, -8, 4], [40, -2, -3], [31, -5, 11], [-14, -14, -1], [44, -10, 16], [-40, 2, 2], [-49, -3, 3], [6, 14, -2], [4, 6, 8], [18, -8, 0], [0, -12, 18], [0, 4, 8], [15, 15, 0], [1, 1, 5], [-19, 11, -4], [-13, -11, -5], [-36, 10, -15], [2, -6, 10], [40, 18, -8], [54, 8, 2], [36, 12, -12], [-24, -4, 4], [-14, 8, 4], [22, -8, 3], [18, -6, 0], [-18, 4, 8], [8, 16, 4], [-19, -5, -5]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4830_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4830_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_4830_2_a_bw(:prec:=3) chi := MakeCharacter_4830_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_4830_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_4830_2_a_bw( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4830_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-16, -12, 4, 1]>,<13,R![8, -28, 2, 1]>,<17,R![-64, -16, 6, 1]>,<19,R![256, -60, -4, 1]>,<29,R![-16, 16, 10, 1]>],Snew); return Vf; end function;