// Make newform 4600.2.a.bg in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4600_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_4600_2_a_bg();". // 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_4600_2_a_bg();". // 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, 9, 7, -7, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [4, 7, -6, -2, 1], [7, 8, -7, -2, 1], [-11, -20, 12, 5, -2]]; Rf_basisdens := [1, 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_4600_a();" function MakeCharacter_4600_a() N := 4600; order := 1; char_gens := [1151, 2301, 2577, 1201]; 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_4600_a_Hecke(Kf) return MakeCharacter_4600_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, 0], [1, -1, 0, 0, 0], [0, 0, 0, 0, 0], [-1, 0, 1, 0, -1], [-1, -1, 1, -1, -1], [1, -2, 0, 0, 1], [0, 1, 0, -1, -2], [1, -1, -1, 1, -1], [1, 0, 0, 0, 0], [-2, -2, 0, 1, -1], [1, 0, 1, 1, 2], [0, 0, 2, 2, 0], [0, -1, -1, 2, 3], [3, 0, -3, 0, 3], [2, 2, -3, 2, 4], [4, -2, 2, 2, 2], [6, -3, -1, 3, 1], [-6, 0, 0, 0, -2], [2, -1, 0, -1, 2], [0, 0, 1, 0, 2], [0, -1, 0, -4, -5], [9, 2, -1, -2, 1], [3, 3, 1, 1, -3], [0, 1, 4, -3, -4], [2, 4, -4, 2, 2], [-2, -3, 1, -1, 1], [-3, 0, 3, 4, -1], [4, -1, -2, 1, 0], [-2, 4, -2, 0, -4], [2, -1, -4, 7, 2], [-7, 4, -3, 1, -6], [5, 4, -4, 3, 6], [10, 1, 0, -1, -4], [8, -3, 1, 1, 2], [-8, 2, -2, 0, -2], [-4, 4, 1, -4, -6], [8, 2, 2, -6, 0], [9, 1, -1, -6, 2], [6, 3, -1, 1, 5], [13, 1, 0, -3, 6], [3, -3, 1, 0, -6], [-2, 0, -2, 2, 2], [-7, -2, 3, 0, -3], [1, -3, 4, -2, 2], [6, -10, 3, 1, 1], [3, 2, -1, 0, 1], [-4, 4, -1, -6, -1], [-4, -3, 9, -5, -5], [4, 4, 4, -4, -8], [-2, 4, -2, 4, 4], [-7, 2, 0, 0, 1], [-1, 10, -2, 1, 0], [-16, -3, 6, 3, -8], [-4, 3, 2, 1, 4], [13, -4, -4, 3, 4], [2, 6, -2, 4, -2], [1, 2, 4, -4, 5], [4, 3, 1, 1, 7], [6, 2, 3, 1, -3], [6, 2, 2, -8, -4], [4, -1, -4, 5, 10], [6, 4, -4, -2, 0], [10, -9, 0, -1, -2], [-3, 2, -3, -1, -6], [-6, 4, -2, 0, -8], [9, 2, -5, 4, 9], [1, -3, 0, 6, 8], [0, 1, -4, 7, -4], [6, -8, 1, 6, 5], [-15, 2, 4, 0, 1], [2, -8, -1, -1, 9], [-3, 6, -3, -2, -5], [7, -4, 5, -4, -7], [-2, 6, 0, 0, 2], [2, 7, -4, -1, 4], [19, -2, 3, -2, -1], [-4, 4, 4, -4, -4], [-11, 6, 3, -1, -8], [0, -7, 0, 5, 2], [12, -5, 1, 0, 5], [3, -3, 11, -5, -1], [-4, 0, 2, -6, 4], [6, 2, -12, 8, 2], [-6, 7, -4, 5, 6], [1, -2, 6, 3, 0], [6, -3, 7, 1, 4], [-8, -2, 1, 8, 3], [2, 7, -8, 7, 12], [-7, 4, 0, -2, -7], [12, -3, -5, 3, 9], [5, 0, -5, 4, 9], [1, 10, -5, 4, 1], [-2, -6, 3, -2, 8], [12, 4, -1, -8, 6], [11, -4, -10, -1, 8], [5, -6, -1, 6, 3], [-15, 2, -8, 5, 8], [-4, 13, -2, -3, 6], [-9, 1, -11, 9, 7], [-10, -6, 5, -5, 7], [4, -1, -9, -1, -2], [-8, 14, -2, -4, -2], [11, -6, 5, -6, -7], [-16, -1, 2, 5, -10], [20, 4, 0, 4, 4], [-8, -9, 3, -2, -9], [-2, -7, 3, -3, 0], [9, -5, -7, -3, 1], [-10, 1, -1, -1, -3], [3, 5, -9, 6, 10], [-24, 10, 0, -2, -4], [10, -4, 4, -4, 2], [0, 6, -4, 10, 12], [0, -16, 6, 4, -4], [-7, 0, 1, 2, 1], [30, 2, -6, 0, 8], [-7, -8, 5, -6, -5], [-3, 8, 1, -9, 0], [6, 0, 8, -5, -5], [5, -3, 3, 1, 5], [-2, -2, -8, 4, 10], [-10, -8, 5, 5, -1], [-26, 8, 6, 0, -8], [-9, -9, 9, -8, -8], [10, 4, -13, 6, 3], [2, 0, 0, 2, 8], [12, 8, -6, 2, 18], [6, 8, -6, -8, 6], [-6, 0, 10, -10, -4], [-16, 10, -2, -4, 2], [7, 0, -1, 3, -10], [21, -4, -3, -2, -3], [-5, -2, 3, -10, -3], [6, -6, -8, 6, 4], [12, 7, -4, 2, -3], [-6, -3, -2, -5, 2], [26, -10, 6, 6, 8], [-23, 12, 1, -4, -7], [-14, 0, 4, -4, 0], [3, 5, -4, 1, 10], [18, 8, 3, -4, 0], [13, -7, -4, -3, -6], [5, 2, -6, -1, 12], [11, -11, 11, 1, -7], [-9, 3, 4, 7, -2], [-13, 8, -7, -4, 5], [23, -10, -3, 0, 3], [7, 9, -1, 2, -2], [15, -7, -7, 2, -8], [15, -8, -6, 3, 12], [-26, 7, 1, -3, 5], [26, -2, 8, -2, -6], [10, 12, -3, -2, 5], [1, -4, 10, -3, -16], [3, -10, -9, 12, 15], [-13, 0, 7, -2, -9], [-8, -2, 6, 10, -2], [-10, 4, -8, 5, 1], [-4, 3, 4, 1, 2], [-28, 18, 2, -6, -6], [-29, 8, -1, -1, -10], [6, -3, 0, -1, 0], [-2, 12, -5, 8, 0], [5, -7, 9, 1, -5], [4, 9, 6, -9, 2], [-7, 8, -1, -8, -1], [-8, 1, -9, 11, 5], [-19, -1, -4, -9, -10], [-12, 1, 4, -9, -12], [32, 0, -9, -8, 14], [1, -13, -3, -5, -1], [-11, -4, -4, 2, -3], [25, -8, -5, -6, 17], [-12, 19, -6, -1, 0], [-7, -2, 6, -9, -20], [-16, 3, -8, 5, 4], [13, 3, -5, 6, -8], [18, -2, -9, -3, 9], [1, -6, -7, 8, 25], [-12, 2, -4, -2, 6], [13, -4, 4, 7, 2], [-21, 1, -1, 7, 7], [10, -4, -1, -9, -7], [3, 14, 4, -2, -20], [15, -8, 1, 2, 9], [-14, 14, -4, -10, -2], [2, -11, 3, -1, 7], [-15, -1, 3, 5, 5], [3, -11, -8, 19, 11], [18, 6, -14, 2, 14], [-8, 7, 0, -6, -15], [12, 4, -3, 6, -3], [-15, 1, 3, 5, -19], [30, 3, 5, -3, -3], [-11, -19, 5, 5, -3], [-12, 10, -2, 4, 2], [4, -17, 4, -1, -6], [8, -12, -2, 6, 12], [20, -3, -4, -7, -6], [23, -6, 6, -7, 0], [6, -10, -4, 0, 20], [14, 21, -11, -1, 11], [-14, -14, -1, -11, 1], [-14, 15, 8, -11, 2], [-23, -8, 11, -16, 1], [-14, -10, -4, 19, 1], [4, 10, 8, -14, -18], [-20, 0, 7, 8, -2], [7, 10, -4, 0, -9], [1, -8, 5, -1, 8], [-18, 12, -4, -13, -5], [3, 10, -4, -12, -5], [9, 8, 1, 0, -15], [27, -1, -3, 1, 11], [0, 3, -13, 9, 17], [10, -7, -2, -13, -14], [-1, 2, 4, -9, 6], [-9, -5, -5, -12, -10], [5, -8, 1, -14, -21], [4, 10, 7, -18, -6], [-6, 2, 16, 3, -11], [-9, 0, 3, -16, -5], [2, 13, -5, 13, 12], [-39, 18, -3, -10, -7], [1, -9, -1, -2, 8], [-18, 26, 3, -11, -3], [2, 5, 12, -13, -2], [21, -12, 2, -7, 0], [-6, 2, -10, 16, -10], [-14, 16, 9, -18, -7], [2, -2, 4, 2, 12], [4, 9, 4, -5, -2], [31, 4, 7, -14, -5], [9, -18, -8, 6, 8], [-20, 8, 18, -12, -8], [15, -20, -5, -2, 11], [26, 11, -8, 17, 12], [12, -6, 4, 4, -4], [-31, -5, -2, 14, 6], [27, -8, 4, 5, 6], [-3, 17, -3, 1, -15], [-13, -13, 24, -14, -8], [-28, -18, 11, -6, -10], [-1, -2, -7, 9, -12], [32, -16, 3, 10, 5], [-16, -16, 1, 0, 2], [-3, 20, 6, -3, -8], [4, -20, -4, 0, 12], [-5, -11, -3, -1, 1], [7, 6, 7, 0, 3], [-48, -2, 4, -8, 2], [24, -7, -2, 5, 12], [15, -8, 5, 0, -5], [38, -21, -2, 3, 0], [7, 13, 8, -3, -6], [-48, -9, 1, 3, 2], [-22, -4, -6, 0, -2], [-3, 7, -7, 1, -9], [-6, -10, 20, -13, 1], [2, 3, -3, 15, 0], [-24, 0, 0, 8, -16], [-5, 33, -5, -5, -1], [-34, 15, -5, -7, -1], [8, 8, -12, 6, -6], [16, -7, 12, 1, -6], [23, -7, 5, -1, 19], [-20, 26, -4, -2, -4], [32, 0, -2, 12, -2], [-31, 3, -3, 3, 7], [-18, -2, 3, 9, -19], [-26, -14, 20, -3, -21], [22, -15, -10, 13, 0], [-20, -9, 4, -15, -18], [-9, -4, 1, 22, 3], [-17, 10, 8, -4, 7], [-15, -2, 5, 5, 12], [-4, 4, 3, -10, -21], [25, -29, 0, 11, 2], [-4, 6, -16, -2, 0], [22, -1, 4, -17, -6], [-23, 6, -11, 3, -10], [-10, 8, -2, -2, 14], [11, 18, -9, 8, -11], [-22, -4, -4, 0, 18], [-10, 18, -19, 8, -7], [2, 12, 2, 0, -26], [30, 0, -8, 0, 4], [0, 8, -20, 0, -2], [16, 9, -25, 3, 25], [19, 1, 8, -15, 2], [-30, 0, -6, -6, -4], [7, 2, 7, 4, 19], [19, 22, -12, 2, 16], [-2, 5, 4, 1, -10], [-13, 8, 5, 11, 0], [24, -18, 6, -6, 8], [-7, -16, 13, -12, -17], [26, -4, 4, 3, -1], [-23, -12, 13, 7, -4], [1, 1, -5, -10, 2], [46, 3, -4, 9, 2], [-34, 14, 16, -4, -14], [1, -20, -1, 0, 1], [26, 3, -3, -13, 0], [-18, -7, 2, 1, 2], [28, -3, -7, 10, -3], [-8, 16, 8, 4, -8], [-20, 0, 0, 20, -2], [-26, 4, 4, -6, -16], [-13, -4, 1, -2, -9], [6, -2, 1, 2, 0], [30, -10, 2, 14, 14], [12, -27, 6, 13, 6], [-30, -12, 3, -16, -4], [16, -16, 0, 4, 18], [24, -8, 20, -6, -12], [-9, 17, -15, 8, -6], [-16, 8, -3, 0, 2], [-10, -17, 12, 5, -8], [-19, 5, 0, 2, -26], [-44, 14, -6, 2, -6], [-8, 11, 6, 5, -8], [-42, 20, 7, -5, -11], [42, -15, 5, -1, -9], [0, 4, -6, -6, 16], [60, -14, 2, 2, 12], [0, 9, 0, 7, 2], [-3, 16, -23, 9, 2], [45, -16, -9, 2, -11], [24, 14, -10, 14, 4], [-20, -2, -13, 14, 22], [41, -8, -4, -6, 1], [60, -2, 3, -2, -6], [-13, 19, 9, -8, -8], [-20, -5, 18, 5, -14], [22, 16, -5, -16, 4], [20, -18, -8, 0, 14], [6, -2, 0, 8, 4], [-4, -7, 16, -10, 15], [28, -12, 12, 0, -2], [-31, 14, -9, -5, 0], [8, 15, -6, 1, 16], [-15, 6, 11, -12, -31], [14, -12, -6, -10, 16], [20, -2, -6, 6, 0], [6, 0, 6, 8, 6], [-52, 2, 13, -14, -6], [62, 4, -5, -2, 11], [-35, -6, -1, -8, -5], [14, 16, -12, 6, -6], [35, 3, 4, -10, 4], [-4, -2, 9, -6, -18], [7, -8, -7, -2, -25], [22, 6, -10, 16, 6], [8, -8, -4, 0, 4], [-49, 7, -5, 5, -21], [23, -20, -7, -2, 1], [1, -5, 5, -11, -21], [6, 16, -16, -3, 5], [26, -31, 7, 3, 9], [46, 4, 2, -2, 16], [-13, -21, 0, -7, -1], [-27, -4, 12, -9, -18], [21, -6, -7, -8, 1], [8, 5, 3, 13, -6], [-10, -19, 21, -7, -32], [45, 0, 7, -3, 8], [4, -8, -10, 8, 24], [4, 25, -4, -9, -4], [41, -20, -2, 13, 8], [-22, 4, -5, 6, 27], [59, -10, 8, 15, 10], [-12, 16, -4, -14, -10], [7, -10, -7, 24, 7], [-14, 4, -8, -13, 7], [-10, -11, 15, -11, -32], [-23, -22, 19, -22, 3], [8, -9, 4, -13, 4], [-22, -17, 4, 23, -2], [-14, 12, -16, -1, 11], [46, 6, -10, 2, 20], [16, 11, 10, -25, -6], [26, -3, 13, 3, -1], [59, -9, -8, -2, 2], [37, -22, 11, 4, 21], [29, 8, 13, -19, -10], [1, -18, 21, -15, -8], [14, 21, -15, 7, 17], [-12, 4, -14, 16, 40], [6, 0, -14, 14, 2], [36, -9, -24, 9, 16], [-27, 1, 12, 21, -10], [-10, -5, 13, -5, -8], [22, 9, -8, -7, 26], [8, -8, 16, -16, -12], [-4, 9, -6, -1, 18], [37, 6, -1, 8, 7], [-26, 11, 12, 7, -14], [31, -11, -1, 21, 9], [-14, 0, 16, -16, 0], [-22, 6, -17, -3, 9], [-2, 3, -15, 3, 20], [-17, -4, 17, -4, -31], [34, 19, -18, 5, 10], [40, 0, -7, 8, -10], [37, -4, -3, -3, -10], [-50, 2, 12, 0, -8], [-11, 8, -4, -13, -20], [-11, 6, 9, 14, 7], [22, -14, -9, 12, -5], [26, 5, -5, -3, 0], [25, -16, -12, 2, 5], [3, 17, -3, -9, -17], [19, 20, -1, -6, 1], [-42, -13, 8, -13, -6], [-2, -2, 8, 13, 23], [14, 2, 2, -6, 14], [6, -5, 17, -11, -25], [6, -3, 4, 5, -14], [9, -2, 15, -10, -31], [40, -20, 12, 4, -4], [-22, 0, -6, 8, 8], [2, -9, 17, -19, -9], [-28, 14, 9, -8, -19], [29, 19, -24, 5, 21], [-3, -2, 8, 4, -23], [-9, -6, -3, 4, -25], [3, -4, 0, 0, 28], [-3, -27, 14, 0, -2], [-21, 30, 7, -17, 4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4600_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4600_2_a_bg();". // 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_4600_2_a_bg(:prec:=5) chi := MakeCharacter_4600_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_4600_2_a_bg();". // 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_4600_2_a_bg( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4600_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-10, -1, 16, -5, -3, 1]>,<7,R![32, 56, -12, -16, 1, 1]>,<11,R![-8, -92, -88, -17, 4, 1]>],Snew); return Vf; end function;