// Make newform 6160.2.a.bj in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6160_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_6160_2_a_bj();". // 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_6160_2_a_bj();". // 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, -3, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-2, -1, 1]]; 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_6160_a();" function MakeCharacter_6160_a() N := 6160; order := 1; char_gens := [5391, 1541, 3697, 5281, 5601]; v := [1, 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_6160_a_Hecke(Kf) return MakeCharacter_6160_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], [-1, 0, 0], [-1, 0, 0], [0, 0, 1], [2, 0, -1], [-4, 2, 2], [-1, 1, 1], [2, -4, 0], [-3, 1, 2], [-5, 3, 1], [3, 1, 4], [1, 3, -1], [-2, -4, -1], [1, -3, 3], [-3, 1, 0], [-1, 3, -4], [1, -1, 1], [0, 4, 4], [-10, 0, 1], [-1, -1, 1], [4, -6, 0], [4, 0, -6], [-8, 0, -6], [7, -7, -6], [0, -4, 7], [-2, -6, -2], [-2, 8, 2], [-2, 0, -6], [6, -6, -4], [-8, -4, -8], [-1, 3, 5], [-4, -4, -6], [-8, 6, 0], [-7, -3, 1], [-4, 0, 8], [3, 9, -1], [-10, 12, 4], [8, 2, -5], [2, -10, 2], [-6, -2, 6], [0, -4, -10], [-1, -5, 5], [3, -9, -7], [9, -9, -8], [-2, -2, 4], [-8, 12, 7], [4, 4, 4], [-8, -6, 4], [-5, 3, -5], [-14, 6, 4], [13, 7, 2], [-1, -5, -4], [6, -6, -10], [-2, -6, 4], [-4, -4, 4], [-8, 0, -6], [-7, -7, -11], [0, 6, -14], [14, -2, 2], [4, 6, 13], [12, 6, -2], [-3, -1, -4], [-20, 2, 2], [-2, 12, -4], [-18, 18, 8], [-19, 13, 7], [5, 11, -9], [-7, -11, 10], [-4, 6, 10], [-19, 1, -1], [0, 14, 5], [13, -15, -1], [4, 8, -12], [-2, -2, -9], [-30, 4, 2], [-2, -2, -6], [2, -12, 4], [-15, 1, -12], [3, 13, 2], [-19, -1, 1], [-5, -1, -11], [18, -12, -2], [-6, -6, -4], [3, -11, 9], [-19, 3, 5], [11, -13, -11], [-1, 13, -4], [5, 7, 5], [16, -4, -3], [-2, -12, -8], [7, -19, -7], [21, 1, -9], [2, -2, -14], [-24, -2, 4], [2, 8, 6], [22, -22, -6], [0, -2, -2], [24, -2, -4], [4, -4, 10], [-1, -9, -1], [6, 2, 16], [-14, 12, 2], [-6, 2, -8], [-10, 6, 0], [-14, 2, -5], [-12, 20, 9], [-6, 2, 10], [13, -1, -2], [-14, 0, -16], [-13, 19, 15], [-29, -5, -7], [-1, 13, 8], [-4, 8, -8], [-21, 25, 7], [24, -2, 7], [-20, -10, 3], [30, 0, 6], [-21, -1, -9], [-10, 6, -4], [25, 1, -1], [-22, -2, -1], [17, -9, -11], [-27, -1, -8], [-8, -14, 8], [6, -4, 4], [-31, -5, -6], [8, 2, -13], [10, 4, 3], [8, 4, 20], [-18, -2, 10], [-20, 0, 2], [-8, 2, -20], [7, -21, 2], [19, -11, 6], [-12, -12, -2], [2, 14, 12], [20, -16, -16], [-20, -6, -18], [30, -16, -6], [14, -16, -6], [-37, 17, 3], [11, 13, -7], [-6, 20, 24], [-15, -5, 10], [0, 16, -11], [28, 6, 23], [1, -15, 6], [39, -15, -7], [7, 3, 3], [26, 6, 2], [31, -7, -1], [-8, 16, -2], [15, -3, 9], [4, -8, -8], [-10, 2, -4], [2, 14, -6], [-28, 8, 13], [15, 15, 8], [5, 7, -1], [-3, 9, 5], [1, -9, 7], [17, 11, 22], [-10, -16, 14], [-10, -8, -5], [22, -6, -4], [18, 16, 13], [10, 16, -2], [12, -8, 14], [-14, 10, -4], [30, -2, 12], [1, 9, 25], [13, -3, 5], [-17, -7, 2], [-6, 24, 6], [-25, 7, -7], [-23, 27, 1], [8, 6, 20], [21, -9, -16], [-15, 7, -21], [10, 4, 24], [0, 26, 4], [-6, -2, 25], [-19, 19, 17], [-6, -10, -26], [-5, -1, -1], [28, -26, -1], [12, -30, -18], [20, 22, -12], [32, -30, -12], [-8, 8, 0], [38, 10, 0], [-8, -12, -6], [-29, -7, 13], [42, 2, -6], [-14, 8, 8], [-13, 19, 25], [28, -32, -11], [-12, 8, -14], [-18, -24, 10], [26, -8, 14], [-8, -6, -4], [27, -5, 2], [-3, 13, 6], [-18, 12, 0], [-5, 3, -6], [30, -2, 8], [0, 22, 16], [-11, -15, -4], [9, 13, 5], [-4, 22, 20], [41, -9, 3], [10, 4, -5], [-26, 16, 12], [-14, 10, 24], [11, 1, -3], [27, 13, 14], [-19, -17, -27], [30, 4, -2], [16, -6, 16], [-2, -4, -4], [-53, -7, -9], [1, -17, 9], [18, 12, 12], [20, 14, -8], [40, -34, -16], [0, -16, -8], [32, 0, -10], [41, -23, -11], [-9, 3, -19], [16, 12, 32], [16, 12, 4], [-4, 6, 22], [6, 8, -11], [2, -24, -22], [-2, -8, 14], [-21, -17, -31], [28, -32, -8], [19, 1, 24], [-33, -3, -13], [-46, 14, 8], [20, 26, -13], [-47, 3, 7], [2, 10, -27], [-22, -28, 14], [38, 0, 1], [-9, 1, -34], [-12, 16, -4], [-14, 18, -16], [39, 3, -15], [-27, 5, 16], [-33, 1, 13], [6, 30, -4], [8, 6, -3], [28, -8, 4], [-37, -15, 1], [38, -20, -14], [4, -26, -22], [6, 10, -7], [-43, -9, -21], [12, -12, -16], [15, -15, 0], [8, -22, 3], [50, -10, 2], [7, -27, -4], [11, -23, 17], [14, 8, -14], [21, 3, 19], [-23, -7, -13], [-40, 10, -2], [-9, -3, -25], [2, 28, 22], [0, 8, -16], [-18, -4, 17], [-14, 18, 9], [-19, 3, -7], [28, 10, 18], [-26, -16, -6], [46, -26, -20], [-38, -8, -28], [10, 14, 44], [-6, -12, 26], [33, 21, -16], [10, 12, -6], [-22, 34, 22], [-11, -15, 15], [-23, -3, -5], [-11, -21, -24], [29, -11, 6], [5, 27, -21], [-10, 26, 31], [17, -3, 29], [-12, -12, -6], [49, -19, -17], [-25, -7, -16], [1, 9, -30], [-18, -18, 4], [0, -16, 2], [30, 18, -8], [-24, -8, -27], [7, 11, -3], [-43, 9, -11], [-17, 33, 1], [-24, -20, 18], [11, 19, 25], [-35, 39, 15], [12, 38, -14], [18, 10, -22], [13, 13, -7], [4, 16, -4], [-46, -4, -18], [-28, 2, 26], [31, -7, 19], [-47, 35, 11], [-27, -13, 24], [3, -23, 4], [-14, -18, -36], [8, 22, -20], [4, -10, 4], [18, -6, 2], [18, 20, 10], [33, -19, -6], [23, -27, -19], [-37, 27, 25], [10, -8, 4], [-54, -10, -4], [-14, 18, -1], [-19, 15, -1], [-35, 9, -7], [-8, 6, 28], [-17, -1, -3], [-38, 36, 10], [4, 24, 10], [-7, 1, -27], [22, 4, -23], [-9, -11, -7], [54, 6, -7], [16, -14, 22], [24, -14, 20], [-8, -18, -36], [-9, -41, 5], [-37, 11, 6], [-2, 2, 16], [44, -34, -2], [12, 36, -20], [-32, 24, 8], [-9, 21, 17], [14, -24, -6], [-40, -6, 4], [6, 16, -24], [-19, 21, 11], [58, 4, -2], [2, 4, -22], [32, 26, 10], [-28, -28, 19], [-37, 5, -2], [-2, 26, 18], [-11, -7, -25], [-9, -11, -3], [55, -13, -7], [7, 25, 0], [-25, -15, -27], [-5, -9, 13], [24, -34, -5], [-33, -13, -11], [8, -6, -13], [22, -30, -4], [-10, 20, 28], [-2, -30, -10], [-65, 7, 6], [-42, 6, 20], [-28, 34, 14], [48, -34, -12], [49, -23, -13], [-5, -39, 8], [-19, -19, 21], [-14, 4, -41], [25, -15, 10], [-2, 4, -13], [-42, -24, 4], [-23, 1, 11], [17, -1, 27], [-13, 31, -13], [-40, 46, 18], [-36, -24, -9], [-30, -34, 20], [-22, -16, -28], [6, 34, 12], [-26, 14, -31], [41, 15, 21], [-30, 50, 36], [0, 0, 10], [-16, 0, -3], [14, 22, -8], [7, -5, 33], [-18, 42, 8], [-2, -40, 8], [8, -40, -10], [-22, -16, -2], [3, -19, 16], [-24, 6, 30], [-28, -12, 4], [2, 8, 28], [-1, 17, 30], [60, -28, -12], [22, 12, 40], [-18, 20, 30], [-4, 24, 7], [-31, 27, -20], [-18, 26, 11], [-45, 11, 5], [49, -17, -3], [-5, -13, 33], [19, 15, 1], [-8, -28, -14], [-8, -32, -24], [-34, -14, -33], [26, 36, -4], [-24, 4, 32], [-6, 0, 24], [-26, -16, -17], [-23, -5, 5], [36, 10, -24], [4, 18, 14], [-42, 26, -12], [9, -13, 29], [10, -40, -20], [15, 27, 18], [12, 18, 46]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6160_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6160_2_a_bj();". // 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_6160_2_a_bj(:prec:=3) chi := MakeCharacter_6160_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_6160_2_a_bj();". // 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_6160_2_a_bj( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6160_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-2, -4, 0, 1]>,<13,R![2, -4, 0, 1]>,<17,R![-2, 8, -6, 1]>,<19,R![-40, 12, 10, 1]>,<23,R![-4, -4, 2, 1]>],Snew); return Vf; end function;