// Make newform 1792.2.a.j in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1792_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_1792_2_a_j();". // 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_1792_2_a_j();". // 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_1792_a();" function MakeCharacter_1792_a() N := 1792; order := 1; char_gens := [1023, 1541, 1025]; v := [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_1792_a_Hecke(Kf) return MakeCharacter_1792_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, -1], [-1, -1], [-1, 0], [-4, 0], [1, 1], [-2, 0], [1, 1], [-2, 2], [6, 2], [0, 0], [2, -2], [-2, 0], [0, 4], [4, -4], [0, 4], [-7, 1], [-1, -1], [-6, -2], [-4, -4], [6, -4], [4, 4], [-7, 1], [2, 0], [2, -4], [3, -5], [8, 0], [-14, -2], [6, 2], [0, -6], [-6, 6], [-3, 5], [-10, 0], [15, -1], [8, -4], [2, -2], [-11, 5], [-12, 0], [4, 4], [-3, -3], [-2, 2], [-23, 1], [0, 8], [8, -2], [-4, 0], [24, 0], [2, 6], [-20, 4], [-3, -3], [9, -7], [2, -8], [14, 2], [-6, 4], [-1, 7], [10, 8], [12, -4], [-7, 9], [-12, -4], [-20, 0], [-6, 0], [9, -7], [-21, 3], [-3, -11], [-8, 0], [-10, -8], [4, -8], [16, 4], [4, 6], [-24, -4], [-11, -3], [-26, -4], [-10, -6], [20, -4], [24, -4], [20, -8], [12, 4], [-6, 6], [17, 9], [4, 2], [-6, 4], [-5, -5], [12, 0], [10, -10], [14, 0], [12, -4], [-2, 10], [6, -8], [16, -2], [-11, 13], [-16, -8], [9, -7], [-16, 0], [-18, 2], [-10, 2], [2, -10], [-20, -4], [-15, -7], [18, -4], [13, 5], [-8, -4], [8, -12], [-12, -8], [-9, 15], [-28, -6], [-4, -16], [6, 12], [-17, -1], [2, 0], [-24, 0], [-22, -8], [-8, -8], [-10, 2], [-28, -2], [21, 5], [0, -8], [12, 10], [7, 15], [0, 8], [26, 6], [-16, 12], [-25, -1], [6, 8], [1, 9], [-26, 2], [-27, -3], [-6, -10], [2, -2], [-28, -4], [-12, -12], [3, -13], [-4, 8], [30, 2], [-6, -10], [30, -6], [-10, 8], [26, 4], [-9, -9], [25, 9], [-31, 1], [8, 6], [-7, 9], [-4, 0], [4, -12], [-22, 6], [7, -9], [40, 0], [-31, 1], [-30, 12], [19, -5], [16, -8], [-22, -10], [6, 12], [-34, 2], [12, 12], [-22, 6], [-50, 2], [20, -12], [-10, 8], [-18, 4], [15, 7], [-48, -4], [14, -8], [-26, -6], [-31, -7], [-18, 0], [8, 16], [4, -12], [-33, -1], [0, 22], [-51, -3], [22, -14], [33, 9], [12, -4], [-6, 16], [-4, 20], [26, -8], [-22, 6], [30, -6], [-24, 16], [-13, -5], [26, -10], [-37, 3], [6, -14], [6, -4], [-10, 10], [9, 25], [-32, -12], [-31, 1], [6, -16], [32, 0], [-10, -20], [-10, 2], [46, 2], [-15, 1], [14, 18], [34, 0], [-28, 6], [-10, 2], [-18, 4], [4, -12], [8, -4], [32, 16], [27, 11], [14, -12], [-19, -11], [27, 3], [16, 0], [-4, -8], [8, -6], [-1, 7], [12, -6], [11, -5], [26, 6], [-31, 1], [20, 20], [-10, 8], [32, -8], [18, -20], [-48, -8], [-22, 6], [-22, -10], [12, -4], [38, 8], [-26, 10], [9, -7], [42, 6], [-6, 24], [-2, -14], [-52, -4], [28, 0], [2, 30], [47, -1], [48, -8], [-14, 8], [-39, -7], [-20, 4], [-42, 8], [0, 4], [14, -6], [16, -8], [-24, -12], [-27, -19], [4, 20], [0, 20], [-18, 4], [0, -24], [32, 0], [1, -7], [40, -4], [-4, -4], [16, 20], [14, -12], [8, 0], [-38, -8], [39, -9], [14, 2], [-6, 6], [7, -9], [35, 3], [2, -4], [-34, 2], [-36, 8], [-21, 11], [51, 3], [2, 12], [19, -5], [-40, 12], [58, 8], [10, -10], [46, -6], [5, 5], [24, 4], [-22, 24], [10, 22], [26, -8], [16, -24], [24, -4], [-4, -32], [24, 10], [-35, -11], [-12, -4], [-20, -4], [52, 4], [-13, -21], [-23, 9], [-42, 10], [22, 8], [-4, 32], [-44, 4], [-34, 20], [40, 12], [-3, 5], [4, 14], [45, -3], [-12, 8], [-15, -23], [12, 4], [17, 1], [11, -29], [3, -13], [-6, -28], [-10, -14], [24, -16], [20, 16], [0, 20], [14, 16], [32, -12], [55, -9], [26, 6], [20, -8], [-36, 20], [2, -2], [6, -24], [-28, 0], [-16, 24], [42, -8], [-33, -9], [-8, -32], [54, -4], [-38, 8], [-33, -1], [30, -8], [-5, -29], [52, 4], [8, 6], [6, -24], [-16, -4], [-19, 29], [-50, 2], [-62, 14], [-60, 8], [18, 14], [-20, -12], [3, 27], [-6, -10], [-43, -3], [-66, -6], [38, -4], [-66, 4], [-48, -16], [12, 16], [12, -14], [33, -7], [8, -16], [18, 14], [24, 4], [5, 13], [-60, -8], [36, 12], [-49, -1], [47, -1], [-58, -16], [32, 4], [24, 8], [-28, 24], [-14, -32], [40, 8], [5, 29], [-50, 0], [-34, 18], [8, 12], [62, 4], [-38, -10], [42, -10], [-7, -23], [28, -26], [-11, -27], [-10, -22], [0, -34], [14, 34], [-49, 15], [36, 4], [-24, 12], [-32, -24], [-48, 4], [17, -15], [6, -6], [6, 28], [-46, -20], [-30, -4], [-11, 13], [-56, 6], [-12, 4], [34, -16], [19, -37], [28, -4], [28, -20], [-17, -17], [-2, 10], [-40, -8], [-28, -10], [-45, 3], [-38, 22], [-33, 31], [18, 30], [6, 0], [-64, -16], [-22, -28], [-22, 6], [46, -22], [-69, -13], [14, 32], [-24, -32], [-46, -4], [57, 9], [-32, -8], [44, -16], [8, -18], [-5, -37], [-41, -1], [-26, -4], [2, 14], [-12, -8], [-70, 14], [-30, 24], [-31, 9], [52, -12], [-36, 12], [42, 4], [-20, 12], [70, -14], [63, 15], [4, 20], [-5, 3], [-2, 32], [-17, -1], [-40, -12], [-6, 0], [-37, 19], [4, 4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1792_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1792_2_a_j();". // 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_1792_2_a_j(:prec:=2) chi := MakeCharacter_1792_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_1792_2_a_j();". // 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_1792_2_a_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1792_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-4, 2, 1]>,<5,R![-4, 2, 1]>,<23,R![-16, 4, 1]>],Snew); return Vf; end function;