// Make newform 1792.2.a.q 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_q();". // 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_q();". // 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 := [-3, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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], [-2, -2], [-5, -1], [2, 0], [-3, 1], [0, -2], [-2, 2], [-4, 0], [-2, 2], [2, 0], [-2, -2], [4, -4], [-12, 0], [1, 5], [-1, 7], [8, 0], [-4, 4], [-6, -4], [4, 4], [3, -5], [2, -8], [2, 4], [-7, 5], [-4, 8], [-12, 4], [-2, -6], [-10, 2], [8, 2], [-9, -1], [-6, -8], [11, -1], [-8, 4], [8, -6], [7, 3], [-10, -2], [-12, 4], [-9, 3], [4, -4], [9, 1], [-16, 0], [10, -2], [-12, -8], [-4, 0], [-4, 12], [0, 4], [9, -3], [19, -1], [-6, 0], [16, -2], [2, -12], [21, -3], [-6, 0], [4, 4], [3, -9], [-8, -4], [4, 0], [-2, 0], [-1, -9], [-7, -3], [7, -9], [8, -8], [-6, -8], [-8, -12], [6, 6], [6, 6], [18, 10], [5, -11], [-6, 4], [-24, -2], [0, -12], [24, 4], [-10, 14], [12, 12], [-6, -10], [17, -7], [10, 2], [10, -4], [29, -3], [12, 8], [32, 2], [2, 8], [24, 4], [-24, 0], [-2, 0], [2, -10], [-19, -3], [-16, 0], [13, -7], [-4, 8], [-32, 6], [-12, -12], [-20, 4], [24, 4], [9, -7], [-6, 12], [-19, 9], [-4, 0], [-22, -14], [-4, 16], [15, 11], [2, 2], [6, -18], [-38, 4], [35, -1], [26, -8], [0, 16], [-22, 0], [0, 16], [-22, -10], [14, 6], [19, 3], [-8, -24], [2, -14], [23, -5], [12, -8], [-42, 2], [-14, 10], [-3, -15], [18, -8], [-15, 17], [4, -20], [-1, 7], [-30, 6], [-10, -22], [-28, 4], [4, 4], [-5, -5], [14, -10], [8, 14], [0, -14], [-14, 14], [-6, 16], [10, -12], [-33, 7], [5, 9], [-13, 15], [-6, -10], [15, 7], [0, 4], [4, -4], [0, 8], [15, 15], [-12, -16], [-21, 15], [10, 12], [1, 17], [-24, 0], [10, 6], [-30, -12], [12, -4], [-12, -20], [16, 16], [8, -2], [-28, 12], [10, 16], [26, -20], [-19, -15], [-14, -6], [6, 8], [16, -18], [33, 13], [-42, 8], [-20, -16], [-4, -12], [7, -1], [-14, 14], [-33, -5], [-28, -12], [-15, -7], [36, -12], [-2, 0], [-28, -12], [18, 24], [4, 20], [-22, -10], [8, 8], [-7, 5], [0, 2], [3, 7], [-2, -6], [-38, -12], [-24, -10], [3, 23], [-4, -8], [-27, -15], [-22, 8], [-4, 24], [-22, 12], [32, 0], [8, -16], [11, 23], [-16, 24], [-30, 16], [-18, -18], [-2, 18], [-30, -20], [24, 4], [0, -12], [-44, 8], [35, 3], [-38, 12], [13, 17], [-7, -11], [44, -8], [-18, -2], [54, 2], [37, -11], [50, 2], [-21, -13], [-40, -6], [7, -25], [24, -20], [34, 0], [-24, 0], [34, -20], [-16, -24], [-58, 2], [10, -2], [32, 4], [-30, -8], [-32, 6], [-43, 1], [22, -30], [-6, -8], [-16, 6], [0, 12], [-38, -14], [18, -10], [-15, -15], [-32, -16], [38, 8], [-41, -1], [-24, 20], [-6, -32], [-4, 16], [48, 8], [16, 8], [26, 10], [-3, 1], [48, 4], [56, -4], [10, 36], [-68, 0], [32, -24], [27, -13], [-18, -10], [-4, -36], [16, -4], [-54, 4], [-24, -16], [-14, 8], [39, -9], [0, -16], [-26, 18], [11, -1], [3, -29], [10, -12], [48, 14], [26, -14], [59, 11], [11, 11], [-6, 28], [-37, 7], [24, 4], [34, -8], [24, 0], [18, 22], [15, 11], [-10, -2], [34, 16], [8, -6], [-30, 0], [-40, 8], [-10, 30], [16, -20], [38, 18], [39, -9], [-64, -12], [20, -12], [-4, 28], [-23, 13], [-25, -33], [24, 14], [-46, 8], [4, -24], [28, -4], [42, 12], [32, 4], [7, 31], [-2, 6], [-45, -13], [-56, 12], [33, -23], [32, 4], [3, 7], [49, 1], [-19, 37], [-38, 4], [-6, -18], [-72, 8], [62, -2], [26, -6], [70, 8], [-42, 6], [55, -9], [-16, -6], [12, 8], [32, 12], [30, 2], [2, 40], [26, 34], [48, -8], [-22, 24], [-9, -5], [-8, 24], [34, -12], [6, 32], [33, 17], [-18, 24], [33, 13], [28, 20], [-30, -18], [10, 24], [22, 6], [19, 19], [8, 46], [-6, -18], [-52, 16], [-66, -6], [-12, 44], [-45, 23], [28, 28], [-19, -7], [-22, -2], [2, -28], [26, 28], [-28, 24], [-8, 36], [-6, 34], [-15, 9], [24, 16], [18, 22], [18, -14], [-59, -11], [78, -10], [72, 12], [-41, 7], [17, -7], [38, -32], [60, 8], [-36, 0], [52, -8], [2, 8], [-36, 8], [13, 33], [14, -24], [0, 6], [-4, 48], [-6, 28], [8, -30], [12, 20], [-35, -15], [-10, 6], [-33, 19], [-32, -10], [2, -26], [-24, 16], [-53, -17], [52, -12], [40, 20], [0, 0], [-28, -24], [-3, -7], [-48, 14], [-22, 44], [10, 20], [-6, 12], [-43, -19], [-30, 14], [28, -20], [-18, 8], [-13, 23], [-8, 4], [68, 12], [-17, -33], [36, 4], [12, -24], [-10, -10], [3, 27], [-64, 8], [17, -15], [8, 50], [-22, 8], [20, 0], [-30, -20], [-20, 44], [-2, -14], [41, -3], [-14, 0], [-16, 24], [-6, -20], [-5, -25], [-44, -32], [-40, -20], [-54, 14], [-35, -27], [31, -29], [18, -20], [18, -10], [-2, -42], [-4, -12], [10, 16], [57, 1], [12, -28], [-60, -4], [-38, -20], [40, -28], [66, -10], [-41, 23], [4, 36], [-15, 49], [10, -24], [-33, 7], [-46, 18], [-34, 16], [-15, -31], [20, -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_q();". // 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_q(: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_q();". // 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_q( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1792_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-2, -2, 1]>,<5,R![-2, 2, 1]>,<23,R![-12, 0, 1]>],Snew); return Vf; end function;