// Make newform 1932.2.a.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1932_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_1932_2_a_e();". // 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_1932_2_a_e();". // 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, -1, 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_1932_a();" function MakeCharacter_1932_a() N := 1932; order := 1; char_gens := [967, 1289, 829, 925]; 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_1932_a_Hecke(Kf) return MakeCharacter_1932_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, 0], [0, 1], [1, 0], [1, -2], [-3, 1], [-3, 0], [-3, -2], [-1, 0], [1, 0], [1, 0], [1, -4], [-1, -2], [-1, -3], [-6, -2], [6, 3], [4, 3], [-3, 5], [-6, 7], [-5, -3], [-3, 6], [-11, 2], [-5, 2], [1, 1], [-11, 0], [2, -5], [-16, 0], [1, 5], [-4, -3], [-8, 5], [1, 1], [-9, 6], [7, -10], [-8, -1], [6, 6], [-2, -2], [0, 10], [-12, -5], [-3, -6], [5, -10], [6, -3], [9, -8], [-2, 10], [-4, 6], [21, 1], [22, -1], [-9, -2], [-16, -5], [22, -1], [-14, 1], [-4, 11], [12, -1], [15, -4], [-8, 2], [-10, 10], [-9, 12], [13, -9], [3, -6], [9, -11], [4, 6], [8, -1], [-4, -4], [-5, 8], [-1, 3], [-16, -4], [0, 1], [10, 4], [1, 5], [15, 6], [-16, 5], [7, 2], [-18, 3], [-9, -9], [13, 8], [4, -4], [-7, -10], [25, -4], [-6, -2], [3, -8], [-3, -8], [-2, 5], [5, 3], [-9, 3], [-12, -10], [5, 12], [4, -12], [24, -7], [20, -9], [-6, 15], [17, 4], [-15, -2], [3, -8], [-29, 4], [13, -11], [2, 15], [-14, -1], [-6, 2], [34, -4], [2, 0], [-6, 2], [10, -5], [12, -18], [-9, 7], [-4, 0], [7, 6], [10, -12], [1, -1], [-12, 10], [-3, 9], [-6, -13], [19, -7], [-3, -18], [9, -7], [7, 11], [25, 0], [-17, 17], [-29, -9], [-30, 3], [4, -5], [25, -2], [1, -4], [-23, 8], [33, -1], [-43, -2], [21, -13], [18, -7], [13, -3], [-25, 0], [-1, 2], [0, 10], [27, 10], [14, -9], [24, 5], [-1, 8], [-4, -4], [8, -22], [-1, -10], [14, -21], [-10, -2], [-27, 15], [-7, 14], [-6, 6], [-26, -11], [0, 15], [-17, 16], [33, -1], [8, 6], [-12, -6], [10, 4], [-24, 2], [-8, -4], [-4, 22], [-3, 7], [47, -5], [-28, 9], [-10, -2], [-3, 24], [0, -3], [-11, 26], [-12, -16], [-30, -8], [-7, -11], [34, -12], [-43, 9], [35, 5], [-31, 14], [11, -1], [31, -8], [4, -9], [9, -9], [-15, -1], [-26, -8], [7, 11], [-9, -20], [8, 0], [19, -2], [-25, -5], [20, -31], [22, -19], [-30, 5], [-14, -5], [33, 3], [26, 5], [-21, 17], [-33, -10], [3, 19], [26, -16], [5, -16], [14, 5], [-9, 2], [20, -19], [40, -1], [-28, 2], [1, 2], [-5, 11], [-19, -19], [-37, 7], [8, 10], [-30, 0], [-54, 3], [18, 16], [8, 0], [4, -1], [41, -19], [3, -8], [-18, 12], [-25, -8], [-24, 3], [-4, -17], [11, 2], [-22, 18], [-28, -5], [19, 6], [0, 20], [-27, 33], [-22, 5], [-37, 26], [-27, -11], [-9, 16], [-9, -10], [-1, 1], [54, 0], [27, 7], [16, 24], [-23, 24], [4, 9], [-15, -6], [40, -16], [-17, 27], [-47, 2], [-19, -12], [13, -7], [22, -1], [-32, -4], [-17, 33], [17, -15], [34, -14], [-57, 6], [-35, -10], [-13, 2], [1, -6], [40, -17], [20, 6], [5, -6], [5, -22], [31, 11], [36, -16], [12, 16], [-1, 28], [9, -4], [13, -32], [30, -8], [11, 6], [-17, -14], [-3, -4], [16, -14], [35, -4], [39, 14], [-5, -1], [1, -16], [-53, 3], [18, -12], [24, -3], [-15, 10], [-53, -5], [23, 0], [42, -8], [43, -18], [-29, 18], [-44, -1], [-25, -16], [-13, -12], [-25, 39], [50, -18], [-33, 4], [50, 12], [23, -38], [-23, 34], [-7, 0], [9, 26], [18, -30], [-11, -19], [21, -46], [-9, -16], [-2, -3], [-17, -22], [-31, 12], [54, 4], [33, -17], [-4, -20], [63, -1], [-39, -13], [-29, 3], [9, -8], [-52, 0], [-69, -6], [12, -28], [-15, -13], [-9, -12], [19, 8], [-11, -29], [26, -4], [-13, -30], [-36, 41], [-68, 4], [-37, -2], [56, -1], [7, -2], [48, -18], [55, -21], [-15, -12], [57, -14], [68, 5], [-13, 12], [-49, 16], [32, -4], [13, -16], [-13, 22], [21, 2], [-6, -22], [58, 0], [5, 21], [32, 3], [-19, 16], [39, 16], [-34, -25], [-7, -18], [-30, 15], [-26, 10], [-56, 10], [-20, -12], [-35, 30], [-13, 19], [54, -25], [-30, -10], [18, 28], [-20, -18], [-46, 34], [-60, 11], [52, 13], [-27, 44], [-6, -8], [31, -7], [23, 16], [2, 4], [39, -21], [24, 3], [-36, 37], [56, -3], [-21, -28], [13, 27], [49, -32], [-47, 38], [-34, -1], [35, 10], [0, -20], [-48, 30], [-39, -4], [-13, -4], [-49, 22], [-65, 2], [-45, 22], [-51, 34], [-12, -4], [6, -22], [56, -8], [-25, 31], [-50, 2], [-43, 37], [63, 1], [-29, 0], [-20, -16], [36, -30], [8, -28], [41, -32], [28, -19], [-36, 3], [-62, -2], [6, 0], [-33, -12], [15, -12], [21, -7], [-58, 6], [-1, -21], [-30, 37], [25, 13], [-39, 9], [-38, 4], [-67, -12], [-40, 34], [-67, 6], [72, -21], [-66, 0], [30, -10], [33, 13], [20, -15], [-3, 2], [73, -4], [-37, -5], [11, 10], [1, -20], [15, 15], [36, -38], [15, -9], [-51, 29], [-25, -5], [64, -8], [-80, 2], [-28, 20], [4, 21], [13, -3], [-18, 30], [-32, -16], [-69, 3], [-35, 15], [-46, 4], [-63, 20], [37, 8], [-23, -22], [30, -14], [-13, -14], [31, -40], [-48, 3], [-9, 23], [-29, -9], [-6, -10], [-9, -31], [32, -10], [-48, 15], [-25, -17]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1932_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1932_2_a_e();". // 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_1932_2_a_e(:prec:=2) chi := MakeCharacter_1932_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_1932_2_a_e();". // 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_1932_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1932_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-3, -1, 1]>,<11,R![-13, 0, 1]>],Snew); return Vf; end function;