// Make newform 3200.2.a.bj in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3200_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_3200_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_3200_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 := [-2, 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_3200_a();" function MakeCharacter_3200_a() N := 3200; order := 1; char_gens := [1151, 901, 2177]; 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_3200_a_Hecke(Kf) return MakeCharacter_3200_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], [0, 0], [-2, 2], [1, 3], [0, 4], [-3, 2], [3, -1], [-2, -2], [-8, 0], [2, -2], [2, 4], [5, 4], [10, 0], [-4, -4], [-2, -4], [2, -4], [-6, 0], [-7, 3], [-4, -4], [3, 2], [10, 2], [-3, 3], [9, -2], [-6, -8], [2, 12], [-4, 4], [-9, 5], [-6, -4], [9, -4], [6, 10], [-6, 8], [-5, 0], [1, 7], [-8, 8], [-2, 6], [-14, -4], [-7, -9], [-8, 4], [-4, -4], [23, -1], [-10, -4], [10, -6], [-11, 2], [-2, -4], [20, 4], [-1, -7], [8, 4], [-10, -8], [-12, 8], [-2, -8], [-4, 16], [7, 2], [13, -3], [-6, 8], [2, 6], [12, -4], [22, -2], [-4, -4], [2, -16], [3, -9], [-6, -8], [1, -3], [-14, -6], [-10, 0], [-8, 4], [13, 7], [13, -4], [-9, -7], [-18, -8], [10, -16], [6, 2], [16, -12], [22, 8], [-7, -7], [-2, -18], [6, -8], [20, -8], [-5, 18], [-11, 12], [-9, 11], [4, 0], [-2, 10], [-19, -2], [12, -8], [7, 11], [-9, -18], [23, -4], [0, -4], [4, -12], [-2, 8], [-38, 2], [6, -22], [6, -16], [-14, 0], [0, -12], [-14, 0], [5, 8], [-21, 1], [12, 4], [1, -11], [32, -4], [10, -12], [-15, -8], [-2, 24], [13, 20], [7, -5], [-7, 0], [-10, 26], [21, 10], [36, 0], [18, -8], [-18, 0], [-2, -8], [30, -10], [-34, -8], [-26, 8], [0, -4], [-6, 0], [-13, -27], [-12, 4], [-14, -8], [-40, 8], [-13, 21], [15, 15], [36, 8], [-12, 24], [2, -18], [32, 0], [32, -4], [-6, -16], [6, 14], [-12, -16], [-20, -4], [-11, -20], [3, 22], [-8, -8], [6, -16], [-42, 0], [18, 16], [-2, -24], [18, 16], [4, -12], [-41, -5], [-20, 0], [4, 0], [6, -16], [11, 0], [33, 15], [-4, 0], [32, -4], [-26, 8], [-39, -3], [28, -4], [34, 8], [8, 20], [-42, 2], [-22, 16], [11, -22], [-24, -12], [-6, 28], [-1, -8], [20, 16], [33, 7], [-7, 22], [14, 14], [14, 34], [0, -12], [-5, -22], [-8, -24], [-7, 29], [-6, -32], [-6, -14], [3, -26], [24, 0], [-7, 2], [-3, 5], [6, -8], [32, 8], [14, 12], [20, 4], [-9, 23], [14, 4], [15, 26], [0, -12], [16, 4], [-18, 24], [-3, 5], [25, 12], [-36, 0], [-15, -24], [-21, -5], [15, -3], [10, -4], [29, 5], [-14, 0], [3, -24], [-26, 24], [-27, -8], [34, 14], [6, 20], [8, -12], [-12, 12], [-21, -4], [14, -8], [14, 0], [50, 6], [14, -16], [-22, 24], [18, 28], [-30, 16], [34, 8], [12, -16], [11, 1], [-30, 6], [41, 0], [-6, 2], [11, -22], [-10, -34], [-30, -20], [2, -4], [32, 12], [39, -14], [-34, -26], [25, -27], [-10, 24], [15, 10], [12, 8], [-32, 4], [-39, 9], [-10, 32], [-21, 9], [22, -26], [-7, 10], [-34, -12], [-22, 22], [27, -2], [8, 0], [-3, -15], [-12, -12], [41, -15], [17, -43], [-2, -38], [-20, -8], [53, 12], [8, 8], [-10, -10], [-30, 8], [33, 25], [-36, -16], [4, -12], [27, 16], [22, -34], [13, -38], [-12, 4], [35, -3], [-44, -4], [31, 15], [10, -44], [67, 4], [30, 14], [41, 15], [-32, 8], [-4, 16], [-3, 10], [7, -29], [-14, -36], [-22, 8], [-45, -3], [-38, 12], [-36, 24], [-19, 31], [63, 10], [60, -8], [14, 32], [-32, 28], [-5, 9], [12, -36], [34, -32], [-9, -29], [12, 4], [14, 18], [20, 16], [-52, 24], [27, 11], [-12, 40], [-10, -16], [4, -28], [28, -8], [-14, 0], [6, -20], [-2, 40], [11, -10], [-3, 25], [40, -28], [-34, 16], [34, 6], [44, 20], [14, 16], [-11, -1], [15, 16], [42, -8], [-28, -32], [45, 13], [5, -15], [17, 30], [-38, 16], [38, -8], [22, 30], [-2, -4], [-18, -2], [-14, 40], [-1, -34], [26, -4], [40, 20], [18, -8], [67, 3], [60, 8], [-39, 16], [-13, 16], [26, -32], [19, 28], [0, 4], [22, 10], [14, 40], [14, 0], [-1, 25], [26, 0], [-2, -30], [-52, -24], [36, 0], [-10, 32], [-42, -14], [-51, 9], [-35, 35], [-66, -12], [-24, 8], [-2, -16], [-15, -2], [54, 10], [-20, 44], [-9, 20], [2, -12], [-20, 28], [62, -12], [26, 8], [8, -36], [-5, -17], [8, -16], [-2, 28], [35, -19], [62, 16], [-14, -20], [-12, 36], [38, -28], [-5, -34], [-24, 20], [-27, -37], [-42, 32], [-68, 12], [-34, 16], [37, -22], [-70, -18], [-58, -16], [-59, 3], [-53, -2], [-20, -20], [12, 32], [21, 24], [-54, -8], [-39, 27], [4, 20], [8, -44], [32, 32], [-30, -24], [-21, -29], [86, -2], [69, -4], [71, -22], [-26, 24], [6, -4], [-1, 4], [16, -8], [49, 28], [-42, -36], [34, -14], [36, 8], [-24, 0], [30, -4], [18, -6], [-21, -12], [12, -48], [-35, 25], [5, 51], [-50, 38], [27, -44], [32, 12], [-7, -16], [17, -17], [62, -12], [28, 20], [21, 46], [16, -20], [39, -16], [28, 20], [16, -56], [30, -8], [-18, -24], [-18, -24], [-41, -3], [-15, 24], [-8, 16], [-33, -1], [-6, 0], [39, -2], [-66, -8], [-22, 10], [-18, 10], [-47, -22], [-58, -10], [88, 0], [6, 52], [34, -22], [-79, 1], [27, 4], [-68, -12], [10, -4], [21, 22], [-27, -33], [-54, 10]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3200_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3200_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_3200_2_a_bj(:prec:=2) chi := MakeCharacter_3200_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_3200_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_3200_2_a_bj( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3200_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-1, -2, 1]>,<7,R![-4, 4, 1]>,<11,R![-17, -2, 1]>,<13,R![-32, 0, 1]>],Snew); return Vf; end function;