// Make newform 4002.2.a.v in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4002_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_4002_2_a_v();". // 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_4002_2_a_v();". // 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 := [-4, -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_4002_a();" function MakeCharacter_4002_a() N := 4002; order := 1; char_gens := [2669, 3133, 553]; 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_4002_a_Hecke(Kf) return MakeCharacter_4002_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 := [[1, 0], [-1, 0], [-1, -1], [0, 0], [-1, 1], [3, -1], [-4, 2], [-4, 0], [-1, 0], [1, 0], [3, 1], [-5, -1], [3, 3], [-6, 2], [6, 2], [-2, -4], [5, -5], [-9, 3], [-1, -3], [-5, 1], [-8, 2], [6, -2], [6, -6], [2, 4], [-10, 0], [1, 1], [-11, -1], [-4, -4], [2, 0], [-14, 4], [-1, -3], [8, -4], [12, -6], [6, 6], [-7, 5], [0, -8], [-6, 4], [-1, 1], [-11, 7], [6, -8], [-4, -8], [-14, 0], [-13, 5], [-8, -6], [-14, 4], [-7, 3], [-19, 3], [12, -4], [-2, 10], [-5, -1], [4, -2], [3, 1], [0, -6], [-13, 5], [14, 4], [8, 4], [-9, -5], [21, -1], [-1, 3], [14, -4], [-1, -11], [28, 2], [5, 3], [10, -2], [-22, 0], [-11, 5], [-20, 0], [15, 3], [-4, -8], [9, 9], [6, -4], [-16, -4], [6, -2], [-6, 16], [30, -2], [-18, 2], [16, -10], [-18, 0], [-3, 9], [-2, -12], [6, -6], [9, 9], [34, -2], [2, 12], [-18, 2], [-6, -6], [-1, -1], [0, -6], [25, 1], [-8, 0], [19, -11], [9, -9], [-4, 4], [-6, 18], [6, 6], [-8, -4], [2, -4], [6, -4], [11, 9], [-26, 8], [36, 0], [9, -3], [1, 7], [0, 14], [17, 3], [8, -6], [8, -4], [0, 10], [0, 0], [18, -16], [0, 0], [-10, -12], [-6, -12], [-6, -6], [19, 9], [-12, -14], [4, -12], [-25, 13], [3, -17], [-20, 16], [4, -18], [-14, 0], [-28, 10], [21, 3], [-22, -6], [-23, 5], [0, -14], [-15, 11], [-2, 6], [-1, -5], [-15, 7], [-17, -7], [32, 4], [43, -1], [18, -8], [15, -13], [28, -14], [-39, 3], [-6, -12], [1, 5], [16, -4], [-36, 10], [-5, 9], [25, -1], [12, -6], [-7, -1], [-4, 18], [-18, 12], [12, 8], [-11, 7], [6, 0], [14, -16], [28, -16], [40, -8], [-14, -6], [7, 1], [-27, -1], [14, -28], [18, -24], [23, -1], [16, -4], [-30, 0], [16, 8], [-19, 19], [-23, 5], [47, -7], [28, 12], [14, -24], [0, -6], [2, 20], [-12, 4], [-14, -12], [42, -2], [2, 12], [40, 0], [-23, 13], [-36, 0], [22, -16], [-42, 14], [6, 12], [-31, -5], [5, -13], [-51, -3], [22, 4], [8, -16], [27, 3], [-6, 20], [-6, 10], [16, -6], [0, -16], [-41, -1], [-26, 14], [-12, 16], [26, 4], [19, -11], [-42, 12], [-53, -1], [-32, -6], [6, -12], [23, -11], [-22, 4], [26, 6], [40, -10], [-34, 0], [-24, 20], [-36, 10], [-22, 18], [-39, 15], [4, -2], [15, 9], [-18, -12], [8, -2], [-12, -16], [-4, -8], [24, 16], [-30, 24], [-6, 14], [-37, -5], [-24, 0], [-45, 15], [-18, 24], [-5, -7], [45, -15], [26, -6], [-24, 28], [-54, -4], [-40, -10], [50, -2], [12, -16], [27, -3], [-16, 22], [-28, 12], [50, -6], [-26, 20], [-27, -1], [36, 0], [-23, -15], [-8, -2], [-1, 17], [0, 0], [12, 0], [4, 0], [-22, -2], [-18, 0], [10, -8], [8, 16], [20, 12], [2, 14], [27, 1], [-17, 25], [-26, 32], [13, 17], [-48, -12], [61, -3], [-30, 12], [27, -3], [26, 12], [-20, -16], [-30, -12], [3, -5], [-60, 8], [-4, -8], [9, -15], [53, -11], [2, -8], [42, -6], [38, 8], [36, -14], [-40, -12], [-36, -14], [-51, 5], [-36, 12], [22, -4], [2, -14], [24, -6], [24, 8], [56, -4], [30, 0], [-33, 3], [12, -12], [-43, 7], [-24, -16], [4, 4], [-48, -6], [0, -20], [30, -14], [50, 12], [4, 10], [32, 0], [30, 4], [2, 8], [-38, -2], [-36, 6], [4, 16], [-24, 30], [-3, 9], [30, 6], [6, 24], [-11, -13], [-28, 24], [25, -15], [-27, 1], [-10, 6], [-30, -6], [52, 0], [-18, -12], [-48, 12], [-12, -18], [-2, -14], [-58, 0], [-33, -3], [18, -28], [-11, 25], [46, -10], [-10, 26], [6, -12], [-12, -8], [41, 11], [-32, 18], [-35, 25], [0, 4], [24, 26], [0, -34], [24, 8], [26, 8], [-17, 27], [-47, -13], [-6, -6], [-3, 3], [-63, -3], [-24, -2], [30, -8], [-14, 30], [-6, -6], [66, -6], [-68, 16], [-4, 30], [-26, -8], [-66, 12], [15, 21], [-44, -6], [51, -21], [-3, 21], [11, -15], [-18, 4], [-52, 16], [32, -18], [-3, -21], [-21, -19], [-20, 2], [34, -30], [-45, 11], [-37, 7], [-27, 7], [26, -12], [51, -5], [33, -9], [66, 6], [-3, 1], [-37, -7], [-16, 42], [6, -12], [62, 2], [-20, 44], [29, 27], [-18, 4], [-54, -12], [51, 17], [-18, -24], [-46, -14], [-18, 22], [16, -8], [74, -4], [63, 5], [21, 21], [6, -18], [33, 23], [-39, -15], [27, -13], [54, -12], [30, 24], [-15, 5], [27, -15], [-3, 9], [-20, 12], [-47, -5], [24, 0], [-34, 0], [18, -30], [-33, 33], [-3, -11], [-42, 0], [-48, -12], [-15, 7], [-9, -7], [68, -18], [-48, 24], [24, -18], [9, 19], [-10, -32], [19, 15], [59, 11], [-6, 18], [-33, 31], [12, -6], [2, -18], [68, 6], [55, -17], [-30, -6], [33, -9], [33, -3], [-22, 4], [-24, 24], [12, -12], [-11, 21], [58, 4], [-54, 6], [14, -14], [12, -30], [-63, 15], [-55, 13], [-18, 0], [8, -16], [-14, 26], [-27, -15], [76, -14], [-92, 4], [86, -4], [44, 0], [-4, -20]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4002_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4002_2_a_v();". // 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_4002_2_a_v(:prec:=2) chi := MakeCharacter_4002_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_4002_2_a_v();". // 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_4002_2_a_v( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4002_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-2, 3, 1]>,<7,R![0, 1]>,<11,R![-4, 1, 1]>],Snew); return Vf; end function;