// Make newform 6400.2.a.ce in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6400_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_6400_2_a_ce();". // 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_6400_2_a_ce();". // 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_6400_a();" function MakeCharacter_6400_a() N := 6400; order := 1; char_gens := [4351, 4101, 5377]; 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_6400_a_Hecke(Kf) return MakeCharacter_6400_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], [-1, 1], [-2, 0], [0, -2], [0, -2], [-4, 2], [-1, -3], [0, -4], [-2, -2], [2, 0], [2, -2], [7, -1], [-5, 1], [-8, -2], [-4, -2], [2, 4], [9, 1], [-2, -2], [4, 2], [-8, 4], [3, -1], [-2, -4], [4, 6], [-4, 4], [7, 5], [-1, -1], [-10, -4], [6, 4], [-15, -1], [6, 8], [-2, 4], [-4, 2], [6, -8], [2, 6], [-10, 4], [5, -3], [-15, 3], [2, 0], [12, -2], [16, 0], [-2, 10], [-4, 2], [16, 2], [-12, 8], [-6, -12], [11, -3], [-9, 11], [-4, 0], [12, -10], [-20, 0], [6, 6], [18, 4], [2, 0], [-3, -5], [2, 4], [-2, -10], [-2, 0], [-14, 2], [1, 5], [2, 8], [1, -15], [-14, -10], [-18, 8], [12, -2], [-14, 0], [6, 8], [-7, 5], [28, 0], [-6, -4], [-12, -4], [-15, 7], [-2, 16], [-12, -14], [-9, -7], [-14, 12], [8, 14], [6, -8], [-6, 10], [-8, -6], [14, -8], [-18, -2], [16, 2], [-20, -12], [5, -15], [-6, -10], [-6, -12], [4, 4], [13, -1], [-17, -5], [-8, 8], [-9, 9], [-10, -4], [-28, -2], [-9, 5], [-12, 8], [30, 8], [17, -3], [20, 4], [35, -1], [-6, 12], [-3, -11], [10, 2], [-14, 12], [2, -24], [9, -3], [2, -20], [0, 20], [-22, 2], [-5, -15], [-16, -6], [-24, 2], [-16, 10], [6, 10], [-10, 6], [27, 7], [15, 1], [-12, 14], [0, 10], [-22, 8], [-16, 2], [-28, 6], [-9, 15], [-18, 0], [-26, 4], [8, -12], [12, -8], [-5, -11], [-42, 4], [-12, -10], [-15, 15], [2, 14], [26, -4], [-22, 16], [6, 4], [12, 6], [1, 9], [12, 22], [-10, -8], [-22, -4], [14, 12], [3, -7], [-11, -7], [-22, -4], [-4, -12], [32, -6], [-6, -8], [4, 14], [-39, -5], [26, -16], [6, 2], [5, -19], [19, -19], [-17, 7], [-18, -18], [32, 4], [-14, -2], [-20, 14], [32, 0], [9, -7], [-38, 12], [25, -5], [30, 4], [28, -2], [-47, -1], [38, 2], [-20, 18], [-26, -12], [0, -22], [4, -30], [8, 16], [-42, -6], [32, -10], [4, -12], [-14, 12], [-2, 0], [48, 0], [3, -15], [-22, -16], [25, 15], [2, -28], [-44, 2], [-8, -10], [-35, -9], [-40, -4], [-42, -8], [-33, 3], [2, 10], [-46, 6], [-26, -4], [9, -11], [2, 20], [-6, 24], [-39, -7], [36, -10], [14, 2], [-8, -22], [-6, 4], [-49, -3], [0, -12], [2, -2], [28, 18], [14, 16], [32, 2], [18, -24], [20, -16], [39, 3], [18, 28], [42, -16], [12, -30], [6, 8], [25, 15], [-23, 5], [48, 4], [34, 14], [29, 11], [30, -8], [45, -9], [0, 14], [2, 24], [20, 4], [-6, 36], [15, 25], [45, 13], [-30, 8], [-24, 6], [0, 4], [-3, -41], [-30, 0], [16, 22], [0, -30], [34, -2], [10, -30], [11, 19], [45, 3], [26, -2], [0, 6], [-24, 26], [-10, 18], [-5, -9], [42, -4], [-11, 11], [-12, 8], [-6, 36], [12, -36], [5, -13], [-30, -8], [-4, 22], [-35, 7], [12, 10], [-10, 0], [35, 13], [6, -38], [-6, -8], [4, 2], [-28, -4], [-15, -35], [26, 4], [46, 0], [29, -9], [23, -13], [20, 0], [-10, 8], [-20, 10], [44, -18], [52, 16], [-18, -16], [-15, -19], [22, -20], [-16, 0], [-1, 11], [0, 10], [48, -12], [-58, -12], [17, 23], [59, -5], [32, 20], [26, 2], [6, 28], [-63, -5], [6, -2], [33, 11], [24, 8], [21, 1], [-18, 2], [12, 10], [-16, -38], [4, 16], [-42, -2], [-10, 0], [-19, 21], [-60, -6], [-58, 4], [-2, -4], [-18, 32], [-10, 10], [-18, 36], [4, -34], [-1, -29], [-14, -8], [30, 28], [-16, 16], [71, -5], [-10, -32], [-4, -10], [27, 27], [-56, 12], [-36, 20], [34, 32], [-33, 17], [36, 24], [-6, 24], [-35, 5], [-3, 7], [-2, 28], [-72, -2], [-30, -22], [-14, -8], [54, 14], [26, 4], [-46, 0], [2, 4], [37, -17], [30, 28], [-2, -34], [-8, 10], [5, 1], [11, 25], [-24, -10], [-18, 8], [-10, 4], [-52, 12], [-25, 35], [10, 24], [-21, 3], [70, 12], [26, 16], [2, -14], [-21, -15], [26, -8], [62, 0], [-66, 4], [26, 6], [42, -12], [-12, 14], [-22, -12], [-19, -7], [-54, -10], [-44, 22], [-2, -28], [46, -24], [-30, -16], [-1, 1], [12, 8], [32, 14], [-40, -12], [58, -8], [0, 26], [-27, -5], [-28, -30], [10, -24], [-15, 7], [-48, -10], [-55, 1], [20, 30], [-56, -2], [49, 7], [-6, -34], [22, -4], [68, 10], [-51, -17], [-50, 20], [-38, -2], [-40, 6], [32, -26], [10, -2], [-2, 24], [-6, 22], [22, -12], [-8, -16], [-34, -4], [-67, 7], [-54, -12], [16, 18], [-41, 29], [50, 14], [-10, -44], [-23, -43], [-9, -3], [-2, -10], [18, -4], [72, 6], [-21, 15], [-18, 30], [10, 0], [-12, -20], [-34, 24], [-22, -12], [34, -4], [22, -24], [36, 2], [-19, -5], [-30, 0], [-30, -28], [54, -10], [68, 6], [14, -8], [-53, -9], [4, -6], [-34, 24], [-34, 20], [-75, 1], [30, 8], [-16, -10], [36, -4], [-52, 8], [-43, 31], [84, -2], [-39, 15], [-38, -8], [80, -14], [-33, -27], [-60, -2], [-30, -44], [-2, -36], [11, 15], [6, 42], [58, 20], [-72, 4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6400_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6400_2_a_ce();". // 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_6400_2_a_ce(:prec:=2) chi := MakeCharacter_6400_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_6400_2_a_ce();". // 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_6400_2_a_ce( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6400_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-2, -2, 1]>,<7,R![-2, 2, 1]>,<11,R![2, 1]>,<13,R![-12, 0, 1]>,<17,R![-12, 0, 1]>,<29,R![-48, 0, 1]>,<31,R![-8, 4, 1]>],Snew); return Vf; end function;