// Make newform 6864.2.a.bk in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6864_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_6864_2_a_bk();". // 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_6864_2_a_bk();". // 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_6864_a();" function MakeCharacter_6864_a() N := 6864; order := 1; char_gens := [2575, 1717, 4577, 4369, 2641]; v := [1, 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_6864_a_Hecke(Kf) return MakeCharacter_6864_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], [-1, 1], [2, 0], [1, 0], [-1, 0], [5, 1], [4, -2], [2, 0], [1, -3], [3, 3], [-2, 0], [-2, -2], [1, 3], [-2, 2], [2, 0], [10, 2], [-2, 4], [7, 3], [0, -4], [-4, 4], [-5, 5], [0, -8], [-7, -5], [2, 4], [3, 3], [-2, -10], [-2, 6], [-8, 0], [10, 0], [-1, -7], [12, -4], [1, -9], [15, -3], [8, 0], [8, -2], [2, -10], [19, -1], [-4, 8], [7, 7], [6, -8], [-2, 6], [12, 4], [0, -4], [8, 0], [-14, 2], [-21, 1], [1, 13], [0, 6], [-12, -2], [-1, 3], [-12, -6], [2, -12], [24, -4], [0, 10], [-6, -2], [0, -10], [8, -10], [-10, 0], [6, 10], [-15, 3], [20, -4], [24, -2], [10, -4], [-6, -2], [5, 7], [-1, -13], [-8, 10], [-6, 6], [10, -8], [-3, 15], [20, -2], [-24, -4], [8, -14], [-7, -3], [-14, -6], [12, -2], [18, 4], [21, 3], [0, -16], [-26, 0], [2, 4], [-8, 12], [-6, 0], [-3, 19], [-16, 12], [3, -3], [-10, -12], [-14, -2], [3, -13], [26, 4], [-24, 2], [17, 5], [-6, -2], [23, 3], [18, -2], [-33, 5], [-34, 4], [23, 1], [6, 12], [13, 15], [-24, 0], [12, -16], [7, -5], [-1, 9], [-2, 4], [-6, -2], [36, 0], [0, -16], [14, 8], [-29, -3], [4, 8], [15, 5], [13, 9], [15, -5], [6, 8], [-11, -11], [-12, 4], [20, 10], [32, -4], [-8, 2], [-20, -2], [3, -25], [4, 12], [3, -17], [-25, 11], [-8, 2], [8, 0], [-22, -10], [-16, -4], [2, 0], [-24, -8], [24, -12], [-6, -2], [20, 20], [30, -8], [-27, -13], [30, 4], [-40, -6], [3, 23], [-12, -6], [-30, 10], [-22, 6], [4, -14], [-34, -8], [4, -16], [-26, 8], [37, -3], [2, -30], [6, -10], [2, 8], [6, -20], [-22, 14], [24, 20], [-4, 4], [-16, -8], [-13, -7], [-39, 7], [-12, -26], [-6, 6], [-18, -10], [9, 1], [-34, -4], [10, 8], [-9, 5], [40, 4], [-26, -18], [24, -10], [-4, -20], [-16, -2], [12, 8], [38, -12], [-24, -8], [-6, 20], [-34, 14], [-28, 10], [10, 12], [33, -1], [23, 9], [-28, 14], [-1, -25], [-12, 20], [-22, -2], [10, -14], [-22, 18], [-50, -4], [36, 2], [-19, 9], [-6, -20], [0, -4], [-46, -6], [12, -20], [14, -6], [39, -15], [12, 28], [12, 10], [10, -4], [10, -16], [18, 2], [0, 24], [-22, -6], [-33, -7], [-26, -12], [-22, 24], [-6, 10], [0, -38], [-5, -29], [52, 0], [-6, -26], [-6, 14], [12, 2], [-18, -24], [-20, -8], [12, 28], [20, -18], [-28, -2], [-11, -21], [-47, 5], [8, 24], [-49, -3], [-10, -28], [2, 20], [-39, -9], [29, 9], [36, -4], [12, 22], [-12, -22], [-12, -12], [-53, -7], [-48, -8], [2, 12], [-3, 3], [28, 26], [19, -13], [10, -18], [-36, -4], [0, 2], [4, 16], [50, 4], [22, 12], [-54, 6], [-35, -9], [4, 0], [-24, 22], [20, 0], [-28, 0], [-11, 25], [6, -14], [-22, 16], [16, 16], [-18, 0], [34, 2], [20, -12], [-2, 34], [31, 23], [-20, -2], [2, -10], [56, 6], [12, 10], [-62, -8], [-11, -37], [-28, -12], [26, 4], [4, 6], [-15, -21], [-24, 28], [45, -21], [-9, -35], [32, -2], [-19, -15], [-70, 0], [-59, -11], [-32, -10], [-29, -15], [-20, 2], [23, -21], [36, -20], [72, -4], [36, -16], [-32, 12], [-38, 6], [13, 17], [-28, -20], [-8, -40], [-2, 36], [-12, 4], [-30, -24], [38, 14], [47, -13], [39, 11], [-32, -10], [14, -28], [-44, 4], [4, -2], [-62, -8], [-25, -5], [18, 22], [50, -16], [6, 18], [-32, 2], [-6, 32], [6, -2], [13, 39], [-10, -8], [-10, -42], [6, 40], [-58, -4], [2, -24], [4, 8], [-64, 2], [-26, -14], [-5, 13], [21, -11], [-36, -8], [-20, 24], [-20, -26], [16, 0], [12, 22], [-68, 2], [-21, -21], [13, 19], [-46, 16], [-15, 17], [-25, 35], [-20, 32], [-2, 30], [-25, -29], [-31, 25], [-6, -22], [-11, 41], [-16, 12], [18, 24], [5, -1], [-12, 10], [10, -20], [10, 2], [26, 32], [14, -2], [-20, -2], [17, 35], [-62, -4], [8, -26], [16, 8], [10, 18], [-13, -7], [70, -6], [6, 32], [55, 19], [-12, -26], [22, -12], [54, 16], [-16, 38], [3, 25], [-55, -5], [54, 20], [-37, -9], [34, 10], [-36, 14], [-33, -21], [-42, 14], [-2, -48], [-39, -19], [0, 16], [-44, 30], [8, -4], [-38, -10], [26, -14], [-72, -2], [-68, -2], [-48, 28], [22, -14], [-28, -26], [33, -7], [1, 7], [46, 24], [-46, 10], [-58, -2], [76, -8], [-1, -41], [-30, -24], [57, -13], [21, 7], [30, -12], [-18, 42], [48, 2], [20, -24], [9, 17], [26, -24], [-2, 28], [-10, -32], [-26, -20], [9, -1], [12, 28], [-20, 40], [-30, 10], [10, 48], [44, -26], [-42, 12], [14, 18], [-20, 36], [26, 36], [84, -10], [22, -16], [-7, 35], [38, -18], [17, -5], [60, 18], [44, 0], [-19, -7], [0, 20], [2, 0], [43, 15], [-12, 8], [21, -1], [60, 6], [-72, 6], [20, 18], [30, 26], [59, -3], [22, -26], [-76, -10], [42, 16], [0, -8], [-18, -18], [6, -4], [63, -19], [-36, -32], [20, 32], [35, -9], [66, -18]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6864_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6864_2_a_bk();". // 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_6864_2_a_bk(:prec:=2) chi := MakeCharacter_6864_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_6864_2_a_bk();". // 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_6864_2_a_bk( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6864_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-2, 2, 1]>,<7,R![-2, 1]>,<17,R![22, -10, 1]>,<19,R![4, -8, 1]>],Snew); return Vf; end function;