// Make newform 6864.2.a.bs in Magma, downloaded from the LMFDB on 29 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_bs();". // 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_bs();". // 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 := [1, -3, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-2, -1, 1]]; Rf_basisdens := [1, 1, 1]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; 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, 0], [-1, 0, 0], [1, 1, 0], [1, -1, 1], [1, 0, 0], [-1, 0, 0], [0, 0, -1], [-1, 1, -1], [-1, -1, 3], [2, -4, -3], [-5, 1, 0], [0, 0, -4], [-1, -1, 1], [4, -6, -3], [-6, 4, 2], [-2, 4, -2], [-10, 2, 0], [-2, -2, 4], [-3, 7, 2], [-2, -4, -8], [-7, -3, -5], [6, 4, 1], [-6, 2, 0], [1, -1, 0], [-4, 0, 4], [-10, 2, 3], [0, -4, -4], [6, -6, -4], [-5, 7, -3], [0, -2, 8], [12, -8, -5], [-2, -6, -2], [13, 3, 4], [2, -2, 9], [7, -1, 5], [-5, 5, -1], [-5, 5, -5], [-5, 7, 2], [6, -6, 2], [-4, -8, 5], [-7, 5, 3], [9, 3, 1], [2, -10, -8], [-5, -1, 1], [-1, 7, -5], [-4, 4, 4], [-8, 2, -1], [1, -5, -4], [-3, -9, -5], [-16, 12, 6], [4, 2, 1], [3, -3, 9], [-2, 8, 2], [-10, 14, 8], [8, -6, 2], [-2, 10, 0], [-2, -8, -6], [-9, 5, 9], [-12, -2, 6], [5, 1, 5], [4, 4, 1], [3, 11, 7], [-11, 3, -7], [5, -7, -3], [-9, 17, 9], [-3, 3, -8], [13, -7, -4], [4, -6, -4], [-16, 10, 2], [-14, 4, 2], [-9, 11, 4], [13, 3, -1], [-16, 8, 6], [0, -14, -8], [1, -11, 2], [-8, 0, -2], [-4, -14, 0], [-12, -4, -16], [-3, 5, -10], [-7, -3, -13], [1, 1, -9], [12, -6, 6], [-6, -10, 14], [-26, -4, 4], [-4, 6, 5], [0, 12, 8], [-5, 7, -10], [-20, -10, 6], [-21, -1, 1], [-7, -3, -20], [23, -5, -5], [5, 11, 19], [1, 13, 10], [-10, -8, 8], [-17, -11, 8], [22, 6, 0], [15, -7, 0], [10, 8, 4], [-20, 2, -13], [-30, 0, 0], [22, 2, 11], [-1, -1, 9], [-14, -14, 2], [0, -4, -15], [-10, -8, 1], [24, -2, 2], [-2, -14, -16], [11, -13, 3], [18, 6, 0], [-10, 12, 8], [-16, 4, -1], [5, -11, -1], [19, 11, 10], [-17, -5, -4], [-11, 1, 0], [12, -10, -10], [-9, 21, 10], [-6, -2, 8], [0, 14, 10], [-8, -10, 8], [-22, 16, 10], [-14, -6, 14], [22, -4, -1], [18, 12, 0], [-35, 5, 0], [-22, -10, 3], [26, -2, 0], [4, -4, -4], [28, 4, -6], [29, 9, 7], [-1, -7, 11], [-12, 0, 10], [-12, 8, -2], [-29, -7, -5], [21, 1, 13], [-18, 8, 16], [5, -5, -20], [-9, -7, -13], [-18, -4, -12], [24, -8, 1], [-19, -5, 5], [33, -7, -3], [-8, 4, 10], [-19, 19, 3], [18, -12, 8], [18, -18, -18], [16, -22, -10], [14, -4, -11], [16, -12, -2], [16, 12, -10], [-6, -16, -2], [-22, -8, -4], [-4, 8, 8], [-12, 10, 0], [20, 0, 2], [-10, -2, -24], [-8, -20, -13], [-5, 17, 18], [4, -24, -16], [35, -1, -7], [4, 18, 4], [8, 18, -13], [1, 11, -13], [-35, -3, -3], [3, 13, 16], [-10, 18, 18], [10, 6, 20], [52, -2, 4], [-7, 5, 23], [0, -22, 6], [28, -4, 6], [12, -16, 8], [-10, 4, -10], [-10, 20, -6], [32, -16, -12], [6, -16, -16], [17, -9, -15], [-13, 13, 6], [8, -10, -17], [12, 6, -12], [3, 17, 12], [-8, -12, -12], [11, 21, -5], [-5, 11, 5], [32, -10, 4], [22, 12, 18], [-6, 22, 18], [5, 9, 22], [-10, -12, -10], [14, 26, -8], [19, 5, -21], [-14, 14, -6], [-4, -4, -20], [-3, 11, -2], [-6, 6, -2], [-26, -8, -16], [-14, 8, 8], [26, -20, -6], [3, 7, -9], [-12, 10, 8], [-27, 25, 13], [-24, 32, 15], [14, -4, 12], [8, 4, -22], [36, 2, 8], [-36, -2, -16], [-1, 7, 30], [14, -22, -2], [25, -3, -19], [-32, 0, 4], [-8, 12, 16], [18, -4, -20], [44, 0, -10], [-14, 22, -10], [13, -1, -17], [-16, -8, -26], [44, -14, -7], [-24, 6, -9], [-10, 16, 8], [39, -29, -14], [-4, 2, 30], [7, 13, 1], [1, 9, -2], [17, -29, -4], [-2, -24, -14], [18, 20, -6], [24, 6, 12], [22, 18, -4], [16, 6, 31], [-18, -6, -16], [-6, -16, -28], [-14, -16, 3], [19, -31, -21], [-14, 16, 3], [16, 4, -16], [4, 4, -24], [8, 18, 16], [25, -15, -11], [33, -23, -5], [7, 15, -17], [-16, -8, -14], [-36, 4, -11], [38, 14, -14], [36, 14, 8], [31, -17, -17], [4, 2, 20], [13, 5, 26], [4, 34, -4], [19, -3, -21], [16, 10, 4], [-4, -14, 28], [-13, -9, -11], [26, -12, -16], [-29, 37, 9], [28, 10, 13], [-13, -19, -35], [-19, 3, -5], [1, -17, -3], [-30, 28, 6], [-8, -2, 32], [18, -10, 15], [22, 16, 34], [32, -10, -12], [4, -26, -28], [-1, -15, 18], [-44, 16, -4], [-5, -9, -12], [31, -1, 14], [5, 15, 13], [-6, -12, 29], [30, -28, -2], [13, -33, 4], [-16, 6, 12], [-16, -8, 27], [-18, -16, 10], [25, -19, 8], [-10, 8, -8], [5, -11, -21], [-13, 3, 21], [-14, 16, -6], [8, -18, 2], [5, 9, -26], [-6, -8, 18], [15, -5, -15], [-37, 5, -19], [28, 12, 0], [-22, -4, 20], [-15, -19, 9], [-7, -35, 0], [-4, 14, -11], [12, 6, 10], [5, -11, 37], [-3, 21, -13], [-9, -7, -7], [24, 20, 10], [16, -20, -27], [-28, -4, -18], [-18, 20, 18], [-46, -16, -8], [19, -35, -29], [48, -2, 22], [13, 5, -33], [-8, 16, -7], [-21, -5, 3], [-22, -2, 2], [-54, -14, 14], [-25, 11, 25], [42, 0, 2], [2, -2, 14], [-10, 36, 10], [0, 24, 18], [-13, -37, 22], [-2, 18, 17], [-2, 2, -4], [-10, 14, 20], [-20, -10, -4], [10, 10, 26], [-15, -17, 15], [38, 8, 26], [-10, -8, 7], [28, -16, 1], [14, -24, -16], [44, 10, -1], [-43, 1, -24], [23, 15, -3], [15, -19, -29], [-45, -7, 4], [-13, 35, 16], [-6, -22, -24], [-4, 10, 19], [-11, 13, 11], [0, 22, -22], [46, -42, -25], [-23, -21, 23], [13, -13, 7], [26, -26, -16], [44, 4, 12], [-21, -5, -5], [-10, -12, -42], [26, 6, 15], [-8, 12, -8], [20, -38, -12], [-37, 35, 9], [22, 6, 28], [-9, 17, -16], [32, -8, 18], [2, -24, -8], [-15, -3, 24], [-55, -5, -1], [-16, 34, -6], [-27, -13, -31], [18, 20, 14], [25, 13, -4], [-30, -12, -9], [-24, -8, -22], [64, 12, -3], [-32, -6, -8], [23, 5, 25], [20, -4, -23], [-10, 8, 34], [8, -10, 18], [-30, -16, 9], [50, -2, 16], [3, 33, -11], [-4, -12, -28], [-3, 39, 21], [41, -15, 13], [-21, -15, 17], [-8, 28, -24], [-6, -6, 8], [-8, -4, 36], [37, -1, -25], [-4, -24, -25], [8, -30, -3], [22, -12, 12], [-10, 32, 14], [26, 22, -4], [-13, -17, 9], [-26, 48, 3], [18, -10, 16], [7, -9, -12], [5, -31, 14], [-15, 11, 25], [5, 1, -13], [-13, -15, 25], [56, -26, -6], [-13, -15, 14], [-26, 4, -6], [11, 13, -17], [7, -17, -27], [28, -12, 4], [25, 15, -12], [-4, -30, 22], [-12, -12, 20], [-6, -18, 16], [-34, 6, -4], [-11, 39, 37], [-6, 32, -24], [48, -4, -12], [3, -21, 13], [8, -22, -14], [12, -22, -10], [3, -27, -33], [-5, -35, -34], [-31, 9, 15], [-22, 18, -1], [22, 6, 20], [-37, 11, -27], [-57, -19, 14], [12, -42, 6], [14, 16, -24], [-26, 14, 15], [14, -18, -20], [-26, 36, 15], [-10, 8, -30], [36, 2, 6], [-45, 17, 9], [24, 16, -10], [5, -35, -16], [28, -14, 2], [-18, -8, -32], [-2, 32, -2], [-14, 20, -16], [-18, 14, 16], [4, 2, -6], [-47, -27, 12], [-16, 0, -12], [-5, 39, 21], [-7, -31, 8], [16, 22, 26]]; 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_bs();". // 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_bs(:prec:=3) 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_bs();". // 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_bs( : 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, -4, 1]>,<7,R![-4, -8, -2, 1]>,<17,R![-2, -4, 0, 1]>,<19,R![4, -8, 2, 1]>],Snew); return Vf; end function;