// Make newform 6720.2.a.ct in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6720_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_6720_2_a_ct();". // 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_6720_2_a_ct();". // 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; Rf_num := [[1, 0], [-1, 2]]; Rf_basisdens := [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_6720_a();" function MakeCharacter_6720_a() N := 6720; order := 1; char_gens := [1471, 3781, 4481, 5377, 1921]; 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_6720_a_Hecke(Kf) return MakeCharacter_6720_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, 0], [1, 0], [-1, -1], [-1, 1], [2, 0], [-3, 1], [0, 0], [2, 0], [1, 1], [2, 0], [2, 0], [-2, -2], [0, 0], [-1, -3], [4, 0], [0, 2], [-6, 2], [-1, -1], [11, 1], [6, -2], [4, 0], [8, -2], [9, -1], [-8, 2], [0, 0], [8, 0], [0, -2], [7, -3], [4, 4], [-10, 2], [17, -1], [5, 1], [4, 2], [4, 4], [-3, -1], [-2, -2], [-16, 0], [-8, 2], [7, -1], [-6, -4], [1, 1], [18, 0], [-3, -5], [1, 1], [-14, 2], [-10, -2], [4, 0], [12, -2], [19, 1], [-9, -1], [12, 2], [12, 0], [12, 2], [-10, -2], [8, 2], [3, -5], [-2, 4], [0, 6], [20, 0], [8, 2], [18, -2], [-16, 0], [21, 3], [11, 1], [6, -2], [-14, 0], [18, 2], [20, -2], [6, -4], [13, 5], [22, -2], [12, 2], [4, -4], [-12, 4], [12, 2], [15, 1], [-6, -8], [-14, 0], [26, -2], [-10, 4], [-13, -5], [9, -1], [13, 5], [-2, 6], [-14, 0], [-4, 2], [2, -4], [8, 0], [-4, -8], [8, 0], [32, 0], [-9, -1], [4, 4], [-20, 4], [20, -2], [6, 4], [10, 6], [-2, 4], [-30, 2], [11, 1], [0, 4], [-6, 0], [20, -4], [-3, -5], [16, 4], [14, 4], [1, 1], [-8, -2], [22, -2], [10, -8], [-1, -3], [13, 1], [6, -2], [0, -2], [14, 2], [-8, -8], [13, -5], [-11, 5], [-40, 2], [-4, 10], [-16, 2], [22, -2], [-33, 3], [22, 4], [26, 0], [-2, -10], [6, 6], [27, 5], [40, 0], [-10, 6], [2, 2], [-44, 2], [-4, -6], [-24, -2], [8, 2], [10, -10], [12, 6], [12, 2], [33, 5], [-22, -8], [4, -4], [-16, -8], [10, -4], [-14, -6], [-21, 5], [-40, -2], [-1, -5], [40, 0], [6, -4], [-32, -2], [-10, -2], [-48, 0], [14, 6], [-19, 5], [-10, -2], [-20, 2], [3, 9], [-10, -8], [10, 10], [15, -3], [16, 0], [-44, 0], [23, -3], [-28, 4], [-34, 6], [-9, -7], [-32, -2], [-24, -6], [19, -5], [-14, -4], [-33, -1], [18, 8], [-25, 7], [-22, 0], [18, 2], [-32, 6], [6, -2], [34, 4], [32, 0], [40, -4], [0, -2], [-6, -8], [-22, -6], [4, -10], [10, -8], [14, 10], [12, 2], [22, -2], [9, -9], [-44, -4], [12, -4], [-34, 8], [42, 2], [2, 0], [-40, 6], [-8, -2], [-20, 2], [-32, -8], [-26, -4], [-3, -3], [-31, 3], [-8, -10], [-24, 4], [20, -10], [-45, 3], [-64, 0], [20, -6], [27, 7], [8, -10], [-24, 2], [20, -12], [-20, 8], [10, 10], [-16, -2], [20, -4], [-32, 6], [-6, 2], [1, -1], [-14, -8], [-19, 5], [-16, 14], [48, 0], [4, 0], [-56, -2], [8, 6], [33, 9], [40, 0], [27, 11], [4, 10], [41, 5], [24, 0], [20, 10], [6, -14], [24, 0], [6, 12], [-39, 7], [-53, 3], [-10, 6], [-8, 8], [-3, -15], [-12, 12], [-20, 2], [-12, 10], [18, -6], [-46, -6], [10, 6], [46, 6], [40, 0], [-38, -8], [20, 2], [8, 0], [14, 4], [40, -6], [7, -17], [-24, -2], [-48, 4], [20, -2], [-7, 7], [-12, 4], [6, 6], [-8, 2], [-31, -5], [-14, 0], [-45, -1], [10, 0], [-6, -16], [30, 6], [-11, 11], [6, -16], [10, -6], [17, 7], [-26, 6], [49, -1], [-70, 2], [20, -12], [-18, -12], [10, 0], [-60, 0], [8, -16], [30, -10], [-28, -12], [-34, 8], [-22, -10], [21, -3], [-36, -6], [-25, 5], [1, -15], [-24, -10], [2, -8], [8, 4], [51, 1], [-14, -2], [0, -18], [18, 12], [-9, -1], [38, -8], [-4, 16], [-30, 6], [35, 1], [-27, 3], [-40, 0], [-66, -2], [72, 0], [-2, -4], [20, -12], [-24, 10], [23, -1], [72, -2], [-12, 12], [-70, 0], [-10, -12], [38, -10], [-20, 4], [-4, 18], [30, 2], [-3, -3], [41, 7], [-10, 4], [-25, -13], [0, 6], [42, -4], [-32, -8], [-3, -5], [0, -10], [28, 4], [12, 8], [0, 0], [63, 5], [0, 6], [-31, -1], [-69, -5], [32, -12], [-2, 14], [-44, 0], [-54, 8], [0, -10], [56, -2], [-6, -6], [-8, -2], [-21, 9], [16, -14], [-12, 12], [-3, 11], [29, -3], [20, -10], [10, 10], [-14, 2], [-48, -6], [-59, 1], [-12, 10], [30, -12], [0, 16], [12, -6], [-6, -8], [38, -2], [-30, 14], [-29, 9], [-38, 10], [-64, 6], [10, 16], [46, -2], [33, 9], [-38, 6], [-28, -6], [38, 0], [-12, 4], [-30, -8], [15, -9], [27, 7], [50, -6], [32, -2], [19, -5], [30, -4], [-10, 2], [17, -7], [1, 15], [14, -12], [19, -15], [-4, 6], [-9, 5], [-48, 8], [61, 3], [-15, 13], [-48, -8], [30, 6], [7, 9], [30, 6], [-36, -4], [28, -6], [10, 4], [-11, -3], [-62, 6], [-17, -9], [-34, -12], [-57, -9], [20, 18], [48, 0], [-62, -8], [-78, 4], [-17, 5], [20, -4], [-48, 6], [18, -12], [3, 3], [14, 12], [-26, -12], [80, 4], [16, -12], [-13, -15], [-33, -11], [-18, 14], [2, 2], [-58, -4], [-8, 2], [79, -1], [54, 6], [26, -8], [-56, 0], [-40, -10], [39, 9], [-6, -14], [-36, -16], [-11, 19], [92, -2], [-64, 0], [18, 8], [-1, -21], [90, 2], [-6, 8], [25, 9], [42, -14], [-28, 12], [13, 7], [-14, 8], [-22, 0], [52, -6], [-26, -10]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6720_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6720_2_a_ct();". // 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_6720_2_a_ct(:prec:=2) chi := MakeCharacter_6720_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(3067) - 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_6720_2_a_ct();". // 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_6720_2_a_ct( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6720_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-16, 2, 1]>,<13,R![-16, 2, 1]>,<17,R![-2, 1]>,<19,R![-8, 6, 1]>,<23,R![0, 1]>,<29,R![-2, 1]>,<31,R![-16, -2, 1]>],Snew); return Vf; end function;