// Make newform 6027.2.a.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6027_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_6027_2_a_l();". // 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_6027_2_a_l();". // 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_6027_a();" function MakeCharacter_6027_a() N := 6027; order := 1; char_gens := [4019, 493, 2794]; 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_6027_a_Hecke(Kf) return MakeCharacter_6027_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], [0, -2], [0, 0], [4, 1], [2, 2], [5, 0], [4, -2], [2, 2], [0, -3], [4, 1], [-1, 0], [-1, 0], [3, 4], [-5, -4], [4, 4], [0, -4], [0, -1], [4, 2], [8, 3], [-4, -3], [-6, 4], [10, 0], [2, 4], [-8, 6], [5, 8], [4, -3], [0, -2], [6, -2], [0, -10], [4, 8], [-14, 2], [8, -3], [-8, -2], [0, -8], [10, 6], [-10, 0], [-11, -4], [-12, -4], [14, -4], [0, 9], [-20, 0], [0, -6], [10, -10], [4, -2], [8, 10], [8, 4], [12, -2], [-7, 4], [2, 4], [0, -8], [-12, -2], [0, -3], [-4, -2], [-3, 0], [-8, -7], [-14, -2], [-4, 9], [-15, -4], [12, -7], [4, -5], [9, 0], [-16, 1], [-12, -8], [4, -4], [28, -3], [-8, 14], [29, 0], [4, -5], [12, -9], [4, -10], [4, 16], [-4, 11], [21, -8], [-24, 0], [-29, -4], [14, -12], [14, 2], [-24, 4], [4, 1], [4, 16], [22, 2], [6, -2], [16, 9], [32, -2], [14, -6], [-6, -10], [-4, -20], [0, 14], [-2, 6], [4, -2], [-13, -16], [9, -4], [-10, 8], [2, -2], [-9, 12], [-15, 4], [7, -8], [0, -18], [10, -4], [-22, -6], [-12, -13], [-1, -4], [-2, 14], [28, -4], [18, -10], [35, -4], [-15, 16], [40, 0], [-16, 14], [20, -14], [2, 20], [-6, 8], [-32, 3], [7, 4], [-20, -13], [14, 14], [4, -4], [-4, -5], [-16, 2], [4, -20], [12, 2], [-10, 8], [12, -14], [10, 18], [-12, -2], [16, 10], [17, 0], [-6, 20], [4, 15], [-13, 12], [24, 2], [0, -14], [2, 8], [14, -14], [8, -3], [17, -8], [4, 13], [6, 8], [-48, 4], [0, 1], [-26, 10], [-2, 24], [-40, -3], [4, -11], [-44, 0], [24, 16], [-10, -14], [-8, 3], [32, -10], [35, -4], [-30, 2], [-2, 24], [7, 8], [-8, 8], [4, -12], [-22, -2], [23, 12], [-12, 4], [-16, -10], [30, -4], [-38, 10], [-12, -14], [-21, 4], [-12, 15], [-6, 12], [14, -22], [10, -18], [-13, 12], [2, 28], [-32, -2], [16, -16], [-12, -26], [-7, 12], [30, -2], [13, 16], [-16, -10], [24, 22], [14, -12], [2, 20], [-11, 20], [-30, 4], [-52, 0], [-6, 16], [38, 0], [-6, 12], [-2, 4], [-8, 10], [8, -28], [-9, 4], [-32, -15], [-8, -10], [20, 0], [-4, 26], [24, 16], [-14, -18], [-14, 0], [62, 2], [6, 16], [-45, 8], [-36, 4], [52, 1], [-18, 16], [16, 14], [-21, 16], [15, 4], [-44, -2], [-40, 9], [2, -6], [-4, 13], [4, -12], [27, -16], [-4, -12], [-20, 8], [3, -16], [20, -19], [-46, 10], [-18, 14], [-36, -11], [48, 6], [-10, -16], [-4, 23], [24, 1], [20, -6], [28, -2], [8, -30], [-6, 12], [-54, -12], [-4, -20], [4, -4], [14, 8], [20, 12], [5, 28], [-4, -8], [-26, -20], [4, -12], [10, 14], [52, 4], [-36, 20], [-11, 12], [-4, -34], [38, -14], [14, -32], [-11, -12], [-42, -14], [8, 10], [8, 26], [41, -4], [3, 16], [22, -8], [-13, -20], [10, -14], [-2, -40], [44, 3], [-2, -12], [2, -20], [0, -18], [20, -6], [-21, -4], [10, -20], [-17, -24], [36, -7], [-34, 20], [-34, -2], [-24, 6], [32, -19], [32, -29], [42, -4], [-28, 0], [-24, 7], [-32, -24], [-3, 32], [4, 5], [-3, -16], [38, 4], [0, 18], [28, -29], [36, 0], [30, -12], [29, -16], [-3, -8], [10, 14], [48, 14], [32, -16], [6, 30], [12, -13], [40, 14], [-6, 12], [-12, -28], [-45, -12], [-4, 11], [6, -20], [-4, -21], [12, 12], [-76, -6], [17, 12], [-26, -2], [-24, -34], [-60, 16], [-62, -6], [28, -16], [4, 4], [13, -4], [-24, 3], [-75, 0], [-38, 0], [-26, 26], [12, 41], [0, -6], [32, 18], [-27, -12], [-16, -32], [-24, -20], [-13, 16], [-42, 20], [-48, -4], [-10, -4], [12, 8], [6, 32], [-20, -21], [-44, 10], [10, 12], [-32, -6], [8, 8], [-24, 36], [4, -18], [0, -28], [-26, 6], [0, 19], [-38, -12], [-20, -8], [20, -2], [-28, 10], [-4, 32], [1, -36], [-25, 40], [14, 34], [-6, 16], [-22, -6], [-8, -37], [26, -10], [12, -8], [42, -4], [74, -12], [52, -8], [28, -36], [-28, 36], [-27, 16], [44, 12], [-12, -4], [-56, -12], [-21, -40], [-24, 25], [20, -23], [-14, -46], [6, -32], [-30, 0], [24, -4], [28, 14], [6, 30], [39, 24], [54, 20], [-32, 17], [-70, -6], [-12, 14], [40, -16], [34, -4], [23, -16], [19, 12], [-60, -2], [-14, 18], [-4, 48], [16, 31], [-64, -18], [8, 30], [38, 6], [-38, -12], [-16, 44], [61, 4], [-72, 6], [63, 20], [-16, -20], [13, 4], [32, 32], [18, -12], [-2, 40], [32, -32], [10, 30], [55, 0], [20, -36], [64, -6], [-3, 32], [16, 18], [-12, 6], [-40, -12], [34, -14], [-26, 8], [-40, 10], [19, -24], [20, 28], [48, 25], [20, 8], [61, -12], [-70, -4], [36, 36], [-36, 10], [37, 16], [40, 21], [-74, -16], [2, -26], [56, 17], [30, 2], [12, 14], [24, 12], [10, -24], [6, 50], [-46, 28], [-46, 0], [-51, -16], [-40, -18], [48, 20], [16, 10], [30, 12], [-29, -8], [56, 3], [12, 0], [40, -13], [28, 3], [-44, -2], [-3, 12]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6027_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6027_2_a_l();". // 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_6027_2_a_l(:prec:=2) chi := MakeCharacter_6027_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_6027_2_a_l();". // 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_6027_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6027_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![0, 1]>,<5,R![-12, 0, 1]>,<13,R![-8, -4, 1]>],Snew); return Vf; end function;