// Make newform 6027.2.a.n in Magma, downloaded from the LMFDB on 28 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_n();". // 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_n();". // 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_n();". // 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_n(: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_n();". // 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_n( : 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;