// Make newform 1638.2.a.y in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1638_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_1638_2_a_y();". // 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_1638_2_a_y();". // 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 := [-14, -1, 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_1638_a();" function MakeCharacter_1638_a() N := 1638; order := 1; char_gens := [911, 703, 379]; 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_1638_a_Hecke(Kf) return MakeCharacter_1638_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 := [[1, 0], [0, 0], [0, 1], [1, 0], [-2, 1], [-1, 0], [4, -1], [2, -1], [2, -1], [-4, -1], [8, 0], [0, -1], [2, -2], [2, -1], [0, 0], [-10, 0], [-8, 0], [-8, -1], [4, 2], [0, 0], [0, -3], [4, 2], [-4, 2], [14, 0], [-2, 4], [6, -2], [6, 1], [8, -2], [-4, 1], [6, 0], [-12, -2], [6, -3], [-4, 5], [4, 0], [-14, -2], [2, 1], [8, -5], [4, 2], [6, -1], [10, -4], [-12, 0], [2, 4], [2, -1], [-10, -2], [-6, 2], [-6, 3], [14, -3], [-4, 6], [12, 2], [2, 4], [-14, 2], [-8, -4], [-2, 4], [6, -7], [14, 0], [-28, 0], [-6, 0], [20, 2], [-10, 0], [2, -4], [-4, 0], [6, 0], [4, 4], [-12, 2], [-2, -6], [10, -2], [-4, -6], [-12, 5], [12, -4], [2, 0], [-10, -4], [32, 0], [-20, 4], [18, 2], [-16, -4], [-26, -1], [6, -8], [-14, 0], [-14, 4], [-20, 3], [2, -1], [-14, 4], [20, -2], [-2, 6], [-2, -3], [-16, 2], [-4, 1], [-10, -2], [-12, 7], [26, 1], [22, 1], [2, 5], [16, -8], [0, -2], [8, 4], [12, -2], [-20, 7], [0, 1], [36, 0], [4, 9], [-4, 0], [2, 10], [14, 5], [14, -4], [-20, -4], [-26, 4], [-4, -10], [-10, 4], [-26, 1], [-2, 6], [22, -3], [12, 1], [8, 3], [34, -1], [2, -3], [-2, 4], [6, -7], [-16, 8], [-16, 5], [16, -2], [-6, -8], [12, 1], [18, -8], [18, -5], [20, 0], [-2, 4], [-6, -4], [4, 6], [14, 5], [6, 2], [12, 6], [-20, 2], [-24, 8], [-18, -4], [-30, -2], [20, 3], [-20, 7], [-6, -1], [-10, 6], [22, 8], [10, -5], [-6, -6], [-8, -4], [22, 1], [16, -1], [-40, 4], [-2, -6], [-2, -4], [-20, 0], [-4, -6], [34, 4], [36, -1], [2, 7], [16, 4], [28, 0], [14, 9], [-4, -2], [-6, -2], [34, -4], [-34, -4], [30, -3], [-6, -6], [30, 3], [12, 4], [-16, 7], [-14, -11], [-4, -10], [-30, 4], [-6, 4], [-26, -2], [30, 5], [-22, -4], [-20, 12], [-38, 0], [-14, 3], [6, 8], [-20, 2], [-30, 6], [-12, 4], [-28, -3], [30, -5], [20, 4], [6, 0], [54, 0], [-16, 8], [-26, 2], [42, -6], [18, 7], [-6, -4], [6, -5], [46, 2], [-50, 1], [-20, 0], [24, -3], [24, -6], [-28, 3], [-2, -6], [-42, 4], [30, -4], [-8, 4], [18, -6], [22, 1], [2, 4], [26, 0], [-56, 0], [-6, -8], [8, 8], [-44, 4], [40, -5], [-20, 4], [0, -13], [-26, -2], [-12, -2], [52, -2], [-10, -13], [12, 3], [-4, 2], [-28, 7], [-8, -4], [10, -2], [-14, -2], [-16, 0], [-60, -3], [0, 4], [18, 7], [6, 0], [20, 3], [-4, 4], [30, 9], [2, 7], [38, 4], [-24, -6], [-26, 3], [2, 14], [-8, 14], [30, 3], [-4, 7], [-14, -2], [-16, -2], [0, 12], [28, 8], [12, -16], [14, -11], [-14, 4], [22, 0], [-8, -12], [-12, 6], [-48, 8], [28, -14], [6, 17], [8, 7], [-46, 2], [-56, 4], [18, -4], [-26, 2], [-14, 7], [-46, 2], [4, -8], [-6, 12], [0, 1], [-48, 0], [-20, 0], [-22, 0], [0, -5], [10, 2], [0, 14], [6, 12], [-10, -16], [0, -4], [46, -4], [10, -8], [36, 10], [8, 9], [-8, 4], [-10, 6], [-32, 0], [-14, -9], [-16, -1], [-2, 2], [4, 0], [-64, -4], [6, -9], [-24, -12], [22, 6], [10, -21], [-36, -8], [-20, 1], [-2, 12], [40, 0], [40, 1], [8, -11], [2, -1], [20, -11], [40, 4], [-2, -4], [6, 14], [-44, -4], [18, 12], [26, 3], [-34, 1], [-50, -6], [26, -2], [52, -6], [20, -12], [38, 1], [-2, 18], [28, -4], [36, 1], [-4, 14], [-42, 0], [32, -4], [-14, -2], [-6, -10], [-62, 7], [12, 10], [54, 2], [-64, 0], [-8, 4], [-28, 7], [22, -4], [-24, 2], [54, -2], [-10, 14], [34, 9], [68, -7], [-6, 0], [-32, -12], [42, -13], [-30, -1], [-24, -3], [40, -9], [8, 1], [44, -4], [20, 2], [-8, 12], [-12, 10], [-24, -1], [2, 14], [-82, 4], [22, 1], [26, -8], [-22, 2], [24, -3], [36, -2], [22, 4], [40, 10], [20, -15], [-52, 6], [-16, 0], [-30, 16], [-28, 4], [-10, 6], [-22, -10], [12, -16], [-74, 4], [68, 3], [18, -7], [20, 10], [-26, 4], [40, -4], [-4, -11], [20, 11], [18, -1], [-20, 16], [64, 6], [-30, 20], [-28, -5], [-18, -1], [16, 15], [40, -14], [68, -4], [56, -4], [-28, -5], [2, -9], [10, 18], [-56, -4], [30, 1], [28, 3], [4, 7], [42, -4], [-34, -4], [8, 15], [14, -13], [2, -8], [28, 0], [46, -5], [-82, 3], [8, -9], [8, 16], [-60, 6], [-22, 4], [-34, 0], [-52, -4], [-20, 0], [16, 4], [42, 8], [20, -18], [-12, 11], [2, 3], [-58, -8], [-34, -10], [-10, -16], [-46, -7], [42, -6], [-2, -8], [-66, 5], [74, -8], [36, -3], [-14, -13], [8, -4], [-38, 16], [-36, 15], [-36, 0], [62, 9], [40, 3], [-18, -18], [4, -2], [-6, -13], [38, 0], [-64, 0], [-32, 13], [-46, -4], [48, 12], [28, -16], [16, -23], [-50, -8], [-64, 2], [-6, 4], [58, -1], [32, -4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1638_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1638_2_a_y();". // 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_1638_2_a_y(:prec:=2) chi := MakeCharacter_1638_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_1638_2_a_y();". // 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_1638_2_a_y( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1638_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-14, -1, 1]>,<11,R![-12, 3, 1]>,<17,R![-2, -7, 1]>,<19,R![-12, -3, 1]>],Snew); return Vf; end function;