// Make newform 2898.2.a.bb in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2898_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_2898_2_a_bb();". // 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_2898_2_a_bb();". // 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 := [-10, -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_2898_a();" function MakeCharacter_2898_a() N := 2898; order := 1; char_gens := [1289, 829, 1891]; 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_2898_a_Hecke(Kf) return MakeCharacter_2898_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], [4, 0], [2, 1], [-4, 0], [2, -2], [-1, 0], [-4, 1], [-2, 0], [8, -1], [4, -1], [2, -1], [0, 3], [2, 2], [0, 2], [-2, 0], [0, 4], [0, 2], [-6, 2], [8, 0], [-6, -2], [4, 2], [10, -1], [-4, 2], [6, -1], [-4, -4], [-4, -1], [4, -3], [-2, -1], [0, 2], [16, 1], [14, 1], [-2, 6], [-2, 7], [6, -2], [4, -4], [6, 2], [-12, -2], [14, 3], [-2, -4], [-8, -4], [-12, -3], [-4, -5], [-6, 1], [-4, 4], [-22, 0], [12, 3], [-22, 2], [26, 0], [4, -4], [10, -7], [-16, -1], [-2, 0], [-6, -1], [-12, 6], [-6, 2], [-14, -2], [4, -5], [-10, 4], [6, 4], [-14, 7], [-6, -2], [16, -6], [12, -1], [-20, 0], [18, -6], [14, -1], [20, -2], [-8, 1], [14, -3], [-6, -3], [14, -4], [10, 1], [0, -8], [6, 0], [16, -2], [-30, 0], [-2, 2], [-22, 6], [-12, -1], [-22, 7], [-2, 9], [-22, 0], [10, 1], [-6, 12], [6, 6], [-20, 6], [-26, 1], [0, 1], [-24, 2], [-10, 1], [-20, 8], [-16, -2], [-12, 2], [-8, -8], [28, -2], [6, 4], [26, 2], [-8, 10], [30, 2], [-8, 3], [-28, -1], [0, -4], [-10, 0], [0, -4], [-40, -1], [16, -8], [-6, -2], [-10, -8], [0, 1], [34, -4], [-22, 4], [0, -8], [-20, 11], [-14, 4], [14, -2], [4, -3], [-8, 0], [-2, 12], [4, 1], [6, 12], [-4, -8], [18, -5], [-10, -10], [38, -4], [-20, 7], [-12, 0], [30, 6], [8, -10], [-32, 0], [0, 8], [18, -12], [6, 8], [-10, 7], [-4, 3], [-26, 2], [-12, -1], [6, -10], [34, 5], [2, -8], [-10, -1], [40, 2], [48, 2], [-20, -6], [-2, 3], [4, -1], [-34, 5], [36, -8], [-10, -2], [16, -6], [-32, 2], [6, 6], [2, 5], [-42, -3], [-8, -12], [20, 7], [14, 3], [8, -9], [10, -7], [-2, -16], [0, -4], [2, -14], [24, -11], [-20, -6], [28, 0], [-16, -2], [18, 0], [-28, 0], [16, 2], [-18, 1], [2, 3], [30, -12], [-2, -10], [28, -10], [16, 6], [-10, 0], [30, 9], [-46, -4], [-4, 0], [36, -7], [-10, 6], [-24, 1], [22, -7], [-4, -1], [38, 4], [14, 10], [20, -3], [24, -5], [-6, 12], [-6, -3], [52, 0], [-8, -2], [4, -10], [0, 0], [-18, -4], [-22, 12], [16, -2], [0, -3], [6, 0], [-22, -10], [34, 5], [-16, 6], [8, -15], [52, -4], [-66, 1], [-12, 0], [14, 4], [56, -4], [64, 1], [4, 6], [-16, 8], [-34, 6], [-4, -15], [32, 2], [-8, 4], [40, -5], [26, -13], [4, 1], [14, 10], [24, -16], [-10, 12], [0, -20], [44, 0], [10, 4], [-8, 4], [4, 8], [-10, 15], [4, -4], [18, 4], [58, 0], [8, 16], [-34, -2], [-14, 6], [12, -4], [-14, -11], [6, -12], [-14, 13], [2, -18], [-12, -14], [20, -16], [-6, 18], [-42, 8], [4, 13], [-20, -11], [38, -12], [16, 11], [10, 15], [-14, 11], [8, 5], [44, 6], [18, -1], [-6, -11], [-32, 8], [-10, 3], [-12, -9], [-22, -4], [8, 10], [-26, -6], [-32, 4], [-12, 6], [50, -3], [-26, 4], [10, -20], [-14, -4], [-6, 12], [0, 0], [10, -9], [-50, 0], [30, -15], [-28, 0], [10, -1], [68, 2], [-10, -4], [6, -16], [30, 3], [14, 4], [-54, 4], [4, 7], [16, -5], [-12, 6], [-4, 2], [-14, 6], [14, 1], [24, 6], [14, 10], [-30, 4], [-46, 0], [20, 15], [14, -22], [-16, 9], [-6, 2], [28, -7], [-42, 0], [-18, -16], [-50, -3], [-12, -8], [-12, -18], [50, -7], [22, -3], [-14, -10], [4, -14], [58, 1], [-38, -9], [12, -13], [14, 3], [26, 10], [-34, 5], [38, 8], [48, -3], [30, -20], [2, -4], [-52, 12], [-16, 12], [-4, 14], [16, -12], [36, -16], [66, -5], [42, -6], [-10, 20], [-50, 12], [44, -2], [-54, 3], [28, 13], [56, -6], [-16, -16], [-26, 9], [30, -3], [22, 10], [-46, -10], [20, 1], [6, 0], [-20, -12], [18, -17], [-8, 0], [24, -13], [-20, 22], [18, -4], [2, 7], [6, -8], [-52, -5], [-8, 14], [-72, 4], [26, -14], [22, -1], [-22, 17], [-80, 2], [-40, 8], [-44, 13], [28, 8], [-92, -1], [-4, -5], [40, 0], [-16, -1], [56, 5], [4, 22], [30, -22], [-26, -6], [4, 2], [34, 8], [12, -17], [-8, 8], [80, 0], [10, -7], [-60, -3], [28, -12], [-16, 4], [6, 18], [-28, 0], [-6, 3], [-12, 26], [38, -6], [28, -12], [66, -2], [-12, 10], [-26, 9], [30, -9], [68, -6], [82, -2], [-54, 8], [6, 20], [-4, 12], [-58, -4], [-2, -10], [-28, 17], [-54, -7], [42, 1], [-66, 1], [20, -2], [-26, 0], [16, -2], [38, 3], [6, -7], [8, -28], [22, -16], [-22, 3], [0, -8], [-2, -1], [-20, 25], [-10, -7], [82, -4], [60, -2], [-24, -16], [40, -14], [-54, 0], [4, -7], [-34, 14], [66, -2], [16, -16], [2, 24], [68, -11], [-56, 16], [20, 16], [86, -4], [-24, -8], [-20, 14], [2, 12], [-12, -18], [20, -14], [-30, -8], [62, 6], [32, -16], [20, 0], [-26, -2], [-72, 8], [-52, -4], [-46, -6], [26, -13], [8, 3]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2898_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2898_2_a_bb();". // 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_2898_2_a_bb(:prec:=2) chi := MakeCharacter_2898_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_2898_2_a_bb();". // 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_2898_2_a_bb( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2898_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-10, 1, 1]>,<11,R![-4, 1]>,<13,R![-4, -5, 1]>],Snew); return Vf; end function;