// Make newform 2205.2.a.w in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2205_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_2205_2_a_w();". // 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_2205_2_a_w();". // 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 := [-1, -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_2205_a();" function MakeCharacter_2205_a() N := 2205; order := 1; char_gens := [1226, 442, 1081]; 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_2205_a_Hecke(Kf) return MakeCharacter_2205_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, -1], [0, 0], [-1, 0], [0, 0], [-2, 2], [0, 2], [-2, 0], [-2, -2], [-4, 0], [2, 0], [-6, -2], [2, 4], [-2, 0], [0, -4], [4, -4], [8, 2], [0, 4], [2, 0], [-4, 0], [-10, -2], [8, -2], [4, -4], [-8, 4], [-2, 0], [-4, 2], [-14, 0], [0, 0], [4, -4], [-2, 0], [4, 2], [4, 4], [4, 0], [-8, -2], [6, 6], [6, -4], [-16, 0], [-4, -2], [-8, 4], [-8, 0], [6, 4], [-2, 2], [-10, -4], [-14, -6], [-14, 0], [-20, -2], [-14, 6], [-8, 4], [4, -4], [-8, -4], [-6, 8], [-4, -6], [6, -2], [10, -4], [8, -4], [10, -4], [-16, -4], [-6, -8], [6, 2], [10, -4], [2, -4], [-12, 0], [14, 4], [24, 4], [8, -8], [4, -6], [8, 10], [-8, -4], [-6, 8], [8, 0], [-6, -8], [-10, -8], [-14, 2], [-12, -4], [6, 0], [-20, 8], [-8, 0], [-2, -4], [0, -6], [-10, 0], [6, 8], [-12, 8], [22, 0], [14, -2], [-12, 2], [-6, -2], [8, 0], [14, 0], [-2, -4], [-14, -8], [12, -4], [0, 4], [0, 8], [-12, 4], [-10, -14], [4, 8], [32, 0], [-6, 8], [-2, 8], [0, -4], [6, 8], [-12, -8], [-16, 2], [-4, -8], [22, -8], [-4, 0], [12, 10], [-4, 0], [6, -8], [10, 10], [-30, 4], [-12, 12], [2, 4], [-12, 2], [-26, 6], [4, -12], [10, 12], [24, 4], [16, 8], [36, 6], [10, 14], [18, -8], [-18, -12], [30, 4], [-16, -8], [-22, 10], [34, 0], [-10, -16], [20, 12], [12, -12], [-24, -6], [12, 8], [-8, 12], [-12, 4], [14, -8], [-10, 8], [26, -12], [-34, 4], [-40, -4], [-10, 4], [-30, 4], [-42, 6], [2, 16], [36, -4], [-4, 4], [22, 4], [28, -12], [0, -14], [10, -12], [-46, 2], [-4, 8], [-30, 12], [-10, 8], [-20, 8], [-4, 12], [36, -8], [-6, 18], [-44, -4], [-34, -8], [48, -2], [-30, 0], [0, -8], [20, -6], [8, -8], [32, -4], [-44, -6], [4, 12], [-4, -4], [-8, -6], [2, -16], [14, -12], [6, -6], [-10, 12], [-10, -2], [-6, -16], [-10, 10], [-2, -24], [0, 12], [6, -12], [12, 20], [22, 12], [-8, 0], [-8, -4], [14, 8], [6, 0], [16, -12], [42, 8], [-46, -4], [8, -4], [34, -8], [-12, -4], [28, 2], [20, -20], [20, 8], [10, 16], [0, 8], [54, 0], [-14, 0], [14, 0], [18, 4], [-8, -16], [-22, 0], [34, -2], [-20, -2], [26, -4], [24, 20], [-42, 4], [-2, 18], [20, 4], [-38, -4], [-22, 10], [38, -4], [-30, 8], [32, 8], [8, -4], [-52, 4], [18, 4], [20, 12], [-10, -24], [-32, 12], [44, -2], [38, 0], [42, -10], [-14, -4], [-8, -8], [-20, 16], [30, 8], [26, 4], [26, 10], [-16, 0], [-26, 18], [-42, 8], [22, 22], [0, -8], [26, 12], [24, -4], [-28, 4], [-2, 8], [28, -2], [-10, 2], [-4, 28], [-4, 20], [-26, -10], [-24, 0], [-2, 0], [18, -12], [4, -20], [44, 12], [-12, 8], [-16, -4], [-12, -16], [-30, -20], [-58, -8], [36, -16], [22, -8], [46, -4], [-26, -14], [38, 0], [-12, 0], [-2, -28], [12, -14], [20, -12], [-32, 0], [-6, 24], [-28, 6], [-2, 0], [-54, -6], [-38, -16], [14, 8], [36, -16], [12, 22], [-50, -12], [8, 28], [4, -30], [-4, 12], [28, 2], [12, 4], [28, 4], [14, -16], [10, -16], [48, 12], [-12, -12], [36, -12], [-12, -4], [-10, -12], [8, -28], [26, -14], [-2, 4], [-32, -6], [18, -2], [-66, 8], [10, 0], [24, 4], [-36, 10], [-12, -16], [6, 24], [50, -8], [14, 18], [66, -8], [-40, 12], [-40, 12], [0, 14], [52, -10], [0, 16], [-72, 0], [-12, 24], [2, -16], [-36, 12], [-22, 0], [22, -2], [54, 0], [12, 28], [66, 8], [-2, -8], [-4, 24], [-20, 0], [-70, 4], [-4, -24], [2, 18], [12, 18], [6, 0], [2, -22], [22, 4], [-54, 16], [40, -8], [-24, -10], [10, -20], [20, -32], [-4, 24], [-52, 0], [56, 10], [30, -16], [44, -18], [-18, 10], [-36, -8], [40, 4], [52, -8], [-10, -24], [-14, 20], [-14, -36], [12, -20], [6, 0], [52, 2], [2, 24], [32, 0], [28, -18], [-46, -10], [2, -32], [-36, 0], [-44, -4], [-2, 4], [38, 22], [-26, 4], [18, 8], [-24, 8], [-10, 16], [22, -16], [-12, 12], [28, 0], [-44, 2], [-24, -20], [6, 16], [-10, -8], [32, 12], [-22, -10], [72, -4], [38, 4], [34, 16], [40, 8], [-6, 16], [-18, 2], [-14, 2], [72, -12], [46, 20], [14, 18], [-38, -12], [-12, 0], [-14, -14], [4, 18], [-74, 8], [-8, 14], [26, -16], [-28, 10], [-12, 20], [-24, 14], [-10, -18], [16, 0], [36, 20], [8, 34], [-60, 0], [-20, 36], [10, 40], [42, 16], [50, -2], [16, 12], [-18, -10], [10, -32], [-10, -22], [-34, 8], [-12, -24], [2, 8], [14, -12], [-36, 10], [16, 24], [2, -4], [42, -16], [30, -6], [-54, -12], [-2, -16], [36, 16], [64, 12], [24, -34], [8, -6], [-32, 24], [-48, -20], [2, 8], [74, -8], [-10, 30], [28, 4], [6, 8], [4, -28], [22, 28], [-8, 10], [8, 36], [48, -4], [-12, -22], [6, -28], [-12, 28], [54, -24], [-54, 18], [-68, 4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2205_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2205_2_a_w();". // 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_2205_2_a_w(:prec:=2) chi := MakeCharacter_2205_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_2205_2_a_w();". // 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_2205_2_a_w( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2205_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-5, 0, 1]>,<11,R![-16, 4, 1]>,<13,R![-20, 0, 1]>,<17,R![2, 1]>],Snew); return Vf; end function;