// Make newform 169.4.a.i in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_169_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_169_4_a_i();". // 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_169_4_a_i();". // 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_169_a();" function MakeCharacter_169_a() N := 169; order := 1; char_gens := [2]; v := [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_169_a_Hecke(Kf) return MakeCharacter_169_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 := 4; raw_aps := [[0, 1], [2, 0], [0, 1], [0, -8], [0, -8], [0, 0], [-117, 0], [0, 66], [78, 0], [-141, 0], [0, -90], [0, 83], [0, 157], [-104, 0], [0, 174], [93, 0], [0, -164], [145, 0], [0, -454], [0, -610], [0, 265], [1276, 0], [0, 456], [0, 564], [0, -116], [-429, 0], [-182, 0], [-1506, 0], [0, -896], [-687, 0], [-286, 0], [-1974, 0], [0, -489], [236, 0], [0, 27], [0, -1022], [1211, 0], [0, -580], [0, -528], [-2574, 0], [-3744, 0], [637, 0], [-2598, 0], [0, -645], [0, 1184], [2522, 0], [1042, 0], [0, -1390], [0, -1390], [0, 1448], [5850, 0], [0, 3108], [0, 2839], [3978, 0], [-2067, 0], [-2052, 0], [3330, 0], [0, -1620], [-377, 0], [0, -21], [7124, 0], [0, -4805], [0, -1282], [-4914, 0], [-518, 0], [0, -2261], [0, 4304], [3575, 0], [-6966, 0], [0, 3840], [0, 3251], [0, -4116], [2, 0], [3499, 0], [0, 3186], [0, 4252], [1209, 0], [0, 6752], [0, 1721], [0, -25], [-9462, 0], [0, -4081], [0, -5732], [6617, 0], [-13988, 0], [2004, 0], [0, 5244], [0, 1457], [0, -11309], [0, 4984], [-5460, 0], [0, 1474], [0, -6252], [-11388, 0], [0, 10206], [3876, 0], [0, -9853], [2121, 0], [-11464, 0], [0, -2751], [6554, 0], [0, 10457], [12168, 0], [7722, 0], [-11440, 0], [0, -8917], [0, -8124], [0, 15553], [-10554, 0], [-14831, 0], [-7954, 0], [0, 14561], [0, 10037], [0, -4740], [0, 7428], [-6201, 0], [0, 9712], [13494, 0], [-11334, 0], [13236, 0], [0, 6843], [-8021, 0], [-21630, 0], [0, -15322], [0, -480], [-30186, 0], [0, -6859], [-18408, 0], [-21112, 0], [0, 13833], [0, -1828], [0, -17380], [-28496, 0], [17422, 0], [0, 23860], [0, -8124], [0, -116], [0, -3986], [31278, 0], [8049, 0], [0, -8098], [0, -4640], [40300, 0], [0, 22820], [12311, 0], [0, 12394], [0, 447], [-13923, 0], [-22358, 0], [0, 1288], [0, 9673], [-17355, 0], [46982, 0], [-8916, 0], [30836, 0], [-27480, 0], [-28442, 0], [0, -4029], [-38465, 0], [0, -2820], [0, -12566], [-6474, 0], [0, -4354], [34998, 0], [0, -14559], [0, -32586], [59282, 0], [-37711, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_169_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_169_4_a_i();". // 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_169_4_a_i(:prec:=2) chi := MakeCharacter_169_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 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_169_4_a_i();". // 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_169_4_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_169_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<2,R![-3, 0, 1]>],Snew); return Vf; end function;