// Make newform 2016.2.s.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2016_s();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2016_s_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_2016_2_s_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_2016_2_s_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 := [1, -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_2016_s();" function MakeCharacter_2016_s() N := 2016; order := 3; char_gens := [127, 1765, 1793, 577]; v := [3, 3, 3, 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2016_s_Hecke();" function MakeCharacter_2016_s_Hecke(Kf) N := 2016; order := 3; char_gens := [127, 1765, 1793, 577]; char_values := [[1, 0], [1, 0], [1, 0], [0, -1]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; 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], [0, 0], [0, 1], [-3, 1], [-1, 1], [0, 0], [-8, 8], [0, -4], [0, -4], [5, 0], [7, -7], [0, -8], [-4, 0], [-10, 0], [0, -6], [-1, 1], [-9, 9], [0, 2], [2, -2], [6, 0], [-2, 2], [0, -9], [-3, 0], [0, -6], [-1, 0], [2, -2], [0, -16], [0, -15], [-10, 10], [-4, 0], [-13, 0], [0, 19], [18, -18], [-10, 0], [0, 6], [7, -7], [-20, 20], [0, 20], [-18, 0], [0, -18], [-20, 20], [20, 0], [0, 0], [11, -11], [-18, 0], [20, -20], [-22, 0], [19, 0], [-7, 7], [0, -24], [0, -8], [12, 0], [-15, 15], [1, 0], [0, 18], [30, -30], [-17, 17], [0, 11], [24, -24], [-6, 0], [-6, 6], [-3, 0], [4, 0], [-28, 28], [0, -1], [0, -5], [0, 20], [1, 0], [-28, 28], [6, 0], [-24, 24], [0, -6], [-3, 3], [0, 4], [28, 0], [0, 34], [-34, 34], [0, -4], [0, -32], [-23, 23], [24, 0], [2, 0], [0, 0], [30, 0], [0, 19], [0, -29], [24, 0], [0, -15], [14, 0], [40, 0], [0, 12], [18, -18], [-31, 31], [-27, 0], [0, 34], [-12, 0], [0, -9], [14, -14], [0, 8], [0, -22], [-8, 0], [9, -9], [-29, 29], [0, -12], [14, -14], [-23, 23], [15, 0], [0, -36], [-30, 30], [19, 0], [0, 25], [-44, 44], [-6, 0], [-10, 10], [-33, 0], [38, -38], [-2, 0], [-10, 10], [0, -31], [16, 0], [-18, 18], [-19, 0], [0, -27], [37, -37], [0, -12], [-27, 0], [0, -14], [0, -10], [-19, 0], [0, 14], [34, -34], [-38, 0], [0, -7], [-26, 0], [0, 0], [-19, 0], [18, -18], [-38, 38], [51, 0], [-32, 32], [2, 0], [0, -1], [16, -16], [-43, 0], [20, -20], [-8, 0], [42, 0], [-6, 6], [0, -18], [0, 46], [0, -32], [-18, 0], [28, 0], [0, -4], [-12, 12], [-6, 0], [0, -32], [0, 34], [-29, 0], [-51, 51], [0, -40], [6, 0], [-7, 0], [0, -59], [50, -50], [24, -24], [-5, 5], [-10, 10], [-1, 0], [0, -3], [57, -57], [42, 0], [0, 4], [35, -35], [-23, 23], [-52, 0], [10, 0], [-51, 51], [-40, 0], [0, -42], [0, -41], [15, 0], [50, 0], [0, 22], [-14, 14], [9, -9], [46, -46], [46, -46], [0, -51], [4, -4], [0, -41], [-57, 0], [0, -12], [0, -18], [19, -19], [-10, 10], [-46, 46], [0, 26], [26, 0], [0, 34], [-15, 15], [0, 0], [0, 38], [-10, 10], [-24, 0], [-69, 69], [0, -16], [0, 9], [48, 0], [-18, 18], [0, 38], [-61, 0], [1, 0], [0, -39], [20, -20], [0, 22], [13, -13], [28, -28], [0, -58], [-2, 0], [0, 0], [56, 0], [0, -42], [0, 19], [-61, 0], [-8, 0], [0, 18], [36, -36], [0, -13], [0, 36], [-2, 2], [44, -44], [33, 0], [42, -42], [-14, 0], [24, -24], [0, 29], [0, 13], [12, 0], [44, 0], [-4, 4], [0, 60], [69, -69], [0, 50], [-24, 0], [0, 0], [-63, 0], [-20, 20], [-32, 32], [-12, 0], [-26, 0], [0, -10], [-14, 14], [-38, 0], [39, -39], [0, 39], [26, -26], [-64, 64], [57, 0], [0, -13], [-8, 8], [69, 0], [38, -38], [-64, 0], [30, -30], [0, 46], [-53, 0], [-72, 0], [40, 0], [5, -5], [0, -64], [-18, 18], [83, -83], [0, 49], [53, 0], [0, -8], [0, 33], [44, -44], [0, 66], [0, -40], [18, -18], [59, -59], [-56, 0], [-16, 0], [0, -2], [0, -32], [65, -65], [27, 0], [17, -17], [20, 0], [6, -6], [-87, 87], [0, 52], [-12, 0], [16, 0], [37, -37], [0, -48], [10, 0], [0, 76], [42, 0], [0, 10], [0, 45], [25, -25]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2016_s_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2016_2_s_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_2016_2_s_i(:prec:=2) chi := MakeCharacter_2016_s(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(1999) - 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_2016_2_s_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_2016_2_s_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2016_s(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![1, -1, 1]>,<11,R![1, 1, 1]>,<13,R![0, 1]>],Snew); return Vf; end function;