// Make newform 2268.2.x.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2268_x();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2268_x_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_2268_2_x_g();". // 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_2268_2_x_g();". // 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_2268_x();" function MakeCharacter_2268_x() N := 2268; order := 6; char_gens := [1135, 1541, 325]; v := [6, 5, 3]; // 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_2268_x_Hecke();" function MakeCharacter_2268_x_Hecke(Kf) N := 2268; order := 6; char_gens := [1135, 1541, 325]; char_values := [[1, 0], [0, 1], [-1, 0]]; 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, 3], [-3, 1], [3, 3], [4, -2], [6, 0], [-1, 2], [6, -3], [-6, -6], [6, -3], [1, 0], [0, -3], [-10, 10], [-6, 6], [0, 0], [0, 6], [8, 8], [0, -2], [-3, 6], [2, -4], [-14, 14], [6, -6], [-9, 0], [4, 4], [18, -18], [-10, 5], [-6, 12], [-11, 0], [0, 0], [2, 0], [0, -18], [-6, -6], [20, -10], [0, 0], [-4, 4], [-12, 6], [-10, 0], [0, 12], [-9, 9], [6, -12], [4, -8], [-3, -3], [0, 2], [12, -24], [1, -2], [0, 20], [-15, -15], [0, 0], [-24, 12], [12, -24], [-12, 6], [2, 2], [24, 0], [0, 9], [-3, -3], [21, 0], [-2, 4], [19, -19], [6, 6], [-12, 6], [0, 6], [-9, 18], [0, -6], [-2, -2], [-18, -18], [-4, 4], [0, 29], [-6, 3], [-16, -16], [-15, 15], [6, -12], [1, 1], [0, -29], [10, 0], [0, -12], [12, 12], [12, -24], [-12, 6], [20, -10], [0, -24], [-5, 5], [21, -42], [6, -12], [-18, -18], [-9, -9], [-12, 24], [17, -17], [15, -15], [0, 16], [18, 0], [24, -24], [-26, 0], [18, -9], [0, 14], [-6, 0], [0, 42], [-33, 0], [7, -14], [17, 0], [40, -40], [12, -24], [0, -18], [24, 24], [0, -20], [20, -40], [-36, 36], [39, 0], [30, -15], [6, 6], [-20, 10], [-19, 0], [36, -18], [19, 19], [26, 0], [12, 12], [-54, 27], [18, 0], [12, -6], [-21, -21], [28, -14], [-22, 22], [27, -27], [-9, 18], [6, 6], [18, -36], [23, -23], [-6, 0], [18, 18], [12, -6], [-16, 0], [-42, 21], [0, 2], [-26, 0], [0, 6], [-4, 2], [-39, 0], [-44, 22], [0, -45], [18, -36], [3, -6], [-18, -18], [0, -20], [27, -54], [-26, 52], [-42, 42], [28, 28], [21, -21], [34, -17], [6, -12], [0, -2], [27, 0], [28, 0], [0, -42], [32, -32], [-6, -6], [16, 0], [30, -30], [32, -64], [0, -21], [15, 15], [-18, 36], [0, 16], [-6, 0], [-48, 24], [42, -42], [10, 0], [0, 0], [35, 0], [0, -33], [6, 6], [-36, 18], [-36, 18], [-59, 59], [22, -11], [0, 21], [-44, 44], [12, -24], [1, -2], [-24, -24], [-32, 32], [60, -60], [0, -5], [-6, 0], [6, -3], [15, -15], [-26, 0], [2, 2], [0, -47], [24, 0], [-26, 52], [30, 30], [28, 0], [-6, 6], [-21, 42], [0, 21], [0, 65], [-14, 14], [9, -9], [12, 0], [36, -18], [5, 5], [-76, 38], [28, 28], [6, 0], [30, 0], [-10, 20], [-6, 3], [6, 6], [20, -10], [-37, 0], [0, -9], [14, -14], [6, -6], [0, -12], [8, 8], [0, -22], [-27, 27], [-3, 6], [24, -12], [0, 38], [36, -18], [36, -18], [-2, 0], [0, -12], [-13, 13], [-15, 15], [33, -66], [-7, -7], [-21, -21], [0, -25], [3, -6], [0, 14], [-24, 12], [7, 7], [54, -54], [32, -16], [-30, 60], [-12, 6], [-6, 0], [27, 27], [7, -14], [-10, 5], [1, 0], [0, 9], [-6, 6], [1, -2], [0, -6], [0, -14], [21, -42], [0, -22], [-18, 0], [-60, 30], [8, 8], [39, -39], [-21, 42], [-55, 0], [-21, -21], [-18, 0], [-6, 12], [58, -58], [-45, -45], [-44, 22], [-14, 28], [0, -63], [-21, -21], [-30, 60], [-54, 54], [0, 4], [36, -18], [-52, 26], [28, 0], [-6, -6], [0, 2], [-84, 42], [29, -58], [42, -21], [-46, 46], [-2, 0], [30, -30], [0, 66], [0, 4], [-48, 48], [10, 10], [-46, 23], [-39, 78], [-17, 0], [60, -30], [17, 17], [-57, 0], [0, 0], [30, 0], [12, -6], [0, -60], [-17, 17], [0, 3], [-23, -23], [-30, 30], [42, 0], [-41, -41], [40, -20], [30, -60], [56, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2268_x_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2268_2_x_g();". // 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_2268_2_x_g(:prec:=2) chi := MakeCharacter_2268_x(); 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_2268_2_x_g();". // 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_2268_2_x_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2268_x(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![9, -3, 1]>,<11,R![27, -9, 1]>,<13,R![12, -6, 1]>],Snew); return Vf; end function;