// Make newform 1134.2.h.r in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1134_h();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1134_h_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_1134_2_h_r();". // 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_1134_2_h_r();". // 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 := [4, 0, 2, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 4, 0, 1], [0, 2, 0, -1]]; Rf_basisdens := [1, 2, 2, 2]; 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_1134_h();" function MakeCharacter_1134_h() N := 1134; order := 3; char_gens := [407, 325]; v := [1, 2]; // 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_1134_h_Hecke();" function MakeCharacter_1134_h_Hecke(Kf) N := 1134; order := 3; char_gens := [407, 325]; char_values := [[-1, -1, 0, 0], [0, 1, 0, 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 := [[-1, -1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1, 0, 1, 0], [0, 0, 1, -2], [-2, -2, 1, 1], [0, 0, 0, 0], [0, 2, -2, 1], [3, 0, 1, -2], [0, 0, 2, -1], [0, 5, 2, -1], [0, -4, 0, 0], [-3, -3, -2, -2], [0, 2, 4, -2], [-9, -9, 1, 1], [0, 0, 1, 1], [0, 0, 0, 0], [-2, -2, 1, 1], [0, -4, 2, -1], [-3, 0, -1, 2], [7, 7, 0, 0], [-5, -5, 1, 1], [0, 12, 2, -1], [0, 3, 4, -2], [0, -4, 4, -2], [12, 0, -1, 2], [5, 0, -1, 2], [0, 6, 4, -2], [-2, -2, -1, -1], [12, 12, -2, -2], [11, 0, -1, 2], [-12, 0, 1, -2], [-3, 0, -4, 8], [-8, -8, -3, -3], [12, 0, 1, -2], [-7, 0, 1, -2], [0, 14, 0, 0], [0, -16, -2, 1], [-3, -3, 5, 5], [-6, -6, -4, -4], [12, 12, -1, -1], [-16, 0, -1, 2], [0, 0, 2, 2], [0, -13, -4, 2], [0, 0, 4, -8], [19, 19, 1, 1], [-2, -2, -2, -2], [0, -1, -6, 3], [18, 0, 2, -4], [8, 0, 4, -8], [0, 12, -4, 2], [9, 9, 3, 3], [17, 0, -2, 4], [6, 0, 3, -6], [-3, 0, 2, -4], [6, 0, 4, -8], [-6, -6, -4, -4], [0, -4, 4, -2], [2, 0, -5, 10], [0, -3, -12, 6], [0, 8, -8, 4], [-6, -6, -4, -4], [-28, 0, 0, 0], [0, 6, 8, -4], [1, 1, -4, -4], [0, 0, -4, -4], [10, 10, -2, -2], [4, 4, 2, 2], [0, 6, 6, -3], [0, 8, -8, 4], [21, 0, 0, 0], [0, 15, 2, -1], [-1, 0, 3, -6], [14, 0, -5, 10], [14, 0, -3, 6], [-3, 0, -5, 10], [6, 0, 1, -2], [0, -16, 2, -1], [21, 0, 2, -4], [0, 35, 0, 0], [12, 12, 1, 1], [0, -10, 2, -1], [33, 33, -1, -1], [-13, 0, 4, -8], [1, 1, 3, 3], [-18, -18, -2, -2], [12, 0, -4, 8], [-20, -20, -2, -2], [0, 30, -2, 1], [13, 13, 1, 1], [0, 0, 0, 0], [-9, 0, -1, 2], [7, 7, 1, 1], [6, 6, 3, 3], [2, 0, -3, 6], [-3, 0, -5, 10], [12, 0, -1, 2], [9, 9, 6, 6], [0, -16, 2, -1], [0, 2, 8, -4], [0, 2, 4, -2], [24, 24, -4, -4], [0, 6, -6, 3], [9, 9, 8, 8], [0, -4, 8, -4], [-2, -2, 8, 8], [0, 18, 10, -5], [0, 3, 4, -2], [0, 6, -8, 4], [0, -1, 8, -4], [-22, 0, -4, 8], [4, 4, -8, -8], [3, 3, -6, -6], [-10, 0, 2, -4], [-16, 0, -2, 4], [21, 0, -6, 12], [34, 34, -3, -3], [-15, -15, -3, -3], [24, 0, 0, 0], [0, -18, 10, -5], [0, -4, 18, -9], [0, -19, 4, -2], [-30, -30, 1, 1], [-12, -12, 7, 7], [-2, -2, 1, 1], [18, 0, -1, 2], [4, 4, 7, 7], [0, -15, -6, 3], [0, -13, 6, -3], [-22, 0, -4, 8], [-2, -2, 2, 2], [-39, -39, -1, -1], [-13, 0, -1, 2], [-4, 0, 4, -8], [21, 0, 0, 0], [4, 4, -2, -2], [24, 24, -5, -5], [0, -10, 10, -5], [0, 0, 0, 0], [9, 9, 0, 0], [2, 0, -6, 12], [-6, -6, -9, -9], [0, 11, -14, 7], [0, 0, 4, -8], [16, 16, -8, -8], [0, 24, 4, -2], [0, -10, -18, 9], [3, 0, -2, 4], [14, 0, 7, -14], [0, 21, 10, -5], [-4, 0, 7, -14], [-45, 0, 2, -4], [20, 0, 7, -14], [27, 0, 3, -6], [-4, 0, -8, 16], [0, -27, -6, 3], [0, 8, -12, 6], [27, 27, 4, 4], [5, 0, 6, -12], [0, 6, 22, -11], [30, 30, 2, 2], [-15, 0, 2, -4], [-23, -23, -7, -7], [0, -18, 4, -2], [0, 3, 8, -4], [-24, 0, 2, -4], [-29, -29, 5, 5], [14, 0, 0, 0], [-16, 0, -4, 8], [0, 0, 0, 0], [24, 0, -4, 8], [4, 4, -9, -9], [-36, 0, 6, -12], [-19, 0, 2, -4], [0, -34, -8, 4], [-36, -36, 4, 4], [0, 26, -10, 5], [-30, -30, -5, -5], [8, 0, 4, -8], [34, 34, 4, 4], [28, 28, 2, 2], [0, -12, -16, 8], [-2, -2, -9, -9], [0, -33, 12, -6], [0, 9, 10, -5], [-12, 0, 1, -2], [22, 22, 4, 4], [-10, 0, 2, -4], [11, 0, 4, -8], [-3, -3, 5, 5], [0, -25, 4, -2], [0, 6, -14, 7], [0, 20, 16, -8], [6, 6, 4, 4], [6, 6, 2, 2], [0, 15, -8, 4], [0, -13, 4, -2], [-32, -32, 4, 4], [0, 36, -8, 4], [0, 39, 10, -5], [0, -18, 8, -4], [0, -4, 4, -2], [2, 0, 1, -2], [-25, 0, -2, 4], [-18, 0, 5, -10], [30, 30, -8, -8], [0, -1, 22, -11], [6, 0, -2, 4], [0, 39, -12, 6], [0, -40, -2, 1], [0, 59, 0, 0], [-36, -36, 4, 4], [0, 11, -2, 1], [-6, -6, 3, 3], [0, -33, -2, 1], [-53, -53, 2, 2], [0, 41, 6, -3], [-27, 0, -10, 20], [0, 0, 24, -12], [42, 42, -3, -3], [-16, 0, 1, -2], [16, 16, 6, 6], [12, 0, 2, -4], [0, -7, -18, 9], [0, 0, 0, 0], [0, 14, 16, -8], [-21, -21, 0, 0], [6, 6, -12, -12], [37, 37, -1, -1], [42, 42, 2, 2], [0, -4, 8, -4], [-56, -56, 0, 0], [-8, -8, 10, 10], [0, 51, -4, 2], [0, 2, 12, -6], [12, 0, 6, -12], [-4, 0, -2, 4], [0, -54, -8, 4], [-12, -12, -4, -4], [-39, 0, -9, 18], [-48, 0, -5, 10], [0, 2, 12, -6], [0, -40, 12, -6], [0, 32, -8, 4], [51, 51, 6, 6], [3, 3, -5, -5], [35, 0, 7, -14], [0, -66, -4, 2], [0, 14, 12, -6], [-3, 0, 1, -2], [-2, -2, 12, 12], [0, 3, -24, 12], [0, -27, -18, 9], [0, -1, 8, -4], [-12, 0, 15, -30], [0, -6, -14, 7], [34, 34, 5, 5], [-28, 0, -7, 14], [-48, 0, 4, -8], [0, 56, 0, 0], [8, 0, -2, 4], [0, 48, 14, -7], [0, 14, 14, -7], [14, 0, 0, 0], [0, 12, -12, 6], [-26, -26, -1, -1], [48, 0, 0, 0], [0, 0, 0, 0], [-50, -50, 3, 3], [0, -30, 0, 0], [-4, 0, 5, -10], [-20, -20, 4, 4], [44, 0, 8, -16], [-40, 0, -4, 8], [-5, -5, 6, 6], [0, 5, -26, 13], [-24, 0, 11, -22], [-10, 0, -4, 8], [0, 41, 4, -2], [-12, -12, -8, -8], [0, -36, -20, 10], [0, 32, 8, -4], [0, -33, -2, 1], [0, 62, 8, -4], [8, 0, 4, -8], [0, -3, -22, 11], [28, 28, -2, -2], [-18, -18, 13, 13], [-64, 0, -4, 8], [-39, 0, -2, 4], [-42, 0, 1, -2], [-30, -30, -6, -6], [-6, 0, -8, 16], [54, 54, -6, -6], [0, -22, 0, 0], [0, 6, 22, -11], [-41, -41, -11, -11], [0, -6, 6, -3], [0, 30, 12, -6], [0, 14, -14, 7], [47, 0, -8, 16], [0, -12, -22, 11], [16, 16, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1134_h_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1134_2_h_r();". // 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_1134_2_h_r(:prec:=4) chi := MakeCharacter_1134_h(); 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_1134_2_h_r();". // 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_1134_2_h_r( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1134_h(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![0, 1]>,<11,R![-18, 0, 1]>,<17,R![0, 1]>,<23,R![-9, -6, 1]>],Snew); return Vf; end function;