// Make newform 1134.2.e.q in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1134_e();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1134_e_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_e_q();". // 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_e_q();". // 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 := [49, 0, 7, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]; Rf_basisdens := [1, 1, 7, 7]; 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_e();" function MakeCharacter_1134_e() N := 1134; order := 3; char_gens := [407, 325]; v := [2, 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_e_Hecke();" function MakeCharacter_1134_e_Hecke(Kf) N := 1134; order := 3; char_gens := [407, 325]; char_values := [[0, 0, 1, 0], [0, 0, 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 := [[-1, 0, 0, 0], [0, 0, 0, 0], [-1, 1, -1, 0], [0, -1, 0, 0], [0, 1, -1, 1], [0, 1, -2, 1], [1, -1, 1, 0], [0, 0, 2, 0], [4, 2, 4, 0], [5, 1, 5, 0], [-2, 0, 0, -1], [0, 3, -4, 3], [0, 3, -3, 3], [-5, 0, -5, 0], [3, 0, 0, -3], [6, 0, 6, 0], [-11, 0, 0, 1], [10, 0, 0, -1], [3, 0, 0, -2], [-13, 0, 0, -1], [0, 4, 0, 0], [-2, 0, 0, 5], [-8, 2, -8, 0], [0, 3, 3, 3], [-3, 4, -3, 0], [0, -1, -11, -1], [4, 3, 4, 0], [0, 0, 6, 0], [6, 1, 6, 0], [0, -1, -5, -1], [4, 0, 0, -1], [-4, 4, -4, 0], [0, 2, 4, 2], [0, -4, -9, -4], [15, -3, 15, 0], [0, 5, -6, 5], [-4, 0, 0, 0], [0, 0, -1, 0], [0, -4, -2, -4], [4, 0, 0, 4], [9, -3, 9, 0], [-8, 0, 0, 2], [-3, 0, 0, 9], [-7, 0, 0, 0], [-10, 0, 0, 2], [-14, -3, -14, 0], [0, -6, -1, -6], [-2, -6, -2, 0], [0, 0, -6, 0], [12, 1, 12, 0], [0, -2, 14, -2], [0, -3, 3, -3], [0, 0, 5, 0], [18, 0, 0, 0], [0, -6, 0, 0], [0, -3, -3, -3], [-22, -2, -22, 0], [0, -5, -8, -5], [0, -3, -10, -3], [1, -7, 1, 0], [3, 0, 0, -2], [0, -3, -15, -3], [-3, 0, 0, 4], [-5, 0, 0, 7], [18, 0, 0, -2], [-3, 0, 0, -9], [-16, 0, 0, -6], [0, 4, -20, 4], [12, 0, 0, 0], [-12, -5, -12, 0], [0, 0, 12, 0], [0, 8, 10, 8], [0, -6, 14, -6], [6, 4, 6, 0], [15, 0, 0, 4], [10, -4, 10, 0], [0, 2, -14, 2], [0, 1, -14, 1], [3, 9, 3, 0], [-1, 0, 0, 6], [0, 3, -21, 3], [20, 2, 20, 0], [15, -9, 15, 0], [-19, 0, 0, 0], [4, 0, 0, 8], [14, 0, 0, 2], [-5, 0, 0, 1], [-3, 0, 0, -2], [11, -5, 11, 0], [0, 2, 24, 2], [0, 5, 13, 5], [0, 1, -1, 1], [-8, 6, -8, 0], [0, 12, 6, 12], [-15, -8, -15, 0], [12, 0, 0, -6], [24, 6, 24, 0], [-24, -6, -24, 0], [0, 0, -1, 0], [0, -8, -20, -8], [-13, 2, -13, 0], [-19, -5, -19, 0], [8, 0, 0, -10], [9, 0, 0, 3], [4, 0, 0, 2], [27, -2, 27, 0], [25, 5, 25, 0], [0, 3, -33, 3], [6, 0, 0, -6], [-11, 6, -11, 0], [-16, -4, -16, 0], [8, -13, 8, 0], [0, 7, 17, 7], [0, 2, -3, 2], [14, 0, 0, -3], [0, -5, 5, -5], [0, -6, -19, -6], [-6, -6, -6, 0], [-6, 0, -6, 0], [-11, 5, -11, 0], [18, 0, 0, -8], [16, 12, 16, 0], [30, 0, 0, 6], [-3, 9, -3, 0], [-7, 0, 0, -12], [16, 0, 0, 4], [14, 0, 0, 9], [0, 3, -15, 3], [-12, -5, -12, 0], [-22, 5, -22, 0], [5, -10, 5, 0], [0, -3, 21, -3], [26, -10, 26, 0], [4, 0, 0, 17], [-8, -4, -8, 0], [0, -4, 30, -4], [-4, 16, -4, 0], [-17, 0, 0, 2], [0, -9, -9, -9], [26, 4, 26, 0], [-24, 0, 0, -2], [-10, 0, 0, -4], [-6, 0, 0, -11], [12, 0, 0, 12], [24, -2, 24, 0], [-44, 2, -44, 0], [-32, 6, -32, 0], [0, -9, 3, -9], [7, 0, 7, 0], [0, 8, -8, 8], [-24, 1, -24, 0], [-12, 0, 0, 6], [-4, 0, 0, -6], [0, -6, 0, 0], [0, -2, 31, -2], [21, -3, 21, 0], [0, 5, 42, 5], [9, 0, 0, 3], [-13, 0, 0, -12], [19, 0, 0, 13], [-29, 0, 0, 7], [-10, 0, 0, -16], [0, 3, 44, 3], [0, -6, -6, -6], [36, 0, 0, 6], [0, 9, 3, 9], [34, 3, 34, 0], [0, -15, -16, -15], [17, 0, 0, 0], [-6, 0, -6, 0], [0, -7, -23, -7], [0, 8, 18, 8], [18, -6, 18, 0], [0, 4, 10, 4], [34, 0, 0, -1], [0, -10, 4, -10], [-1, 8, -1, 0], [13, -13, 13, 0], [56, 0, 0, 3], [-28, 0, 0, 3], [34, 0, 0, 2], [-26, 2, -26, 0], [0, -7, -42, -7], [0, -1, 1, -1], [-20, 0, 0, -8], [0, -6, 18, -6], [26, 5, 26, 0], [0, 0, -10, 0], [-25, 8, -25, 0], [33, -9, 33, 0], [0, 8, 6, 8], [21, -3, 21, 0], [0, -4, 33, -4], [-31, 0, 0, 11], [-3, -9, -3, 0], [26, 0, 0, -10], [-10, 0, 0, -6], [-12, 0, 0, -11], [-30, -6, -30, 0], [0, -12, 18, -12], [-54, 0, 0, 6], [20, 5, 20, 0], [28, -3, 28, 0], [0, 4, 25, 4], [21, 0, 0, 3], [39, 3, 39, 0], [0, -3, -16, -3], [-30, 6, -30, 0], [36, 0, 36, 0], [-31, 0, 0, -6], [0, 0, -1, 0], [0, 2, 64, 2], [2, 5, 2, 0], [55, 0, 0, 1], [6, 0, 0, 12], [23, 0, 0, 0], [6, 0, 0, -2], [0, -5, -49, -5], [0, 1, -25, 1], [0, -11, -19, -11], [24, -11, 24, 0], [0, 5, 0, 5], [12, 0, 12, 0], [0, -10, 24, -10], [0, 21, 15, 21], [22, -9, 22, 0], [14, 0, 0, -16], [5, -11, 5, 0], [-8, 0, 0, -4], [15, 0, 0, -9], [-32, 0, 0, 2], [6, -20, 6, 0], [0, 5, -18, 5], [-3, 0, 0, 21], [25, -12, 25, 0], [0, -12, 6, -12], [-36, -8, -36, 0], [0, 3, -51, 3], [0, 12, -30, 12], [-23, 0, 0, 19], [0, 7, -25, 7], [0, 0, 2, 0], [-16, 0, 0, -12], [0, 1, 4, 1], [0, 1, 23, 1], [-39, 0, 0, -9], [4, 0, 0, 17], [26, 0, 0, -10], [44, 0, 0, -6], [-9, 0, 0, -9], [0, -7, -24, -7], [0, 6, 36, 6], [6, 0, 0, 12], [45, -8, 45, 0], [0, 10, -28, 10], [0, -8, 2, -8], [16, -15, 16, 0], [0, -2, 49, -2], [-50, 0, 0, -8], [0, 12, 26, 12], [0, 2, 18, 2], [-37, -11, -37, 0], [6, 0, 0, -17], [42, 0, 0, 13], [-6, 0, 0, 0], [-15, 0, 0, 4], [-18, 0, 0, 0], [23, 19, 23, 0], [0, 20, -30, 20], [0, 0, 0, -12], [8, -10, 8, 0], [-17, 0, -17, 0], [0, -8, -17, -8], [22, 3, 22, 0], [0, -10, -18, -10], [0, -5, 46, -5], [-5, 11, -5, 0], [0, -17, -26, -17], [0, -16, -27, -16], [-6, 0, 0, -12], [63, 0, 0, -3], [16, 0, 0, -1], [9, 9, 9, 0], [22, -9, 22, 0], [23, 14, 23, 0], [0, -2, -58, -2], [-43, 2, -43, 0], [0, 7, 35, 7], [0, -3, -28, -3], [-48, 0, 0, -12], [0, -3, -3, -3], [33, 9, 33, 0], [3, -9, 3, 0], [0, -10, -32, -10], [-8, -15, -8, 0], [-24, 0, 0, -18], [20, 0, 0, -21], [-42, 6, -42, 0], [0, -15, -33, -15], [-19, 8, -19, 0], [9, -14, 9, 0], [0, 12, 12, 12], [-18, -14, -18, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1134_e_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1134_2_e_q();". // 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_e_q(:prec:=4) chi := MakeCharacter_1134_e(); 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_e_q();". // 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_e_q( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1134_e(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![36, -12, 10, 2, 1]>,<11,R![36, 12, 10, -2, 1]>,<17,R![36, 12, 10, -2, 1]>,<23,R![144, 96, 76, -8, 1]>],Snew); return Vf; end function;