// Make newform 114.2.h.f in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_114_h();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_114_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_114_2_h_f();". // 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_114_2_h_f();". // 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, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]; Rf_basisdens := [1, 1, 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_114_h();" function MakeCharacter_114_h() N := 114; order := 6; char_gens := [77, 97]; v := [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_114_h_Hecke();" function MakeCharacter_114_h_Hecke(Kf) N := 114; order := 6; char_gens := [77, 97]; char_values := [[-1, 0, 0, 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, -1, 0], [0, -1, 1, 1], [0, 1, 0, 0], [2, 2, 0, -1], [-1, 0, 2, -1], [-4, 0, 2, 0], [-2, -2, -2, 0], [0, -3, -1, 3], [0, 5, 0, -5], [0, -1, 0, -1], [0, 0, 0, 3], [-2, 0, 4, -3], [-3, 0, 3, 0], [4, -2, -4, 4], [4, -7, -2, 7], [0, 2, -6, 2], [9, -1, -9, 2], [0, 1, 4, 1], [-2, -3, 1, 3], [6, 0, -6, 0], [1, 4, -1, -8], [0, 6, 0, 0], [-9, 0, 18, -1], [0, 2, 12, 2], [-3, -6, -3, 0], [0, -7, 0, 7], [2, 0, -4, -6], [0, 4, 0, -2], [0, 0, 0, 0], [-9, 8, 0, -4], [-12, 0, 6, 0], [7, 7, 7, 0], [-2, -4, 1, 4], [0, -3, 7, -3], [2, 1, 2, 0], [-8, 0, 16, 3], [4, 6, -4, -12], [11, -6, 0, 3], [0, 2, 0, 2], [6, 2, -6, -4], [-15, -6, 0, 3], [20, -3, -10, 3], [8, 0, -16, -4], [-10, 6, -10, 0], [-6, 0, 12, -7], [0, -2, -2, -2], [4, 6, 4, 0], [-8, 3, -8, 0], [3, 2, 0, -1], [-4, -4, 0, 2], [-7, 4, -7, 0], [-4, 0, 8, -10], [2, 6, -1, -6], [-6, -1, 3, 1], [0, 0, -15, 0], [6, 1, 6, 0], [0, -10, 0, 20], [-2, -9, 2, 18], [8, -10, 0, 5], [0, -8, 3, -8], [7, -1, -7, 2], [-12, 6, 0, -3], [5, -9, 5, 0], [0, 0, 0, 11], [0, 0, 19, 0], [0, 0, -6, 0], [11, 0, -22, 3], [-7, 6, -7, 0], [3, 1, 3, 0], [8, -20, 0, 10], [3, 0, -6, 14], [0, -17, 0, 0], [0, 3, 16, 3], [0, 0, 0, -18], [-4, 0, 8, 6], [18, -1, -18, 2], [-24, -4, 12, 4], [-2, -3, 2, 6], [3, 4, -3, -8], [14, 6, -7, -6], [-6, 0, 12, -10], [-10, -9, -10, 0], [0, 6, 18, 6], [-4, 12, 2, -12], [-4, 3, -4, 0], [6, 17, -3, -17], [21, -8, 0, 4], [-13, 0, 0, 0], [-14, 10, -14, 0], [-10, 8, 0, -4], [13, 0, -26, -7], [24, 14, -12, -14], [12, 0, -24, 3], [-6, 4, -6, 0], [-5, 1, 5, -2], [0, 11, 0, -11], [0, -12, -12, -12], [-21, 4, 0, -2], [12, 12, -6, -12], [0, 6, 4, 6], [-4, 24, 2, -24], [-16, -10, 8, 10], [9, -26, 0, 13], [0, 28, 0, -14], [11, -18, 0, 9], [-1, -16, 0, 8], [-6, 4, -6, 0], [-6, -16, 3, 16], [0, 7, 12, 7], [-13, 0, 26, 6], [14, 0, -28, -9], [-38, 4, 38, -8], [14, 2, -7, -2], [-16, 12, 0, -6], [0, -5, 22, -5], [3, 12, -3, -24], [7, 7, -7, -14], [2, 0, -4, 8], [8, 0, -16, -4], [0, 0, -24, 0], [0, -21, 0, 21], [2, 0, -4, 0], [-18, 12, 0, -6], [-12, -24, 0, 12], [-40, 0, 0, 0], [0, 7, 0, 0], [0, -3, 10, -3], [6, -14, 6, 0], [-8, 0, 8, 0], [2, 18, 0, -9], [-17, 3, 17, -6], [-12, -3, 12, 6], [4, -12, -2, 12], [10, -12, -10, 24], [-3, 0, 6, 26], [0, 8, 16, 8], [0, 1, 0, 1], [-3, 0, 6, 9], [30, -6, 0, 3], [3, 0, -6, 2], [4, 0, -2, 0], [-4, 2, 2, -2], [0, -6, 34, -6], [0, 5, -9, 5], [16, 0, -32, -6], [6, -3, -6, 6], [10, 0, -10, 0], [-3, -8, 3, 16], [0, 13, 13, 13], [24, 6, 0, -3], [-12, 15, -12, 0], [19, 0, -38, -16], [0, -13, -11, -13], [0, -18, 12, -18], [5, 33, 5, 0], [-36, -6, 0, 3], [-22, 12, 0, -6], [-3, 16, -3, 0], [0, 0, -23, 0], [0, -20, -6, -20], [-6, -2, -6, 0], [-15, 4, 15, -8], [22, -12, -22, 24], [9, 7, -9, -14], [45, 4, 0, -2], [-48, 4, 48, -8], [20, 12, 20, 0], [0, -21, -8, -21]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_114_h_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_114_2_h_f();". // 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_114_2_h_f(:prec:=4) chi := MakeCharacter_114_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(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_114_2_h_f();". // 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_114_2_h_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_114_h(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![4, 0, -2, 0, 1]>,<7,R![-2, -4, 1]>,<17,R![16, 48, 52, 12, 1]>],Snew); return Vf; end function;