// Make newform 630.2.j.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_630_j();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_630_j_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_630_2_j_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_630_2_j_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 := [9, -3, -2, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -2, 2, 1], [3, 2, 0, -1]]; Rf_basisdens := [1, 1, 6, 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_630_j();" function MakeCharacter_630_j() N := 630; order := 3; char_gens := [281, 127, 451]; v := [1, 3, 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_630_j_Hecke();" function MakeCharacter_630_j_Hecke(Kf) N := 630; order := 3; char_gens := [281, 127, 451]; char_values := [[0, 0, -1, 0], [1, 0, 0, 0], [1, 0, 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, 0, -1, 0], [0, 0, 0, -1], [0, 0, 1, 0], [1, 0, -1, 0], [1, 0, -1, 0], [-1, 2, 1, -1], [3, 0, 0, 0], [1, 1, 1, 1], [0, 0, 2, 0], [2, 0, -2, 0], [0, 0, 2, 0], [4, 2, 2, 2], [-1, 2, 4, -1], [-6, -1, 7, 2], [3, 1, -4, -2], [-6, 2, 2, 2], [3, -6, -4, 3], [2, -4, 2, 8], [1, -2, -2, 1], [-6, -3, -3, -3], [1, 4, 4, 4], [-7, -3, 10, 6], [-3, -1, 4, 2], [-2, -4, -4, -4], [-7, 4, 3, -8], [2, 2, -4, -4], [0, 0, -8, 0], [3, -5, -5, -5], [-12, 3, 3, 3], [0, 0, 6, 0], [2, -2, -2, -2], [0, 0, -4, 0], [2, -1, -1, 2], [-1, 2, 6, -1], [5, -10, -1, 5], [7, -5, -2, 10], [-5, 10, 5, -5], [8, -2, -2, -2], [1, -2, 13, 1], [18, 0, -18, 0], [-16, 3, 3, 3], [-4, 2, 2, 2], [-8, 4, 4, -8], [-1, 2, -2, -1], [0, -2, -2, -2], [-24, 0, 0, 0], [9, -18, -5, 9], [-7, 5, 2, -10], [7, 0, -7, 0], [-2, 4, 2, -2], [5, -1, -1, -1], [2, -4, 8, 2], [16, -1, -15, 2], [-1, 7, 7, 7], [2, -4, -11, 2], [4, -8, 4, 16], [-14, -4, -4, -4], [0, 4, 4, 4], [2, 4, -6, -8], [9, 5, -14, -10], [-7, 14, -5, -7], [-4, 8, -10, -4], [5, -6, -6, -6], [-6, 12, -12, -6], [6, 3, -9, -6], [6, 6, -12, -12], [3, -7, 4, 14], [-3, 6, -6, -3], [-5, 10, 20, -5], [4, 4, -8, -8], [2, 5, -7, -10], [-12, -4, -4, -4], [9, 7, -16, -14], [2, -4, -16, 2], [17, 0, 0, 0], [3, -6, 1, 3], [25, -1, -24, 2], [-24, -2, -2, -2], [-7, 14, -2, -7], [-1, 2, 24, -1], [-4, 8, 28, -4], [29, -3, -26, 6], [-20, 7, 7, 7], [3, -1, -1, -1], [-12, 2, 10, -4], [-10, 3, 7, -6], [9, 1, 1, 1], [2, -1, -1, 2], [-26, -2, 28, 4], [0, 0, 28, 0], [-15, 0, 0, 0], [-10, 0, 10, 0], [-6, -8, -8, -8], [-8, 16, 5, -8], [12, -24, -1, 12], [18, 6, 6, 6], [10, -20, 2, 10], [37, 1, 1, 1], [-4, -3, -3, -3], [-8, 5, 5, 5], [-2, -1, 3, 2], [-40, 2, 2, 2], [-6, 12, 5, -6], [7, -8, 1, 16], [8, -16, 7, 8], [-7, -8, -8, -8], [-4, -7, 11, 14], [-2, -12, -12, -12], [3, -6, 13, 3], [-16, 9, 7, -18], [8, -16, -24, 8], [-18, -4, -4, -4], [11, -22, -20, 11], [-6, 5, 1, -10], [-22, -5, -5, -5], [-22, 5, 17, -10], [0, 0, -35, 0], [-2, 6, 6, 6], [2, -4, -20, 2], [39, -3, -36, 6], [2, -4, 38, 2], [-8, 2, 6, -4], [-7, -5, 12, 10], [-9, 5, 5, 5], [14, 2, -16, -4], [-14, -5, -5, -5], [10, -12, 2, 24], [16, -10, -10, -10], [1, 15, -16, -30], [-10, 20, 30, -10], [-7, 3, 3, 3], [14, -28, -8, 14], [0, 0, -20, 0], [-34, 0, 0, 0], [8, -16, 14, 8], [-12, 24, 2, -12], [26, -7, -7, -7], [9, -18, 11, 9], [1, -2, -17, 1], [1, -8, -8, -8], [7, -11, -11, -11], [25, -5, -20, 10], [0, 0, -12, 0], [12, -2, -2, -2], [-32, 0, 0, 0], [8, -14, 6, 28], [10, -10, 0, 20], [-13, -9, 22, 18], [-7, 14, 14, -7], [14, 4, 4, 4], [-4, 8, -18, -4], [30, 2, 2, 2], [-3, 9, 9, 9], [-2, 4, -16, -2], [34, -3, -31, 6], [-25, 3, 22, -6], [38, 2, 2, 2], [26, 8, -34, -16], [-34, 5, 5, 5], [4, -8, -20, 4], [12, 13, -25, -26], [11, -17, -17, -17], [-16, 32, -2, -16], [20, 2, 2, 2], [1, -2, 40, 1], [33, -5, -28, 10], [-12, -7, -7, -7], [34, 0, -34, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_630_j_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_630_2_j_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_630_2_j_g(:prec:=4) chi := MakeCharacter_630_j(); 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_630_2_j_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_630_2_j_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_630_j(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![1, -1, 1]>,<13,R![64, 8, 9, -1, 1]>],Snew); return Vf; end function;