// Make newform 576.2.i.i in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_576_i();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_576_i_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_576_2_i_i();". // 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_576_2_i_i();". // 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_576_i();" function MakeCharacter_576_i() N := 576; order := 3; char_gens := [127, 325, 65]; v := [3, 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_576_i_Hecke();" function MakeCharacter_576_i_Hecke(Kf) N := 576; order := 3; char_gens := [127, 325, 65]; char_values := [[1, 0, 0, 0], [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 := [[0, 0, 0, 0], [-1, 0, 0, 1], [0, 0, -1, 0], [-1, 1, 1, -2], [-1, -1, 1, 2], [0, -2, 1, -2], [0, -4, 0, 2], [-4, 0, 0, 0], [0, -1, 3, -1], [5, 2, -5, -4], [0, -1, -5, -1], [-4, -4, 0, 2], [0, -2, 7, -2], [5, 3, -5, -6], [-1, -3, 1, 6], [-4, 4, 0, -2], [0, -3, 7, -3], [-3, 2, 3, -4], [0, -3, 5, -3], [-2, -8, 0, 4], [0, 4, 0, -2], [-11, 1, 11, -2], [-3, -1, 3, 2], [-8, 4, 0, -2], [-1, -2, 1, 4], [3, -4, -3, 8], [0, -1, -15, -1], [-4, -8, 0, 4], [4, 4, 0, -2], [0, 4, 1, 4], [8, 0, 0, 0], [0, 5, -1, 5], [15, 2, -15, -4], [0, 1, -1, 1], [0, 2, 9, 2], [-3, 5, 3, -10], [0, 4, -1, 4], [-4, -8, 0, 4], [0, -1, -7, -1], [3, 0, -3, 0], [12, 0, 0, 0], [4, 12, 0, -6], [15, 1, -15, -2], [0, -2, -9, -2], [-2, -16, 0, 8], [2, 8, 0, -4], [0, 1, 13, 1], [-1, -7, 1, 14], [15, 3, -15, -6], [0, 0, -1, 0], [-6, 0, 0, 0], [0, 7, 3, 7], [-7, 0, 7, 0], [-12, 8, 0, -4], [0, -4, 1, -4], [-3, 1, 3, -2], [-12, -4, 0, 2], [0, -24, 0, 12], [-5, 4, 5, -8], [15, 2, -15, -4], [0, 1, -1, 1], [0, -4, -9, -4], [12, -8, 0, 4], [0, -5, 3, -5], [17, 0, -17, 0], [21, -2, -21, 4], [-1, -5, 1, 10], [0, -4, 1, -4], [0, -7, -11, -7], [-5, 4, 5, -8], [-1, -6, 1, 12], [8, 8, 0, -4], [-3, -3, 3, 6], [0, -6, 17, -6], [26, 0, 0, 0], [0, -1, -31, -1], [3, 4, -3, -8], [-4, -12, 0, 6], [0, -6, -1, -6], [0, 8, -7, 8], [0, 5, -19, 5], [5, 2, -5, -4], [24, 8, 0, -4], [16, 20, 0, -10], [29, -3, -29, 6], [21, -1, -21, 2], [16, 4, 0, -2], [7, -10, -7, 20], [21, -6, -21, 12], [0, -5, 19, -5], [-20, 16, 0, -8], [-3, 9, 3, -18], [-8, 8, 0, -4], [0, 5, 23, 5], [0, -3, -9, -3], [-30, 8, 0, -4], [0, 10, 9, 10], [-18, 0, 0, 0], [6, 16, 0, -8], [-10, 16, 0, -8], [-9, -1, 9, 2], [4, -12, 0, 6], [0, -3, -3, -3], [1, 0, -1, 0], [0, -3, -11, -3], [-24, -4, 0, 2], [-3, 11, 3, -22], [24, -4, 0, 2], [0, -1, -31, -1], [-25, -6, 25, 12], [0, 11, 1, 11], [12, -12, 0, 6], [0, 6, -9, 6], [-9, 7, 9, -14], [-16, -8, 0, 4], [-7, 8, 7, -16], [0, -3, -3, -3], [0, -8, 0, 4], [0, 0, -17, 0], [15, 3, -15, -6], [0, 12, -1, 12], [7, 14, -7, -28], [-5, -12, 5, 24], [-12, -24, 0, 12], [-11, 11, 11, -22], [-12, -20, 0, 10], [-5, 0, 5, 0], [-8, -32, 0, 16], [21, 1, -21, -2], [0, 2, -7, 2], [14, 0, 0, 0], [0, -1, 33, -1], [0, -17, 1, -17], [42, 0, 0, 0], [0, -14, -9, -14], [0, -8, 17, -8], [12, -20, 0, 10], [0, 5, -9, 5], [0, -6, 33, -6], [-8, 20, 0, -10], [-28, -16, 0, 8], [-19, -14, 19, 28], [0, -1, -13, -1], [28, 0, 0, 0], [28, 4, 0, -2], [23, -3, -23, 6], [-3, 2, 3, -4], [23, -2, -23, 4], [0, 1, -19, 1], [-8, 8, 0, -4], [0, 2, -23, 2], [0, 20, 0, -10], [-20, 16, 0, -8], [0, -13, 3, -13], [13, -5, -13, 10], [-3, -19, 3, 38], [0, 8, 0, -4], [-33, 6, 33, -12], [32, -20, 0, 10], [0, 0, 7, 0], [-33, -5, 33, 10], [48, 4, 0, -2], [0, -5, 3, -5], [10, 0, 0, 0], [0, -4, -23, -4], [-27, -3, 27, 6], [8, 24, 0, -12], [3, -12, -3, 24]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_576_i_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_576_2_i_i();". // 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_576_2_i_i(:prec:=4) chi := MakeCharacter_576_i(); 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_576_2_i_i();". // 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_576_2_i_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_576_i(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![1, 1, 1]>,<7,R![25, -10, 9, 2, 1]>],Snew); return Vf; end function;