// Make newform 441.2.a.i in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_441_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_441_2_a_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_441_2_a_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 := [-2, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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_441_a();" function MakeCharacter_441_a() N := 441; order := 1; char_gens := [344, 199]; v := [1, 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; function MakeCharacter_441_a_Hecke(Kf) return MakeCharacter_441_a(); 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], [-2, 1], [0, 0], [2, 0], [4, 1], [-2, -3], [0, 2], [2, -4], [4, -2], [-4, -2], [-4, 0], [-2, 3], [0, -4], [0, -2], [2, 0], [4, 2], [8, -3], [0, 4], [2, 8], [4, -7], [8, -4], [4, 8], [10, -3], [4, 1], [-10, -5], [-4, -6], [6, 4], [0, 4], [-6, -8], [4, 4], [4, -8], [0, 10], [-12, 4], [-6, 8], [12, 0], [16, 3], [0, 8], [0, -14], [-14, 5], [14, -4], [-4, -1], [18, 0], [-6, -8], [-2, 0], [16, -4], [-4, -12], [8, -12], [-4, 14], [4, 3], [-8, -10], [10, -4], [-12, 3], [4, -6], [-2, 15], [6, 8], [2, -9], [4, 10], [2, 8], [8, 6], [0, 6], [-18, -1], [16, -10], [8, 10], [-8, -7], [-10, 0], [-4, 0], [-24, 4], [-22, 8], [0, 7], [2, -9], [6, 4], [8, 8], [-22, -8], [20, 8], [24, 4], [4, -10], [16, 13], [-8, -10], [-20, 1], [-36, -2], [-6, -16], [10, -12], [-16, 3], [24, 8], [-18, 12], [-6, -8], [-18, 0], [18, 1], [-4, 16], [-20, -14], [-16, -10], [4, -16], [-22, -12], [-8, -8], [0, 0], [10, 11], [34, -1], [-20, 4], [6, -8], [-20, 12], [-26, 0], [4, 2], [8, -6], [28, 4], [-24, 7], [20, -2], [14, 5], [2, 0], [-8, 5], [-24, -4], [16, 8], [16, -6], [12, -12], [12, -8], [8, -2], [8, 30], [-24, 2], [-28, 6], [2, 8], [-8, 11], [12, -8], [-30, 1], [14, -4], [-12, -8], [20, 2], [-8, 20], [-8, 24], [4, -6], [20, -21], [16, 8], [34, -12], [8, -4], [32, 4], [-14, -23], [0, -21], [-6, -11], [36, 8], [-10, -13], [14, 24], [-12, 4], [22, 8], [-8, -12], [-42, 4], [12, 9], [-48, 2], [0, 35], [-14, 1], [32, -18], [-6, 8], [0, 8], [-26, -3], [-16, -8], [0, -2], [-16, 0], [18, 12], [20, 20], [18, -15], [32, 1], [-14, 15], [-42, -8], [-2, 0], [28, -4], [28, -16], [24, 2], [8, -32], [4, 12], [8, -13]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_441_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_441_2_a_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_441_2_a_i(:prec:=2) chi := MakeCharacter_441_a(); 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_441_2_a_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_441_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_441_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-1, -2, 1]>,<5,R![2, 4, 1]>,<13,R![14, -8, 1]>],Snew); return Vf; end function;