// Make newform 624.2.a.j in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_624_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_624_2_a_j();". // 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_624_2_a_j();". // 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 Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_624_a();" function MakeCharacter_624_a() N := 624; order := 1; char_gens := [79, 469, 209, 145]; v := [1, 1, 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_624_a_Hecke(Kf) return MakeCharacter_624_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 := [[0], [1], [4], [0], [2], [-1], [2], [-8], [-4], [-6], [4], [6], [-12], [-4], [6], [-2], [14], [10], [4], [-2], [-2], [8], [-14], [0], [-10], [-6], [8], [0], [10], [-18], [8], [-8], [-12], [12], [-16], [-20], [-2], [-24], [-18], [18], [12], [6], [0], [26], [12], [8], [20], [-28], [6], [10], [-2], [-2], [-18], [-8], [14], [20], [26], [-8], [-2], [20], [-20], [0], [12], [12], [6], [-12], [28], [30], [12], [6], [-28], [30], [8], [26], [16], [-30], [-26], [-10], [0], [6], [24], [26], [-14], [-2], [0], [-8], [-8], [-10], [28], [16], [0], [-14], [-28], [0], [-36], [-36], [12], [-10], [20], [-6], [4], [-8], [-4], [-2], [-12], [2], [-22], [-8], [12], [22], [40], [14], [12], [-28], [20], [-14], [36], [-32], [-6], [12], [38], [-14], [-26], [2], [-20], [6], [2], [20], [24], [2], [28], [-34], [48], [-22], [20], [26], [-48], [44], [-2], [-10], [8], [24], [16], [10], [34], [-30], [38], [30], [36], [2], [-42], [38], [-44], [-32], [20], [-16], [-16], [20], [-18], [48], [-42], [6], [28], [60], [0], [-30], [32], [-26]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_624_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_624_2_a_j();". // 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_624_2_a_j(:prec:=1) chi := MakeCharacter_624_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_624_2_a_j();". // 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_624_2_a_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_624_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-4, 1]>,<7,R![0, 1]>],Snew); return Vf; end function;