// Make newform 3234.2.a.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3234_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_3234_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_3234_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 Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_3234_a();" function MakeCharacter_3234_a() N := 3234; order := 1; char_gens := [1079, 199, 2059]; v := [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_3234_a_Hecke(Kf) return MakeCharacter_3234_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], [-2], [0], [1], [-2], [-6], [8], [4], [2], [-8], [6], [-6], [8], [-4], [10], [-4], [14], [-4], [-4], [14], [-8], [-4], [14], [-18], [6], [16], [-12], [10], [10], [8], [4], [-6], [16], [10], [16], [2], [12], [8], [14], [-12], [10], [-4], [-6], [-6], [8], [8], [-16], [4], [10], [-18], [-8], [-2], [20], [-26], [24], [-2], [-8], [-6], [30], [0], [6], [8], [-12], [6], [18], [4], [-22], [-36], [22], [-26], [24], [0], [-22], [20], [28], [-30], [18], [34], [22], [12], [-26], [32], [-2], [40], [4], [-6], [10], [-2], [-24], [-20], [-24], [-24], [20], [-28], [24], [-2], [30], [-16], [-38], [40], [18], [-36], [38], [32], [-18], [12], [-30], [-12], [-42], [-24], [2], [-46], [-4], [-16], [-6], [-28], [-12], [10], [-28], [26], [-22], [6], [-36], [-12], [-30], [-34], [-12], [-40], [14], [-16], [40], [8], [6], [-30], [38], [22], [-8], [22], [-10], [-56], [10], [8], [28], [26], [-36], [-10], [18], [4], [-44], [2], [-18], [44], [-32], [12], [20], [16], [22], [22], [-18], [-36], [22], [-48], [12], [2], [4], [-40], [-18], [2], [-18], [4], [-14], [24], [-30], [32], [22], [0], [-62], [56], [-26], [-8], [4], [-26], [18], [-12], [46], [-62], [4], [10], [8], [-50], [60], [-44], [-42], [36], [-66], [-14], [6], [-38], [40], [50], [48], [-38], [-26], [-44], [46], [40], [-52], [6], [-60], [42], [30], [32], [36], [0], [-58], [64], [66], [-4], [42], [10], [-64], [66], [-16], [12], [10], [54], [36], [-8], [-60], [-50], [32], [-16], [70], [44], [-48], [-34], [-22], [28], [44], [-44], [0], [-64], [38], [2], [48], [64], [4], [0], [-16], [-14], [50], [-44], [38], [14], [52], [22], [40], [70], [62], [24], [12], [-10], [6], [14], [60], [74], [-42], [-8], [-22], [18], [-28], [-10], [-64], [-10], [-24], [-12], [-70], [-22], [28], [24], [40], [0], [62], [-72], [-36], [34], [26], [-40], [18], [26], [52], [-18], [-12], [34], [46], [80], [54], [4], [-32], [14], [18], [-64], [4], [4], [-62], [-12], [-86], [76], [26], [0], [42], [-42], [-44], [72], [-18], [44], [-24], [-66], [78], [40], [26], [-34], [64], [46], [-50], [44], [60], [-24], [-54], [82], [42], [16], [-44], [-8], [84], [6], [-14], [-58], [-88], [-50], [66], [-90], [-88], [-94], [36], [58], [52], [-24], [6], [16], [26], [34], [72], [50], [-30], [52], [44], [-18], [84], [34], [-86], [12], [36], [-84], [74], [-18], [16], [34], [-4], [84], [8], [-54], [24], [-34], [-60], [-72], [-18], [-70], [-10], [70], [34], [-48], [-66], [8], [20], [8], [-14], [24], [-84], [82], [-2], [-52], [-28], [76], [-38], [80], [54], [20], [-70], [-66], [-78], [32], [-42], [38], [-40], [-2], [-42], [-20], [-60], [30], [-22], [-76], [24], [2], [-2], [-104], [-16], [78], [40], [74], [-90], [-76], [-20], [-10], [-50], [76], [-66], [52], [0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3234_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3234_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_3234_2_a_i(:prec:=1) chi := MakeCharacter_3234_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(2999) - 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_3234_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_3234_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3234_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![2, 1]>,<13,R![2, 1]>,<17,R![6, 1]>],Snew); return Vf; end function;