// Make newform 3200.2.a.v in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3200_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_3200_2_a_v();". // 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_3200_2_a_v();". // 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_3200_a();" function MakeCharacter_3200_a() N := 3200; order := 1; char_gens := [1151, 901, 2177]; 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_3200_a_Hecke(Kf) return MakeCharacter_3200_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], [2], [0], [0], [-2], [-2], [-6], [-6], [0], [10], [8], [-2], [-6], [-2], [-12], [-10], [6], [-6], [-14], [4], [10], [-8], [10], [14], [-6], [-6], [8], [-2], [2], [-2], [12], [-6], [6], [14], [-6], [12], [-10], [-14], [16], [-10], [2], [-14], [24], [-6], [-18], [28], [10], [20], [-6], [-14], [10], [8], [-2], [-18], [-18], [-24], [2], [-32], [-2], [30], [14], [6], [-22], [28], [-26], [-10], [-2], [-2], [30], [-22], [-2], [12], [-28], [-18], [-26], [4], [34], [38], [-2], [-22], [-30], [10], [0], [26], [20], [-2], [14], [6], [-22], [28], [-6], [-16], [-16], [22], [2], [16], [-14], [10], [6], [34], [2], [22], [34], [-38], [22], [-18], [-18], [14], [-28], [14], [28], [14], [-6], [-42], [-4], [-2], [-6], [-16], [-10], [-30], [-38], [-22], [38], [-18], [10], [42], [-6], [-8], [-8], [30], [-6], [-8], [-32], [-42], [10], [-50], [-10], [18], [14], [26], [46], [18], [-16], [6], [10], [-4], [-58], [38], [-26], [4], [-18], [2], [18], [8], [22], [48], [20], [-18], [-38], [-22], [42], [6], [-32], [-42], [10], [-48], [-24], [-42], [-18], [-18], [-18], [26], [-28], [-54], [40], [30], [-50], [-22], [40], [10], [28], [34], [-34], [58], [12], [-30], [-26], [-46], [-6], [-8], [-66], [-2], [-30], [18], [-38], [42], [-62], [-2], [30], [-8], [18], [32], [22], [50], [-10], [-18], [16], [34], [30], [14], [10], [-54], [16], [-42], [-36], [-22], [68], [-2], [-8], [54], [-22], [-44], [-50], [-52], [-54], [34], [10], [0], [-16], [-10], [6], [-6], [-56], [-18], [46], [52], [-50], [62], [-18], [36], [26], [-10], [72], [-46], [-2], [-28], [-52], [-38], [-50], [12], [54], [-78], [64], [14], [-66], [-30], [18], [-18], [-18], [-42], [-44], [42], [26], [-34], [-22], [-78], [26], [-2], [6], [78], [-14], [58], [-54], [32], [46], [16], [-42], [2], [-34], [34], [-28], [-60], [-24], [-14], [-58], [64], [-38], [-42], [20], [50], [82], [-38], [-54], [-34], [-50], [10], [-8], [6], [-34], [50], [-26], [6], [-8], [26], [70], [10], [62], [-30], [-76], [-2], [28], [2], [30], [-22], [32], [-34], [34], [-24], [-34], [66], [10], [38], [-30], [-28], [22], [-2], [26], [-50], [-20], [14], [-22], [-66], [80], [42], [-58], [46], [-30], [-66], [-50], [52], [54], [22], [18], [-12], [54], [26], [-38], [46], [0], [-50], [50], [70], [58], [-52], [-46], [-54], [48], [-42], [74], [-80], [-90], [-50], [-12], [70], [-22], [42], [14], [-40], [26], [90], [-42], [-36], [-14], [-4], [38], [-30], [-8], [14], [-82], [-42], [42], [6], [-56], [78], [-22], [-24], [-32], [46], [-26], [-68], [-46], [-50], [-58], [-62], [-52], [-58], [-64], [-54], [22], [2], [-14], [-54], [20], [-42], [90], [-12], [62], [62], [-38], [-62], [46], [-34], [6], [-46], [-6], [-30], [-80], [-64], [-38], [-16], [-6], [22], [-92], [-42], [6], [-42], [82], [30], [22], [36]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3200_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3200_2_a_v();". // 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_3200_2_a_v(:prec:=1) chi := MakeCharacter_3200_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_3200_2_a_v();". // 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_3200_2_a_v( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3200_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-2, 1]>,<7,R![0, 1]>,<11,R![2, 1]>,<13,R![2, 1]>],Snew); return Vf; end function;