// Make newform 4600.2.a.g in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4600_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_4600_2_a_g();". // 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_4600_2_a_g();". // 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_4600_a();" function MakeCharacter_4600_a() N := 4600; order := 1; char_gens := [1151, 2301, 2577, 1201]; 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_4600_a_Hecke(Kf) return MakeCharacter_4600_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], [0], [0], [-4], [6], [2], [-6], [-6], [-1], [-6], [0], [8], [6], [2], [8], [8], [4], [-4], [-2], [-8], [-6], [12], [-10], [10], [18], [6], [-8], [-2], [-12], [-2], [-8], [-12], [2], [4], [8], [-8], [24], [8], [8], [-2], [0], [12], [-4], [22], [6], [20], [20], [16], [-14], [28], [6], [8], [14], [10], [10], [0], [26], [16], [-10], [-6], [-22], [0], [-16], [16], [2], [-10], [-28], [22], [-24], [2], [-34], [12], [28], [8], [2], [24], [-28], [2], [6], [-14], [10], [8], [-36], [-14], [-16], [-24], [-18], [-14], [10], [-16], [6], [8], [-32], [-40], [32], [-4], [42], [14], [6], [2], [28], [28], [10], [6], [-22], [-30], [-12], [26], [48], [-26], [-16], [28], [-18], [-10], [-40], [-10], [2], [24], [42], [26], [12], [2], [16], [-20], [-44], [-12], [32], [-32], [12], [-16], [-16], [-24], [-32], [-36], [50], [-34], [-44], [-22], [36], [-46], [-20], [30], [-24], [-18], [-54], [24], [2], [22], [-8], [8], [6], [18], [-36], [48], [34], [-20], [-44], [46], [-10], [-48], [32], [-6], [-16], [46], [18], [-52], [24], [14], [-26], [-6], [42], [26], [8], [54], [32], [26], [16], [-10], [32], [-16], [-32], [-22], [54], [6], [-36], [-4], [-54], [-6], [22], [48], [-22], [36], [22], [-50], [-18], [14], [-38], [32], [22], [48], [60], [48], [-10], [34], [30], [-2], [-4], [4], [-50], [8], [18], [-2], [24], [50], [-48], [-42], [-8], [42], [56], [18], [-18], [16], [-6], [-24], [24], [6], [54], [48], [-8], [-56], [-58], [46], [40], [38], [66], [20], [-58], [36], [-12], [24], [26], [40], [64], [14], [-18], [40], [-8], [-54], [-26], [-8], [12], [42], [8], [-70], [18], [-24], [52], [-2], [-2], [62], [32], [-34], [-62], [44], [30], [-10], [60], [-30], [-14], [-74], [10], [2], [30], [72], [30], [-16], [-68], [-34], [34], [54], [0], [-8], [-20], [-20], [-40], [48], [26], [-20], [-64], [30], [48], [-26], [-6], [86], [-2], [-40], [52], [-30], [-40], [60], [-14], [60], [-4], [4], [-22], [-50], [24], [52], [-32], [58], [8], [-48], [-26], [28], [60], [-54], [-40], [8], [-34], [6], [-74], [10], [-18], [-24], [18], [-74], [-34], [68], [-48], [68], [82], [-14], [32], [-44], [86], [-72], [16], [-66], [58], [64], [-62], [-30], [62], [52], [84], [0], [-10], [24], [-76], [-80], [0], [74], [42], [76], [20], [-74], [0], [10], [-42], [-40], [-68], [-42], [-72], [30], [-92], [42], [-54], [12], [-26], [16], [-48], [24], [-20], [40], [-30], [44], [4], [-10], [30], [54], [36], [-38], [-32], [58], [26], [-96], [80], [-82], [-58], [84], [58], [6], [20], [-4], [44], [-46], [-32], [6], [94], [-42], [-34], [18], [-60], [34], [-42], [0], [-28], [-50], [98], [-24], [10], [-30], [78], [2], [78], [-54], [48], [88], [42], [-68], [-20], [88], [48], [60], [14], [-38], [-6], [26], [-52], [-56]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4600_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4600_2_a_g();". // 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_4600_2_a_g(:prec:=1) chi := MakeCharacter_4600_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_4600_2_a_g();". // 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_4600_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4600_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![0, 1]>,<7,R![4, 1]>,<11,R![-6, 1]>],Snew); return Vf; end function;