// Make newform 7728.2.a.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_7728_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_7728_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_7728_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_7728_a();" function MakeCharacter_7728_a() N := 7728; order := 1; char_gens := [4831, 5797, 5153, 6625, 6721]; v := [1, 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_7728_a_Hecke(Kf) return MakeCharacter_7728_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], [0], [1], [2], [-6], [2], [6], [-1], [-6], [0], [0], [6], [6], [-8], [-4], [0], [-8], [2], [0], [-2], [-8], [2], [-2], [14], [-14], [4], [-2], [-12], [6], [8], [0], [-6], [0], [4], [24], [-20], [-16], [-16], [6], [8], [-8], [12], [10], [10], [-16], [-12], [-16], [-26], [24], [26], [-24], [-2], [2], [-2], [24], [6], [-8], [-22], [-10], [-30], [-24], [-12], [16], [-26], [2], [-20], [-6], [-8], [-2], [-14], [28], [-16], [-8], [34], [0], [0], [-14], [2], [10], [-38], [0], [36], [-34], [16], [32], [6], [-18], [-10], [-16], [-22], [24], [8], [0], [-16], [-12], [14], [-30], [-18], [-22], [-4], [-24], [6], [-46], [2], [6], [-48], [14], [40], [-34], [-48], [-20], [-42], [-14], [-4], [-30], [26], [0], [22], [6], [-8], [-50], [-8], [-44], [8], [24], [32], [40], [-40], [20], [-8], [32], [-20], [-28], [42], [-34], [28], [10], [-52], [26], [48], [-42], [8], [-42], [-34], [0], [34], [42], [20], [56], [18], [30], [36], [-48], [38], [-44], [0], [30], [-38], [-48], [-32], [58], [56], [6], [10], [36], [24], [38], [-10], [58], [30], [-50], [0], [34], [16], [-26], [-16], [-18], [28], [28], [8], [18], [-14], [58], [-36], [-4], [-58], [10], [-10], [48], [14], [44], [38], [-6], [-18], [-42], [50], [-16], [6], [-24], [16], [32], [38], [-38], [46], [-34], [64], [4], [70], [-44], [14], [66], [-44], [-2], [-64], [-18], [-48], [-46], [-8], [-10], [-2], [36], [66], [28], [-36], [-58], [46], [24], [44], [-24], [-54], [-14], [4], [54], [-38], [28], [38], [-16], [4], [0], [58], [12], [-40], [38], [-70], [56], [0], [50], [22], [56], [-20], [30], [48], [26], [2], [-48], [68], [-2], [22], [-22], [-44], [6], [14], [-16], [-30], [26], [80], [54], [22], [34], [-2], [38], [-46], [-76], [2], [16], [-4], [-50], [26], [54], [0], [-4], [-52], [-40], [-4], [72], [-34], [40], [8], [30], [60], [-30], [6], [-18], [26], [16], [24], [-78], [-60], [16], [-34], [0], [-32], [-28], [-38], [34], [-56], [40], [-24], [14], [-72], [60], [-14], [76], [-20], [-14], [20], [40], [-14], [-18], [-30], [-10], [34], [-16], [-6], [-2], [-10], [-64], [-48], [8], [-62], [46], [-24], [0], [30], [60], [-24], [-42], [74], [-20], [54], [74], [-6], [8], [0], [-64], [66], [32], [36], [40], [-4], [-26], [42], [-32], [-28], [-6], [56], [-54], [34], [80], [-44], [-74], [88], [10], [48], [54], [-46], [-32], [46], [-24], [28], [0], [-72], [-76], [94], [8], [-20], [66], [-54], [82], [20], [-54], [56], [14], [30], [40], [-72], [-98], [74], [-68], [90], [-66], [-12], [-16], [52], [-58], [-28], [-46], [-34], [-18], [-6], [46], [40], [-78], [66], [-8], [68], [-34], [2], [20], [62], [-74], [-10], [98], [74], [-18], [-48], [88], [-30], [92], [-72], [4], [-88], [-32], [42], [66], [90], [-22], [-72], [40], [62], [30], [44], [-60], [-6], [-94], [70], [-106], [-36]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_7728_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_7728_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_7728_2_a_g(:prec:=1) chi := MakeCharacter_7728_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(3067) - 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_7728_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_7728_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_7728_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![0, 1]>,<11,R![-2, 1]>,<13,R![6, 1]>,<17,R![-2, 1]>],Snew); return Vf; end function;