// Make newform 8550.2.a.bg in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8550_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_8550_2_a_bg();". // 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_8550_2_a_bg();". // 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_8550_a();" function MakeCharacter_8550_a() N := 8550; order := 1; char_gens := [1901, 1027, 1351]; 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_8550_a_Hecke(Kf) return MakeCharacter_8550_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], [0], [0], [2], [-2], [4], [-6], [1], [-8], [-6], [-8], [8], [-12], [0], [0], [-10], [-6], [-6], [-12], [12], [10], [-8], [8], [-8], [14], [2], [14], [-4], [-2], [18], [-18], [2], [6], [-20], [-2], [20], [-24], [-4], [16], [-2], [2], [14], [12], [2], [26], [-24], [4], [14], [-16], [30], [10], [-28], [-18], [18], [10], [-16], [10], [0], [-4], [0], [20], [-30], [-16], [20], [-14], [18], [-28], [-6], [12], [-22], [10], [8], [-14], [-28], [12], [-16], [-26], [-20], [24], [10], [10], [10], [0], [34], [-28], [-12], [-8], [-10], [30], [-10], [-32], [-12], [26], [18], [-12], [24], [-14], [-24], [36], [-38], [-8], [-18], [-16], [-24], [12], [-2], [0], [-18], [8], [30], [-22], [20], [-22], [-12], [20], [28], [4], [-24], [2], [30], [-18], [2], [-6], [4], [-36], [42], [14], [24], [22], [-36], [-28], [-24], [16], [36], [28], [-2], [-6], [12], [-2], [-16], [4], [-38], [-26], [-48], [-10], [16], [12], [38], [20], [-40], [-12], [-52], [36], [8], [-8], [0], [-20], [36], [-2], [-18], [12], [18], [2], [-10], [-26], [-24], [-60], [56], [14], [14], [-30], [-30], [-56], [10], [0], [-12], [20], [-18], [10], [38], [-34], [54], [40], [26], [40], [-18], [20], [44], [-50], [-12], [34], [-20], [52], [38], [-24], [26], [-30], [56], [30], [8], [46], [0], [-52], [-50], [-34], [-46], [4], [-32], [-4], [28], [50], [-42], [-30], [8], [60], [10], [10], [-40], [8], [46], [-38], [-44], [44], [-26], [36], [54], [34], [-24], [26], [-10], [-40], [-20], [20], [-16], [-36], [48], [34], [10], [-42], [-68], [-16], [-44], [-18], [30], [6], [-60], [2], [10], [-68], [-24], [-52], [64], [-48], [10], [78], [-46], [58], [-52], [-42], [-14], [-26], [8], [-50], [56], [30], [28], [6], [-40], [20], [74], [2], [-48], [-54], [-52], [-22], [-46], [76], [6], [-54], [10], [40], [72], [56], [42], [-8], [-44], [-34], [-22], [-16], [28], [30], [-32], [54], [50], [-48], [38], [-88], [-26], [10], [-56], [66], [-78], [-24], [20], [68], [-2], [60], [-6], [-60], [56], [-8], [42], [24], [56], [80], [38], [14], [-12], [-86], [0], [-20], [50], [-30], [2], [74], [14], [-36], [48], [-88], [-66], [-10], [22], [28], [0], [-4], [24], [50], [-26], [58], [82], [-20], [-22], [-54], [-4], [30], [66], [22], [32], [20], [-42], [-84], [62], [-6], [-74], [14], [14], [96], [30], [-50], [-32], [88], [56], [-16], [66], [36], [-34], [-18], [-26], [-58], [-42], [20], [-8], [34], [-80], [36], [-78], [-24], [-34], [52], [14], [-2], [6], [-78], [-6], [-20], [56], [12], [60], [48], [-48], [14], [14], [30], [-76], [64], [-58], [-100], [4], [-76], [54], [50], [54], [-2], [-22], [-86], [-20], [-84], [-48], [28], [-14], [22], [-22], [-84], [44], [-38], [2], [56], [-26], [-26], [24], [70], [12], [88], [-42], [50], [10], [-52], [-48], [-28], [-48], [-50], [-98], [-68], [-96], [76], [-56], [38], [-34], [28], [100], [36], [-68], [10], [-84], [26], [-90], [-64], [-72], [18], [58], [-44], [-84], [-84], [4], [-46], [-50], [-10], [-18], [-56], [-30], [-68], [-40], [-110], [-30], [104], [-46], [24], [44], [-40], [-20], [46], [-96], [-8], [50], [-90], [8], [46], [-60], [-16], [74], [-34], [12], [78], [-46], [-106], [-60], [-42], [-66], [-4], [32], [16], [-64], [90], [-102], [2], [58], [-52], [-42], [64], [-4], [114], [74], [-58]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8550_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8550_2_a_bg();". // 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_8550_2_a_bg(:prec:=1) chi := MakeCharacter_8550_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(3593) - 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_8550_2_a_bg();". // 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_8550_2_a_bg( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8550_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-2, 1]>,<11,R![2, 1]>,<13,R![-4, 1]>,<17,R![6, 1]>,<23,R![8, 1]>,<53,R![10, 1]>],Snew); return Vf; end function;