// Make newform 8550.2.a.bf 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_bf();". // 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_bf();". // 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], [-2], [-1], [4], [0], [-8], [-8], [8], [6], [-12], [-6], [0], [2], [-8], [8], [-14], [0], [4], [0], [12], [-2], [-4], [-12], [10], [-6], [-8], [18], [-22], [0], [-10], [-8], [-18], [-14], [8], [14], [0], [2], [18], [-24], [18], [-20], [12], [-4], [-12], [10], [-6], [-10], [2], [-2], [18], [24], [0], [12], [-18], [-12], [-14], [14], [32], [18], [6], [18], [12], [-28], [-12], [-10], [-6], [10], [-18], [36], [-20], [24], [-30], [22], [-12], [30], [30], [22], [-12], [16], [-40], [24], [0], [-18], [-2], [6], [8], [10], [-28], [-22], [-40], [-16], [-20], [28], [-4], [-18], [32], [-42], [4], [0], [32], [22], [8], [-46], [-20], [-18], [12], [6], [-42], [-40], [32], [-32], [-14], [-32], [-46], [-20], [42], [-4], [18], [-36], [-8], [-2], [10], [-50], [-18], [26], [20], [24], [-8], [2], [18], [50], [34], [-28], [-2], [30], [12], [38], [-34], [28], [10], [40], [-14], [-42], [20], [-16], [12], [18], [26], [48], [-28], [-32], [0], [30], [2], [8], [-52], [34], [-58], [-12], [38], [24], [-8], [62], [50], [-26], [60], [22], [-22], [6], [-40], [30], [-8], [-42], [-44], [-50], [42], [28], [36], [-22], [44], [30], [32], [-4], [30], [38], [16], [64], [12], [8], [8], [34], [62], [26], [-42], [44], [0], [32], [12], [50], [50], [18], [0], [44], [50], [52], [22], [58], [-34], [-12], [20], [22], [62], [-52], [-72], [34], [-58], [0], [0], [-54], [-12], [-10], [-46], [-60], [-48], [-22], [26], [60], [32], [-32], [46], [-12], [-50], [54], [-70], [-12], [4], [-68], [-14], [-70], [14], [30], [42], [-72], [20], [-16], [-38], [-2], [8], [-50], [34], [30], [42], [-28], [18], [-78], [-44], [-52], [70], [-44], [-42], [-60], [-40], [-42], [76], [-6], [-58], [32], [-14], [-20], [52], [6], [-12], [50], [22], [-62], [-56], [-8], [-52], [62], [-58], [18], [6], [58], [80], [40], [-42], [-52], [-26], [-52], [76], [10], [-48], [34], [0], [-58], [26], [18], [0], [84], [72], [-68], [-52], [-50], [-30], [-14], [-76], [-70], [68], [16], [48], [50], [-70], [88], [86], [10], [52], [42], [88], [16], [-26], [-78], [20], [-4], [-72], [-6], [-18], [-62], [-60], [-36], [32], [-52], [-90], [-26], [-18], [2], [36], [18], [60], [-8], [-6], [60], [2], [-8], [-12], [-42], [-68], [32], [18], [-4], [10], [-26], [30], [-42], [78], [-56], [-18], [58], [48], [60], [2], [-64], [38], [76], [-58], [-22], [20], [24], [-60], [-48], [22], [-80], [-42], [-74], [-90], [-28], [-52], [-66], [22], [18], [-20], [24], [-68], [-58], [-14], [88], [-50], [34], [-50], [2], [-12], [-64], [40], [20], [-28], [-62], [-50], [-26], [-48], [18], [40], [-8], [22], [-72], [-64], [-10], [-4], [18], [-76], [-48], [2], [38], [-20], [-8], [18], [-16], [-60], [12], [-92], [-40], [96], [78], [-36], [30], [72], [-10], [-18], [-102], [-20], [-16], [82], [-82], [10], [82], [-88], [60], [44], [90], [-10], [0], [62], [38], [-34], [-32], [70], [-18], [52], [108], [44], [30], [62], [-52], [10], [-52], [46], [98], [-60], [-8], [-40], [-58], [42], [-94], [-80], [-36], [-30], [-68], [96], [-12], [-60], [22], [-12], [56], [30], [-28], [-32], [-46], [56], [0], [32], [88], [-34], [-92], [10], [-12], [60], [-88], [-8], [88], [-50], [14], [30], [-98], [-88], [38], [-40], [-68], [18], [86], [-66]]; 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_bf();". // 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_bf(: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_bf();". // 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_bf( : 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![2, 1]>,<23,R![-4, 1]>,<53,R![6, 1]>],Snew); return Vf; end function;