// Make newform 1088.2.a.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1088_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_1088_2_a_e();". // 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_1088_2_a_e();". // 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_1088_a();" function MakeCharacter_1088_a() N := 1088; order := 1; char_gens := [511, 69, 513]; 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_1088_a_Hecke(Kf) return MakeCharacter_1088_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], [2], [2], [-6], [-2], [1], [0], [-6], [10], [-2], [-6], [-6], [-8], [0], [10], [-8], [-14], [4], [-2], [-14], [10], [8], [-10], [2], [-10], [-8], [-18], [-6], [10], [-4], [14], [6], [14], [10], [12], [2], [-22], [6], [-6], [8], [2], [0], [-6], [10], [14], [-6], [12], [2], [14], [10], [8], [-22], [20], [6], [-12], [-14], [8], [-22], [10], [-22], [-6], [12], [30], [-6], [-22], [8], [2], [-6], [34], [18], [-12], [-6], [-26], [6], [28], [14], [18], [10], [-6], [-6], [6], [10], [38], [-2], [-36], [18], [-18], [34], [8], [8], [-10], [2], [8], [-14], [-22], [34], [10], [4], [26], [-42], [-2], [-24], [-22], [18], [-18], [0], [2], [16], [-14], [-26], [-6], [-6], [-22], [-44], [50], [14], [-16], [-30], [-44], [-22], [-30], [34], [-18], [-38], [30], [-6], [-26], [-8], [34], [40], [38], [-18], [-50], [6], [-2], [14], [22], [-30], [-30], [14], [-30], [-14], [-14], [2], [-54], [-46], [-38], [-8], [24], [-22], [-30], [4], [6], [-38], [-46], [48], [42], [42], [-22], [-58], [6], [20], [16], [18], [-50], [38], [58], [18], [18], [36], [-2], [50], [-26], [-60], [18], [22], [18], [-20], [22], [-16], [-50], [26], [-6], [-44], [-42], [-22], [-20], [-46], [22], [-30], [34], [64], [22], [22], [34], [-54], [26], [-14], [-56], [-6], [-46], [-54], [6], [-12], [6], [56], [-8], [-14], [20], [34], [30], [50], [40], [10], [-22], [16], [-30], [42], [30], [-6], [14], [-30], [-34], [36], [-74], [50], [-46], [-20], [-54], [-14], [-18], [-44], [-2], [-12], [-12], [10], [-6], [-10], [-60], [2], [20], [8], [-74], [-54], [30], [-18], [18], [-32], [-36], [2], [10], [-44], [18], [-26], [-4], [-22], [38], [-30], [-50], [-26], [-36], [66], [18], [82], [-28], [66], [-54], [-14], [-6], [34], [-28], [58], [4], [-30], [52], [16], [-2], [22], [-48], [16], [78], [-30], [42], [22], [48], [2], [34], [36], [6], [42], [-34], [-34], [-62], [26], [-54], [-40], [6], [-22], [72], [-26], [-42], [26], [38], [86], [-78], [-4], [-14], [40], [-38], [58], [18], [-62], [-56], [48], [10], [8], [6], [-70], [-30], [-30], [10], [-30], [-16], [42], [-46], [30], [18], [-50], [66], [50], [34], [-26], [-20], [34], [66], [-30], [10], [-70], [84], [-38], [-70], [-54], [-8], [14], [-14], [66], [-4], [-42], [18], [32], [-14], [50], [70], [-2], [6], [-20], [-42], [-22], [-60], [-62], [10], [8], [-6], [80], [26], [58], [-88], [34], [-56], [90], [34], [54], [-16], [2], [-34], [-30], [-10], [34], [58], [-62], [26], [-34], [-22], [66], [-14], [20], [62], [86], [-80], [26], [-22], [60], [-92], [84], [42], [72], [90], [2], [54], [10], [14], [-24], [-38], [26], [62], [-42], [-18], [-8], [-90], [82], [78], [-68], [-50], [-38], [-102], [-46], [78], [66], [88], [34], [34], [-18], [-24], [74], [-46], [-10], [-78], [-76], [82]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1088_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1088_2_a_e();". // 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_1088_2_a_e(:prec:=1) chi := MakeCharacter_1088_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_1088_2_a_e();". // 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_1088_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1088_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![2, 1]>,<5,R![-2, 1]>,<7,R![-2, 1]>],Snew); return Vf; end function;