// Make newform 1122.2.a.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1122_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_1122_2_a_f();". // 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_1122_2_a_f();". // 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_1122_a();" function MakeCharacter_1122_a() N := 1122; order := 1; char_gens := [749, 409, 1057]; 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_1122_a_Hecke(Kf) return MakeCharacter_1122_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], [-1], [-2], [0], [1], [-4], [-1], [-4], [-4], [0], [2], [0], [6], [-12], [-10], [-4], [4], [-2], [-12], [8], [-10], [8], [4], [-6], [-2], [-2], [8], [20], [-2], [6], [22], [-8], [6], [-20], [-10], [22], [-14], [20], [2], [8], [16], [4], [6], [-22], [-4], [-14], [4], [12], [-16], [-14], [-6], [0], [-6], [-16], [-14], [12], [14], [-6], [10], [30], [-28], [6], [8], [20], [-10], [-18], [-4], [2], [-4], [-28], [-30], [-20], [26], [-12], [-20], [-6], [24], [-28], [26], [14], [20], [-22], [-10], [-2], [28], [-24], [-14], [-26], [18], [-8], [-28], [10], [14], [12], [-24], [-6], [-40], [-30], [24], [18], [28], [18], [20], [18], [-28], [22], [8], [-14], [-18], [-26], [-16], [12], [-6], [40], [-4], [-2], [16], [18], [-6], [-20], [-18], [10], [-4], [-44], [44], [-30], [-48], [-16], [-28], [-28], [-16], [-30], [-10], [22], [-14], [54], [0], [28], [12], [14], [-20], [-36], [-6], [-32], [30], [4], [26], [-10], [-4], [-54], [46], [-14], [-12], [54], [-36], [8], [26], [46], [-14], [-12], [-12], [18], [42], [-28], [-2], [0], [-2], [18], [14], [-50], [60], [30], [-2], [-38], [48], [10], [52], [62], [-34], [32], [20], [40], [16], [38], [-22], [-48], [46], [-28], [-26], [46], [34], [36], [20], [-8], [-64], [-18], [-30], [-36], [42], [32], [24], [4], [-38], [42], [36], [-68], [16], [-36], [-22], [-60], [-2], [28], [-66], [12], [-66], [6], [18], [26], [-48], [-64], [-52], [48], [6], [34], [-28], [-44], [-50], [24], [22], [-24], [-14], [-4], [-42], [-18], [28], [0], [-34], [24], [36], [50], [44], [-52], [-24], [18], [74], [50], [-18], [-60], [0], [20], [52], [-22], [30], [-54], [14], [-76], [72], [60], [-66], [38], [-44], [-76], [-6], [-42], [-14], [-28], [-56], [18], [-20], [42], [0], [-28], [-42], [10], [-62], [8], [-32], [0], [-38], [36], [-60], [-26], [-66], [28], [-44], [-18], [-34], [36], [-12], [-18], [54], [60], [-6], [-12], [42], [60], [0], [60], [-12], [68], [-82], [38], [8], [4], [-64], [-18], [-48], [-16], [26], [-48], [-78], [10], [-2], [52], [-48], [10], [68], [-58], [34], [-62], [-44], [-42], [38], [-40], [38], [-50], [-84], [16], [-66], [60], [26], [42], [16], [60], [68], [-36], [54], [26], [-38], [-86], [-50], [-90], [40], [20], [36], [-60], [-28], [-76], [10], [-78], [4], [78], [-80], [64], [52], [22], [62], [24], [94], [54], [14], [-78], [-6], [-8], [68], [46], [52], [86], [-10], [52], [0], [-74], [-42], [-82], [44], [-68], [-18], [-38], [-22], [-10], [-42], [30], [92], [18], [-28], [60], [80], [14], [-44], [10], [26], [-66], [84], [-20], [-66], [-10], [18], [70], [-16], [14], [30], [-66], [-2], [22], [-18], [20], [-10], [46], [-4], [84], [82], [54], [16], [44], [-70], [-18], [50], [46], [10], [56], [-96], [2], [8], [-44], [54], [88], [36], [18], [36], [-30]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1122_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1122_2_a_f();". // 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_1122_2_a_f(:prec:=1) chi := MakeCharacter_1122_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_1122_2_a_f();". // 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_1122_2_a_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1122_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![2, 1]>,<7,R![0, 1]>,<13,R![4, 1]>,<19,R![4, 1]>],Snew); return Vf; end function;