// Make newform 1856.2.a.n in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1856_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_1856_2_a_n();". // 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_1856_2_a_n();". // 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_1856_a();" function MakeCharacter_1856_a() N := 1856; order := 1; char_gens := [639, 581, 321]; 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_1856_a_Hecke(Kf) return MakeCharacter_1856_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], [-4], [-6], [-2], [2], [-6], [-4], [1], [6], [-2], [2], [10], [2], [-10], [0], [-10], [-12], [-8], [10], [6], [16], [2], [10], [14], [4], [-8], [10], [-6], [-2], [-18], [-6], [-8], [6], [16], [6], [-18], [8], [-18], [20], [-22], [-10], [2], [-6], [-20], [-18], [-4], [0], [-10], [-22], [12], [2], [-6], [-2], [-18], [6], [10], [-10], [-6], [4], [-2], [2], [-10], [6], [-18], [-2], [-14], [4], [30], [22], [22], [-10], [-6], [2], [16], [6], [2], [-30], [-6], [20], [-2], [4], [2], [-32], [-2], [2], [-2], [22], [-40], [42], [2], [28], [-18], [-20], [2], [30], [14], [0], [-10], [32], [-22], [22], [42], [-40], [26], [-8], [6], [38], [2], [18], [-14], [-6], [-26], [-16], [-38], [4], [8], [-34], [-30], [-42], [-30], [46], [48], [32], [30], [6], [44], [-2], [30], [-14], [-38], [-14], [-26], [6], [-14], [-26], [-4], [-10], [-6], [36], [-42], [-2], [22], [-34], [-54], [22], [54], [14], [16], [14], [18], [-44], [30], [-26], [34], [-12], [-18], [-2], [-22], [42], [-22], [-22], [26], [-50], [50], [-16], [-2], [14], [-58], [12], [26], [-52], [-22], [-12], [-18], [40], [54], [-2], [-46], [-58], [-42], [10], [-10], [4], [26], [-42], [-46], [66], [12], [2], [42], [-54], [22], [6], [42], [-30], [18], [18], [0], [14], [0], [54], [18], [-10], [-46], [-18], [-24], [-18], [-34], [-6], [54], [10], [34], [-18], [54], [56], [-22], [56], [-26], [-50], [-52], [-70], [6], [-60], [-2], [-54], [-30], [-22], [-32], [-18], [20], [-50], [10], [12], [-6], [50], [-58], [56], [30], [38], [56], [16], [-34], [-46], [0], [-12], [20], [-28], [22], [-26], [-26], [18], [10], [54], [0], [-50], [42], [-30], [-6], [26], [-54], [58], [-10], [-6], [26], [14], [42], [14], [18], [-50], [40], [74], [22], [-62], [-34], [-18], [50], [-78], [-12], [60], [72], [28], [-10], [66], [18], [-14], [-50], [-28], [-14], [-14], [68], [-6], [66], [22], [-50], [-14], [54], [-40], [22], [50], [46], [10], [-6], [78], [30], [54], [-34], [-80], [58], [-88], [-26], [-78], [72], [-24], [-22], [-86], [-84], [-2], [-6], [-22], [70], [66], [2], [-6], [-62], [-68], [-52], [14], [-30], [-2], [-18], [-16], [-34], [-50], [-12], [-78], [-30], [42], [92], [-74], [58], [30], [-12], [62], [66], [-18], [-82], [-90], [-42], [-76], [-18], [54], [64], [-10], [34], [22], [-20], [-78], [60], [90], [42], [54], [64], [42], [42], [70], [-8], [-62], [-18], [-56], [-36], [86], [0], [-6], [-34], [-30], [-6], [-74], [-58], [-18], [6], [-86], [90], [-40], [36], [-50], [86], [70], [-50], [-62], [6], [-14], [38], [-90], [-10], [-40], [-62], [-44], [14], [94], [50], [-70], [-22], [22], [20], [26], [-70], [26], [-28], [-34], [-42], [-96], [52], [82], [-10], [42], [84], [90], [-34], [42], [-58], [-78], [6], [10], [42], [-12], [-62], [84], [58]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1856_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1856_2_a_n();". // 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_1856_2_a_n(:prec:=1) chi := MakeCharacter_1856_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_1856_2_a_n();". // 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_1856_2_a_n( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1856_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-2, 1]>,<5,R![-2, 1]>,<17,R![-2, 1]>],Snew); return Vf; end function;