// Make newform 8670.2.a.g in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8670_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_8670_2_a_g();". // 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_8670_2_a_g();". // 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_8670_a();" function MakeCharacter_8670_a() N := 8670; order := 1; char_gens := [2891, 6937, 6361]; 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_8670_a_Hecke(Kf) return MakeCharacter_8670_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], [1], [4], [0], [2], [0], [-4], [0], [6], [-8], [-2], [6], [-4], [0], [-6], [0], [10], [-4], [0], [-2], [-8], [12], [18], [-2], [18], [-4], [12], [10], [18], [20], [0], [6], [4], [-6], [8], [2], [4], [0], [-18], [24], [-14], [-24], [22], [6], [-8], [-20], [20], [12], [-10], [18], [24], [-2], [-24], [-18], [0], [6], [-16], [-2], [18], [28], [-6], [20], [0], [-2], [-18], [-28], [-26], [-12], [-10], [6], [-24], [28], [26], [4], [0], [-6], [22], [6], [26], [0], [-10], [0], [26], [-8], [12], [6], [26], [-30], [-4], [-36], [24], [28], [24], [4], [-24], [-6], [-18], [20], [10], [28], [18], [-12], [18], [-20], [2], [-12], [30], [-24], [22], [4], [2], [-30], [-44], [32], [30], [4], [24], [-18], [48], [14], [-26], [6], [12], [-44], [-6], [-38], [0], [-28], [-22], [-52], [24], [40], [2], [18], [2], [42], [4], [-30], [54], [4], [-18], [-20], [-12], [38], [-24], [46], [18], [-4], [24], [-2], [54], [-4], [0], [-44], [48], [-16], [6], [26], [-18], [-36], [6], [-4], [24], [-42], [24], [16], [-26], [-2], [54], [48], [14], [-24], [50], [56], [-18], [52], [-42], [-28], [14], [-52], [-48], [-50], [6], [0], [-6], [-2], [-52], [-26], [24], [22], [-12], [-28], [-30], [36], [-6], [-50], [46], [-30], [-24], [-18], [-8], [-22], [-46], [0], [-30], [-40], [-12], [30], [-28], [22], [42], [-44], [60], [48], [-26], [20], [-6], [-24], [66], [-10], [40], [-30], [52], [12], [-58], [-6], [0], [20], [-48], [2], [-20], [-16], [18], [-28], [-24], [-50], [54], [-48], [-48], [-36], [20], [68], [38], [-6], [48], [-68], [-24], [-28], [-24], [-22], [-66], [-48], [-26], [66], [-48], [34], [28], [-42], [2], [-20], [-12], [58], [-26], [18], [44], [-30], [42], [76], [-6], [34], [20], [-46], [-40], [26], [-52], [-36], [38], [-22], [0], [72], [40], [72], [-10], [4], [24], [46], [-42], [8], [66], [30], [-60], [-66], [0], [-26], [-18], [56], [42], [0], [-28], [50], [-78], [-80], [-12], [-68], [46], [-12], [-14], [48], [74], [-72], [-66], [54], [-28], [-72], [26], [-72], [-24], [70], [-30], [-44], [-2], [-54], [-4], [66], [2], [-20], [-44], [-48], [6], [-62], [-66], [88], [60], [52], [-36], [-34], [42], [-74], [20], [74], [78], [78], [-40], [18], [-24], [-38], [-4], [72], [-42], [20], [70], [42], [28], [14], [-66], [-48], [48], [66], [-48], [70], [-42], [0], [0], [92], [-22], [30], [-52], [-26], [-24], [-44], [48], [-78], [-40], [22], [-24], [-48], [-46], [42], [26], [-18], [6], [-92], [42], [-68], [48], [8], [-94], [-92], [-24], [-2], [6], [-48], [-52], [24], [22], [32], [42], [-20], [-54], [-14], [30], [-28], [-54], [-30], [16], [74], [-6], [-4], [48], [-2], [-54], [-60], [-44], [2], [6], [96], [-92], [-78], [96], [-78], [-26], [48], [-72], [-2], [-78], [84], [6], [20], [24], [-22], [24], [4], [-72], [22], [-78], [-26], [-34], [28], [-16], [36], [-42], [86], [72], [-2], [102], [-52], [72], [46], [14], [-52], [-96], [12], [66], [-22], [90], [-10], [-96], [22], [90], [76], [16], [-24], [-38], [-100], [26], [80], [-36], [30], [-4], [76], [-12], [-96], [-98], [-72], [-26], [78], [56], [48], [42], [74], [90], [-2], [-18], [-68], [12], [62], [48], [28], [80], [26], [-48], [-74], [-18], [0], [-14], [28], [90], [-56], [-76], [102], [20], [18], [-44], [-94], [-42], [48], [112], [50], [52], [24], [-24], [-22]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8670_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8670_2_a_g();". // 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_8670_2_a_g(:prec:=1) chi := MakeCharacter_8670_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(3673) - 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_8670_2_a_g();". // 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_8670_2_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8670_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-4, 1]>,<11,R![0, 1]>,<13,R![-2, 1]>,<23,R![0, 1]>],Snew); return Vf; end function;