// Make newform 8112.2.a.be in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8112_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_8112_2_a_be();". // 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_8112_2_a_be();". // 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_8112_a();" function MakeCharacter_8112_a() N := 8112; order := 1; char_gens := [5071, 6085, 2705, 3889]; v := [1, 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_8112_a_Hecke(Kf) return MakeCharacter_8112_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], [1], [2], [0], [4], [0], [2], [-4], [8], [6], [8], [-6], [6], [-4], [0], [-2], [4], [-2], [-4], [8], [-10], [8], [-4], [6], [-2], [-18], [-16], [12], [2], [18], [8], [4], [6], [12], [-14], [-16], [-2], [12], [24], [6], [-12], [6], [0], [-2], [18], [-16], [20], [-8], [12], [-22], [10], [-16], [-18], [-20], [2], [8], [-10], [8], [-26], [-26], [28], [18], [12], [24], [-6], [-6], [20], [18], [12], [-30], [-2], [-24], [8], [-10], [20], [0], [-2], [-14], [30], [6], [-12], [10], [32], [-14], [0], [-20], [14], [22], [26], [8], [36], [-16], [-32], [12], [12], [-24], [-6], [26], [-4], [18], [-44], [26], [-28], [10], [-36], [-2], [-44], [14], [-24], [-38], [40], [-38], [-42], [-44], [16], [-14], [12], [-8], [6], [-12], [10], [34], [-2], [4], [-4], [6], [10], [32], [-48], [-14], [-4], [-8], [-24], [38], [22], [-2], [18], [28], [22], [26], [4], [-30], [16], [-28], [-50], [-24], [10], [42], [12], [-32], [18], [50], [4], [-8], [-4], [-16], [-16], [-50], [42], [-6], [12], [-54], [-16], [-36], [30], [-24], [-40], [-26], [46], [-50], [36], [-30], [40], [-10], [40], [26], [-28], [-30], [-16], [14], [8], [52], [54], [54], [48], [-18], [-50], [-36], [-10], [0], [-62], [-12], [20], [-22], [36], [-22], [14], [46], [-2], [-24], [-38], [-8], [-38], [34], [-60], [6], [56], [-12], [22], [-4], [18], [-18], [32], [-28], [8], [22], [40], [-62], [-24], [-54], [-10], [0], [-18], [8], [-60], [-10], [-6], [-48], [-48], [52], [-66], [-28], [-40], [10], [60], [-64], [-50], [50], [-20], [-40], [-36], [28], [-64], [-46], [-18], [40], [-56], [60], [52], [16], [-78], [30], [8], [10], [54], [12], [38], [4], [78], [-58], [8], [-60], [-22], [-66], [14], [4], [-38], [-26], [-76], [62], [-50], [-52], [38], [-40], [-14], [-64], [-12], [-62], [-10], [-76], [80], [16], [8], [58], [-28], [-16], [50], [18], [-16], [2], [-26], [-12], [38], [-12], [-18], [-74], [40], [-66], [12], [44], [74], [58], [40], [20], [-84], [-2], [-36], [-34], [56], [-58], [-48], [-30], [-46], [20], [-8], [-22], [-84], [48], [-2], [-78], [20], [38], [-58], [-56], [-58], [18], [12], [84], [32], [30], [2], [-42], [-24], [-36], [-12], [84], [2], [-66], [22], [-72], [-22], [-54], [-78], [48], [26], [20], [-42], [52], [0], [-34], [-84], [22], [-22], [24], [-10], [26], [64], [-76], [34], [-24], [-38], [42], [-48], [20], [68], [42], [42], [-32], [-38], [52], [-36], [80], [-2], [-16], [-82], [60], [96], [-34], [66], [26], [42], [70], [48], [30], [-20], [24], [24], [-58], [4], [48], [62], [98], [-12], [-60], [72], [58], [24], [-6], [28], [-46], [-62], [18], [40], [22], [-46], [-64], [2], [-66], [-100], [12], [-46], [46], [76], [-76], [42], [22], [0], [-32], [30], [40], [-10], [58], [-16], [76], [-10], [10], [-28], [6], [-12], [-88]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8112_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8112_2_a_be();". // 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_8112_2_a_be(:prec:=1) chi := MakeCharacter_8112_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_8112_2_a_be();". // 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_8112_2_a_be( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8112_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-2, 1]>,<7,R![0, 1]>,<11,R![-4, 1]>],Snew); return Vf; end function;