// Make newform 2112.2.a.n in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2112_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_2112_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_2112_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_2112_a();" function MakeCharacter_2112_a() N := 2112; order := 1; char_gens := [2047, 133, 1409, 1729]; 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_2112_a_Hecke(Kf) return MakeCharacter_2112_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], [4], [-2], [-1], [-4], [-2], [0], [-6], [-10], [-8], [2], [2], [-4], [-2], [-4], [0], [8], [12], [2], [-6], [10], [-4], [10], [-2], [-2], [4], [12], [-20], [-6], [-22], [-12], [-2], [0], [10], [2], [-18], [-4], [-12], [6], [0], [-2], [22], [14], [-18], [20], [-12], [-16], [12], [-10], [-6], [-20], [-18], [8], [18], [-16], [20], [22], [-8], [22], [-4], [-14], [-8], [2], [-6], [32], [28], [-22], [12], [20], [-6], [-20], [8], [-4], [20], [-6], [20], [-18], [-18], [-10], [0], [18], [32], [-6], [10], [-24], [30], [-2], [18], [4], [-28], [40], [28], [-12], [20], [4], [20], [-18], [-44], [-12], [32], [-38], [36], [-10], [-32], [-2], [-8], [14], [-30], [42], [-22], [16], [-22], [20], [-48], [-18], [-44], [-22], [-24], [20], [-42], [14], [22], [16], [-52], [18], [-10], [10], [28], [-4], [-20], [44], [-8], [42], [22], [-50], [-4], [52], [-28], [10], [-12], [38], [24], [12], [-30], [30], [-24], [-22], [20], [54], [-28], [-38], [-44], [-12], [12], [2], [-10], [30], [58], [-42], [-28], [-6], [-22], [-32], [38], [-46], [-8], [52], [-10], [56], [-60], [38], [12], [-6], [-40], [-30], [8], [48], [14], [-60], [-32], [-12], [-54], [58], [-46], [0], [32], [-44], [-10], [-48], [-26], [-44], [-52], [-12], [-28], [34], [-38], [46], [18], [-56], [10], [-18], [-18], [10], [40], [-8], [20], [-4], [30], [28], [38], [48], [-16], [52], [60], [-58], [-22], [-18], [-22], [-24], [-32], [-30], [-10], [24], [-68], [20], [-46], [30], [18], [-12], [-14], [20], [22], [-18], [-44], [8], [50], [-14], [0], [-18], [36], [28], [-16], [-10], [14], [20], [28], [-12], [40], [-36], [-8], [2], [-2], [-10], [-54], [60], [-22], [52], [72], [-2], [-26], [-68], [60], [-44], [-42], [60], [0], [42], [-64], [46], [-42], [-68], [-66], [-70], [18], [-36], [12], [-80], [2], [28], [-16], [12], [-12], [28], [-48], [42], [-66], [-58], [60], [30], [-12], [-8], [-26], [28], [36], [-30], [-68], [36], [-20], [52], [54], [42], [30], [36], [-12], [58], [12], [10], [30], [36], [-76], [40], [-38], [-44], [-72], [10], [-20], [72], [-46], [-30], [48], [38], [58], [-36], [54], [62], [-60], [-4], [68], [6], [-72], [12], [70], [-84], [-52], [12], [30], [-66], [-38], [-22], [46], [-82], [10], [-88], [36], [-20], [38], [-28], [-28], [92], [68], [18], [-12], [-6], [-40], [74], [90], [-92], [18], [14], [-48], [22], [18], [60], [12], [-6], [2], [-46], [-18], [-72], [20], [24], [-10], [2], [82], [-60], [-28], [34], [-30], [-2], [-12], [14], [-62], [38], [60], [54], [-48], [42], [-44], [-82], [-10], [-24], [0], [-68], [-18], [-26], [-10], [70], [28], [-42], [-40], [54], [38], [78], [90], [-98], [42], [-38], [-4], [-20], [-46], [2], [-44], [48], [-102], [8], [-60], [28], [-82], [44], [40], [32], [18], [60], [54], [-28], [-4], [90], [28], [-80]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2112_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2112_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_2112_2_a_n(:prec:=1) chi := MakeCharacter_2112_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_2112_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_2112_2_a_n( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2112_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-4, 1]>,<7,R![2, 1]>,<13,R![4, 1]>,<17,R![2, 1]>,<19,R![0, 1]>,<23,R![6, 1]>],Snew); return Vf; end function;