// Make newform 2112.2.a.ba in Magma, downloaded from the LMFDB on 29 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_ba();". // 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_ba();". // 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], [2], [2], [1], [2], [0], [-2], [0], [8], [4], [6], [-4], [-6], [-8], [6], [4], [6], [-4], [-8], [-10], [-14], [8], [2], [14], [0], [8], [8], [2], [14], [-14], [-20], [-2], [-10], [12], [10], [2], [-16], [-12], [-8], [20], [10], [24], [18], [8], [-12], [-18], [-16], [0], [6], [-12], [12], [14], [28], [18], [4], [-26], [-10], [-22], [-12], [-22], [4], [-34], [8], [-2], [-6], [16], [-18], [8], [-26], [-6], [-8], [8], [22], [32], [24], [-14], [-34], [-18], [-18], [12], [26], [20], [-34], [-2], [-4], [-30], [-18], [-12], [-40], [28], [4], [4], [-40], [-28], [36], [22], [-10], [-6], [10], [26], [-20], [-36], [-32], [-22], [34], [-44], [12], [-24], [-22], [10], [-2], [-2], [40], [-20], [34], [16], [32], [6], [-36], [-38], [10], [0], [-12], [40], [44], [38], [24], [-28], [18], [-14], [-40], [-20], [10], [-24], [10], [-10], [-22], [-22], [-24], [38], [0], [48], [8], [38], [0], [14], [16], [-40], [24], [-42], [6], [4], [0], [48], [48], [-10], [-10], [26], [-48], [-12], [48], [58], [4], [38], [-16], [4], [-26], [38], [14], [4], [46], [24], [46], [32], [-10], [10], [-18], [10], [14], [12], [36], [58], [64], [-56], [-58], [2], [-24], [-26], [52], [10], [-16], [-12], [18], [4], [-14], [58], [50], [32], [-24], [60], [-30], [-6], [-58], [12], [42], [-24], [20], [-24], [20], [-30], [30], [-36], [28], [-32], [62], [38], [-12], [48], [6], [58], [-6], [-66], [-32], [68], [-50], [6], [0], [38], [0], [-10], [14], [2], [72], [52], [-40], [-54], [36], [12], [-16], [36], [10], [68], [18], [-44], [0], [52], [-60], [-54], [-24], [10], [44], [-72], [-74], [20], [0], [46], [54], [-22], [2], [22], [-28], [-70], [78], [-54], [-44], [-66], [-74], [-34], [-16], [-2], [0], [50], [-50], [-50], [28], [-4], [78], [-14], [20], [-68], [76], [-72], [-2], [10], [8], [-18], [-48], [20], [-28], [2], [52], [-52], [4], [-66], [-60], [-20], [-2], [88], [-10], [66], [28], [74], [-36], [-20], [-50], [28], [18], [-80], [54], [84], [-22], [-12], [68], [-68], [-34], [44], [-48], [26], [8], [62], [26], [36], [-4], [44], [82], [12], [-20], [36], [32], [-62], [50], [-14], [72], [-30], [68], [-2], [36], [-38], [-62], [22], [66], [-20], [-16], [54], [32], [10], [32], [64], [-42], [22], [90], [-10], [-54], [-50], [-48], [40], [-32], [24], [40], [-42], [28], [24], [0], [-92], [-70], [-12], [34], [-6], [-52], [-16], [72], [0], [-82], [34], [-12], [100], [-14], [-48], [-18], [-22], [-78], [-46], [48], [-78], [24], [-44], [62], [82], [-24], [-34], [-42], [60], [-28], [64], [14], [-6], [-78], [40], [-24], [6], [102], [42], [42], [76], [-30], [54], [20], [4], [-20], [-62], [84], [4], [50], [-26], [10], [60], [64], [18], [76], [6], [38], [0], [-40], [58], [46], [36], [20], [-8], [-36]]; 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_ba();". // 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_ba(: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_ba();". // 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_ba( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2112_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-2, 1]>,<7,R![-2, 1]>,<13,R![-2, 1]>,<17,R![0, 1]>,<19,R![2, 1]>,<23,R![0, 1]>],Snew); return Vf; end function;