// Make newform 2112.2.b.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2112_b();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2112_b_Hecke();" // 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_b_e();". // 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_b_e();". // 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 poly := [3, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_2112_b();" function MakeCharacter_2112_b() N := 2112; order := 2; char_gens := [2047, 133, 1409, 1729]; v := [2, 2, 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; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2112_b_Hecke();" function MakeCharacter_2112_b_Hecke(Kf) N := 2112; order := 2; char_gens := [2047, 133, 1409, 1729]; char_values := [[1, 0], [1, 0], [-1, 0], [-1, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; 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, 0], [-1, 1], [1, -2], [0, 0], [-1, 2], [0, 0], [0, 0], [0, 0], [-1, 2], [0, 0], [5, 0], [7, 0], [0, 0], [0, 0], [2, -4], [4, -8], [1, -2], [0, 0], [13, 0], [5, -10], [0, 0], [0, 0], [0, 0], [5, -10], [17, 0], [0, 0], [-4, 0], [0, 0], [0, 0], [-1, 2], [0, 0], [0, 0], [-7, 14], [0, 0], [0, 0], [0, 0], [-23, 0], [16, 0], [0, 0], [0, 0], [-5, 10], [25, 0], [-7, 14], [0, 0], [0, 0], [20, 0], [0, 0], [-1, 0], [0, 0], [-5, 0], [0, 0], [0, 0], [0, 0], [-5, 10], [8, -16], [0, 0], [4, -8], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-10, 20], [-19, 0], [7, -14], [-35, 0], [0, 0], [0, 0], [0, 0], [11, -22], [0, 0], [-37, 0], [0, 0], [25, 0], [-1, 2], [-11, 22], [-2, 0], [8, -16], [0, 0], [10, -20], [10, 0], [0, 0], [29, 0], [0, 0], [-11, 22], [5, -10], [0, 0], [0, 0], [-31, 0], [13, -26], [0, 0], [-43, 0], [0, 0], [40, 0], [0, 0], [1, -2], [-13, 26], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [47, 0], [-2, 4], [0, 0], [-10, 20], [0, 0], [0, 0], [0, 0], [8, -16], [1, 0], [-7, 0], [-7, 14], [-41, 0], [-13, 26], [1, -2], [0, 0], [13, 0], [0, 0], [0, 0], [-14, 28], [-17, 0], [0, 0], [19, 0], [5, -10], [53, 0], [0, 0], [0, 0], [0, 0], [23, 0], [-38, 0], [0, 0], [0, 0], [4, -8], [0, 0], [-17, 34], [0, 0], [0, 0], [0, 0], [-49, 0], [0, 0], [-29, 0], [11, -22], [0, 0], [0, 0], [31, 0], [14, -28], [0, 0], [5, -10], [-56, 0], [0, 0], [-8, 0], [2, -4], [0, 0], [-16, 32], [0, 0], [0, 0], [7, -14], [0, 0], [0, 0], [13, -26], [17, -34], [11, -22], [20, 0], [0, 0], [0, 0], [19, -38], [0, 0], [-50, 0], [0, 0], [0, 0], [41, 0], [-19, 38], [0, 0], [-5, 10], [0, 0], [0, 0], [-28, 0], [0, 0], [-59, 0], [0, 0], [-19, 38], [-20, 40], [0, 0], [64, 0], [0, 0], [0, 0], [-61, 0], [0, 0], [-47, 0], [-20, 40], [0, 0], [-1, 2], [0, 0], [1, 0], [0, 0], [0, 0], [0, 0], [0, 0], [58, 0], [0, 0], [-5, 10], [13, -26], [65, 0], [0, 0], [0, 0], [40, 0], [0, 0], [-17, 34], [-19, 0], [-2, 4], [0, 0], [53, 0], [0, 0], [0, 0], [-22, 44], [19, -38], [0, 0], [0, 0], [-1, 2], [29, 0], [0, 0], [0, 0], [-16, 32], [14, -28], [0, 0], [0, 0], [31, 0], [0, 0], [0, 0], [0, 0], [-71, 0], [0, 0], [59, 0], [0, 0], [19, -38], [23, -46], [-11, 22], [0, 0], [-76, 0], [61, 0], [0, 0], [0, 0], [-52, 0], [10, -20], [0, 0], [0, 0], [0, 0], [0, 0], [17, -34], [-10, 0], [0, 0], [0, 0], [70, 0], [0, 0], [-23, 46], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [8, -16], [-80, 0], [-5, 10], [-25, 50], [0, 0], [0, 0], [67, 0], [-83, 0], [26, 0], [0, 0], [0, 0], [-79, 0], [-17, 34], [0, 0], [0, 0], [0, 0], [0, 0], [-85, 0], [0, 0], [0, 0], [0, 0], [-13, 26], [-34, 0], [0, 0], [71, 0], [0, 0], [25, -50], [22, -44], [0, 0], [0, 0], [0, 0], [0, 0], [-73, 0], [-11, 22], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2112_b_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2112_2_b_e();". // 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_b_e(:prec:=2) chi := MakeCharacter_2112_b(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(1999) - 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_b_e();". // 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_b_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2112_b(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![11, 0, 1]>,<7,R![0, 1]>,<17,R![0, 1]>,<29,R![0, 1]>,<31,R![-5, 1]>,<83,R![0, 1]>],Snew); return Vf; end function;