// Make newform 9216.2.a.y in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9216_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_9216_2_a_y();". // 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_9216_2_a_y();". // 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 := [2, -4, -6, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, -4, -1, 1], [3, 2, -1, 0], [-3, 4, 2, -1]]; Rf_basisdens := [1, 1, 1, 1]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; 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_9216_a();" function MakeCharacter_9216_a() N := 9216; order := 1; char_gens := [8191, 2053, 4097]; 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_9216_a_Hecke(Kf) return MakeCharacter_9216_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, 0, 0, 0], [0, 0, 0, 0], [-1, 0, 0, -1], [1, 0, 1, 0], [0, 1, -1, -1], [2, 0, -1, -1], [0, 1, 1, -1], [0, 1, 1, -1], [0, 2, 0, 0], [-3, 2, 0, 1], [3, 0, -1, 0], [4, 2, 1, -1], [0, -3, 1, -1], [0, -1, -1, -3], [0, -2, 0, 0], [-5, -2, 0, -1], [0, 4, 0, 0], [4, 2, -1, 1], [-4, -2, -2, -2], [2, 0, 2, 0], [-2, 4, 2, 0], [3, -4, -1, -4], [0, -5, 1, -3], [2, -6, 2, 2], [0, -6, -2, -2], [-5, 0, 2, 1], [1, 0, 1, 4], [0, 4, 4, 0], [6, 0, 1, 1], [2, 4, 0, 4], [3, -4, -1, 0], [-4, -2, 2, 2], [0, -3, 1, 3], [-4, 6, 2, 2], [-7, 4, 0, 1], [1, -8, 1, 4], [8, -4, 3, 1], [0, -1, 3, 1], [10, 2, -2, 4], [-9, -6, 2, 1], [0, -2, -2, 6], [6, 2, 3, -1], [10, 6, 2, 0], [-4, 0, 6, 4], [-9, -2, 0, 3], [7, 4, -1, -4], [-4, -10, 2, 2], [-3, 8, 1, -4], [0, -5, 1, -3], [-2, 6, -3, 5], [2, -2, -6, -2], [14, -10, 2, 4], [-4, 2, -4, 2], [0, 3, -3, 5], [-10, 2, 2, 2], [16, -4, -4, -4], [5, 8, 6, -1], [13, -8, 1, 4], [2, 2, 5, 1], [-10, 2, -2, 2], [12, 4, 0, 0], [-1, 6, -8, -5], [-4, -2, -2, 6], [4, -12, 0, -4], [-4, -4, -6, 0], [-1, -8, 6, 1], [12, 6, -6, 2], [10, 0, 2, -4], [16, 9, 3, -1], [-8, 2, -7, -5], [2, -6, -2, -6], [6, 4, 6, -8], [7, 12, 3, 0], [4, 4, -5, 5], [0, -3, 1, 7], [10, 10, -6, 0], [11, -10, -2, 1], [-8, 4, -1, -3], [0, -5, -5, 5], [4, -2, 6, 2], [0, -7, 3, 7], [-2, 6, 1, -7], [4, -10, -8, -4], [4, 8, 0, 0], [11, 12, -5, 4], [0, 13, -1, -5], [16, -1, -1, 5], [-16, -2, 2, 2], [5, 10, -2, 3], [-9, -12, 3, 0], [16, -9, 5, 9], [6, -4, 2, 12], [19, -4, 3, 0], [-4, 0, -4, 0], [12, 0, 4, 4], [18, -12, 2, 8], [19, -2, -4, -5], [0, 17, 1, 3], [-8, -15, 1, -1], [-2, 10, 1, 9], [0, -7, -7, 3], [9, -8, 4, -3], [16, -13, -3, -3], [16, -3, 1, 7], [-4, 12, 0, 0], [-2, 16, 2, -4], [-20, 10, 2, -2], [2, -12, 8, 4], [12, 14, 4, 0], [-2, -4, -2, 0], [-1, 8, -5, 0], [0, 4, 3, 1], [6, 4, 8, 4], [12, 0, 8, 0], [15, 16, -1, 0], [-8, -11, 1, 3], [8, -7, 5, 7], [-12, -2, 4, 8], [-3, 10, -6, -1], [-20, 4, 0, -4], [-16, 0, 1, 3], [-12, 12, -4, -4], [9, 20, 4, -3], [16, 7, -3, 1], [-8, -5, 7, -3], [-3, -4, -2, 7], [-18, -6, 7, -5], [10, 8, -10, -4], [-1, -8, 7, -4], [-10, 10, 3, -9], [-4, 4, 0, -8], [22, 2, 6, 0], [-3, 4, 1, -12], [2, -10, 3, 11], [-8, 11, -5, 1], [12, -8, 2, 12], [7, -24, -2, 1], [8, -9, -5, -3], [3, 8, 12, 7], [8, -1, -5, -11], [-8, 3, -9, 1], [-7, -4, 6, -5], [-25, 12, 7, 8], [16, -6, -2, 2], [-18, -16, -7, 1], [14, -4, 2, -12], [16, 12, 7, 5], [8, 7, 3, -7], [-32, 9, -3, -1], [6, -10, -10, -8], [8, -22, 7, 5], [-2, -2, -10, -10], [8, 11, -5, -7], [-8, -4, 8, 0], [-32, 13, 1, 3], [-2, 6, 2, 12], [19, 0, 3, 8], [-8, 1, 1, 3], [10, 0, 0, 16], [-11, -14, 2, 7], [8, -2, 6, -10], [8, 9, 1, 3], [-23, 8, 1, 0], [-8, -11, -1, -5], [16, -3, -3, -9], [12, 0, 8, -4], [5, -4, -7, -12], [-4, 6, 1, 15], [-12, -10, -2, -6], [-5, 2, -2, 9], [-20, -16, 4, -8], [12, -12, -7, -17], [-10, 6, 2, 4], [22, -12, -4, 4], [-3, -8, -7, 4], [-26, -18, 2, -2], [28, 8, -4, -4], [-11, 20, -4, -7], [15, 8, -9, 4], [-16, 6, 1, -13], [-5, -12, -1, -4], [24, 15, 5, -7], [18, -4, -11, -7], [-10, 6, 2, 14], [14, -2, -2, 0], [11, 10, -14, -3], [6, 2, 5, -11], [4, 20, 0, 8], [-4, 34, 0, -6], [-24, -8, -8, 8], [-6, 4, 0, -20], [24, -16, -4, -12], [28, 16, -4, -4], [3, 10, -6, -3], [-4, 2, -2, -2], [-18, 16, 4, 16], [-2, 4, -10, 0], [42, -6, 5, 1], [14, 10, -2, -6], [-8, 4, 0, 16], [13, 16, 12, 1], [1, 4, -3, 8], [2, 4, -7, -3], [10, -4, 6, 8], [-8, -3, 3, 19], [5, -18, 4, -11], [21, -4, 1, 4], [-32, 7, 1, 9], [18, -20, -8, -4], [-8, 1, 9, 11], [-12, -12, 6, -8], [-23, 4, 4, -3], [-25, 8, -1, -12], [-32, 0, -4, -4], [-30, 8, -6, -8], [-16, 8, 4, 8], [-25, -8, -5, 0], [8, 3, -9, 9], [-38, -14, 2, 0], [-35, 4, 4, -7], [42, 0, 5, -7], [3, -12, 3, 8], [8, 35, -1, -3], [11, 8, 7, -20], [-32, -1, -3, 9], [4, 4, -17, -7], [-6, 0, -8, 8], [-2, 4, -2, -8], [-15, -4, 1, 8], [20, -8, 4, 8], [10, -24, 1, -3], [-8, 33, 1, 11], [-17, 20, 3, 8], [-18, -14, -18, -6], [16, 1, -7, 19], [-50, 0, 2, -4], [-8, -18, 10, -6], [-51, 8, 0, 5], [16, -17, -3, 17], [8, -10, 8, 8], [-32, 3, 5, -3], [20, -22, 2, 10], [-13, -4, -5, -4], [8, 4, 9, -13], [-26, 0, -4, -16], [-10, -12, -10, 8], [21, 8, 9, -12], [0, 4, -4, 0], [4, 10, -6, -6], [-20, 4, 4, 16], [10, 8, 3, 11], [2, -8, -12, -16], [-24, -10, 8, -8], [6, 20, -8, -4], [-17, 18, 0, 3], [16, 22, 10, -2], [18, -4, 11, 11], [-28, 0, 0, 16], [-19, 10, -16, -7], [-4, 8, 0, 16], [-15, 8, -11, -8], [-8, 0, -4, 4], [-2, -4, 11, 15], [48, 0, -3, 7], [-8, -3, 1, -1], [-4, 6, -10, 14], [11, -24, -8, -5], [10, 8, 4, -16], [0, 15, 3, 13], [-35, -20, 4, -7], [20, 18, -15, 7], [-28, 12, 4, -4], [26, -12, 0, -4], [-15, -8, 5, 8], [6, -8, -8, 8], [11, -12, -5, 0], [-24, -7, -1, 7], [-40, -12, -1, -3], [14, 8, 2, -20], [-8, 22, 10, 6], [6, 12, 14, 16], [-21, 12, 3, 20], [-16, 22, 8, 8], [16, -24, -1, 5], [40, 7, 3, -3], [-24, -20, 16, -8], [20, 10, -4, 18], [-13, -8, 0, 11], [1, 16, -7, 8], [10, 22, 10, 6], [-11, -10, 2, 15], [8, -10, -18, -2], [-30, -8, 8, 0], [48, -2, -2, 6], [24, -18, -1, -11], [37, 6, -10, 11], [-9, -16, -5, -4], [-21, -8, -10, -11], [8, -21, 5, -3], [-8, -13, 23, 9], [-36, -28, -10, 8], [-25, 2, -2, 5], [37, 16, -15, 4], [44, 24, 0, 0], [-8, 11, 3, 1], [28, 2, 4, 2], [0, -9, -7, -7], [-6, -10, -9, 15], [14, 24, -10, 0], [32, -20, 1, -5], [-2, -26, -6, -4], [39, 20, 6, 1], [-32, 13, -7, -5], [8, -3, -7, -17], [-24, 8, -8, -16], [2, 16, -4, 8], [28, 18, -14, 14], [-10, -36, 2, 4], [18, -12, 14, 0], [26, 2, -6, -14], [4, -4, -24, -8], [12, 4, -10, -8], [9, 30, 0, -3], [11, -16, -9, 16], [16, 31, 3, 1], [20, 22, -2, -14], [8, 41, -3, 3], [8, -5, 11, -7], [-44, -4, -8, 12], [-15, 14, 14, -1], [26, 12, 7, 7], [9, -36, -8, -7], [-23, 4, 5, 4], [16, 4, -4, 8], [4, -22, 6, -10], [16, 15, 9, -15], [-48, 8, -3, -9], [50, -20, -4, -12], [10, 0, -14, 20], [7, -24, -13, -12], [20, 8, 1, 15], [0, -1, -1, 25], [13, -2, 4, 21], [-23, -4, 1, 0], [-13, -2, 0, -17], [8, 47, 1, 1], [-22, 4, -3, -7], [20, -2, 14, -10], [26, 20, 2, -8], [5, 14, 0, 1], [-64, -5, 7, 1], [12, -10, 6, 2], [-7, 18, -4, -15], [21, 12, 1, 4], [12, -36, -3, -5], [-18, 22, -10, -18], [-14, -6, 2, -16], [-28, -22, 2, -2], [-24, 9, 13, 15], [-16, 24, 4, -4], [8, -22, -15, -5], [34, 30, -18, 6], [2, -18, 6, 20], [-16, -17, 1, 9], [12, 16, 16, -8], [-14, -16, -2, -4], [7, -8, 6, 1], [13, 16, -3, 8], [0, -8, -12, 0], [40, 11, 13, 5], [0, -11, -7, -1], [-20, 20, -16, -4], [35, -2, 4, 11], [-17, -8, 7, -4], [6, 6, 7, -13], [-8, -5, 29, 13], [18, 34, -6, 0], [2, 20, 0, -4], [8, -3, -23, -17], [-4, -26, 12, 6], [27, -10, -6, -3], [-40, 11, -13, -15], [17, 40, -7, 12], [14, -36, -4, -4], [32, -11, 1, -13], [14, 2, -22, -4], [29, -28, -7, -12], [-18, 10, 5, -11], [-20, -18, -14, -6], [-28, 14, 4, 8], [-6, -4, 8, 4], [-9, -10, -2, 5], [4, 2, -10, -18], [8, -3, -3, -13], [-18, 6, 26, 12], [-48, -10, -6, -6], [41, -16, -3, -8], [-32, -25, -1, 1], [-48, 5, 5, 3], [25, -14, -2, 7], [-48, -8, 1, 11], [-8, -15, -19, 3], [-45, 4, -1, 12], [16, 29, 9, -9], [53, 8, -10, -1], [37, 24, 13, -12], [-10, 0, 11, 7], [8, 11, -5, -11], [32, -19, 13, 19], [-40, 7, 1, -7], [-14, 40, -2, 12], [-25, -4, -16, 15], [-24, -13, 21, -3], [0, 15, -17, 5], [-22, -12, -18, -8], [23, -28, -10, -15], [-10, 10, 10, -4], [5, 16, 5, 16], [32, 21, 5, -1], [-10, -8, -18, -8], [43, -8, 20, 15], [-20, 8, -7, -17], [-16, -8, -12, -20], [32, -13, -7, -11], [4, -12, 2, 16], [51, -4, 10, 1], [-16, 5, 7, 19], [-6, 10, -6, -14], [8, -31, 17, 11], [8, 34, -8, 8], [36, -18, -8, -2], [12, -8, -8, 8], [-16, 3, 7, 5], [2, -16, -2, -4], [-20, -6, 3, -3], [24, -21, 3, -11], [18, 8, -6, -4], [-4, -14, -3, 19], [32, 17, 5, 11]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9216_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9216_2_a_y();". // 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_9216_2_a_y(:prec:=4) chi := MakeCharacter_9216_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(3067) - 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_9216_2_a_y();". // 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_9216_2_a_y( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9216_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![8, -16, -4, 4, 1]>,<7,R![4, 24, -8, -4, 1]>,<11,R![32, -32, -24, 0, 1]>,<13,R![4, 48, 4, -8, 1]>,<17,R![16, 64, -32, 0, 1]>,<19,R![16, 64, -32, 0, 1]>,<67,R![256, -256, 0, 16, 1]>],Snew); return Vf; end function;