// Make newform 9216.2.a.bb 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_bb();". // 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_bb();". // 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 := [1, 0, -4, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [-2, 0, 1, 0], [0, -3, 0, 1], [0, 5, 0, -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], [0, 0, 0, 1], [0, 0, -1, -1], [-3, -1, 0, 0], [0, 0, -2, 1], [0, 2, 0, 0], [3, 1, 0, 0], [0, 0, 3, -1], [0, 0, 0, -1], [0, 0, 4, 0], [0, 0, 2, 1], [0, -4, 0, 0], [3, -3, 0, 0], [0, 0, 0, -4], [0, 0, 6, 1], [-9, 1, 0, 0], [0, 0, -2, -1], [1, 3, 0, 0], [0, 0, -3, 1], [-6, 2, 0, 0], [0, 0, 4, -4], [-3, 3, 0, 0], [6, -2, 0, 0], [0, 2, 0, 0], [0, 0, -6, 3], [0, 0, -5, -1], [-3, -9, 0, 0], [0, 0, 8, 3], [6, 4, 0, 0], [0, 0, -4, 0], [-9, 5, 0, 0], [-12, 0, 0, 0], [-5, -3, 0, 0], [0, 0, -12, -1], [0, 0, 5, 1], [0, 0, 2, 3], [-15, -1, 0, 0], [0, 0, -15, 1], [0, 0, -6, -1], [-3, 3, 0, 0], [0, 0, -4, -7], [0, 0, 12, 0], [-8, -6, 0, 0], [0, 0, 0, 1], [0, 0, -1, 7], [-5, -3, 0, 0], [0, 0, 4, 0], [-9, -7, 0, 0], [0, 0, 4, 3], [6, -2, 0, 0], [0, 0, 12, 4], [-8, 6, 0, 0], [-9, -7, 0, 0], [-6, -4, 0, 0], [0, 0, -3, -7], [0, 0, 0, -3], [0, 0, -8, 4], [0, 0, -2, 3], [-6, -6, 0, 0], [-7, 3, 0, 0], [0, 0, 0, -7], [-5, -15, 0, 0], [0, 0, -9, -1], [-8, 12, 0, 0], [0, 0, -12, 3], [11, 9, 0, 0], [6, -16, 0, 0], [-9, 1, 0, 0], [0, 0, 8, -7], [-6, 4, 0, 0], [0, 0, -3, 9], [0, 0, 16, 4], [0, 0, 2, 7], [-15, 7, 0, 0], [0, 0, 0, -8], [0, 0, 6, -1], [0, 0, -4, -9], [24, -6, 0, 0], [24, -4, 0, 0], [-9, -3, 0, 0], [0, 0, 4, -13], [0, 0, 12, 4], [24, 6, 0, 0], [0, 0, -7, 9], [-9, -3, 0, 0], [0, 14, 0, 0], [0, -12, 0, 0], [0, 0, -12, -7], [0, 0, 20, 4], [-27, -1, 0, 0], [0, 0, -12, -8], [0, 0, 7, 7], [-27, 3, 0, 0], [11, -3, 0, 0], [0, 0, 3, 15], [0, 0, -6, 9], [-12, 0, 0, 0], [-21, 1, 0, 0], [0, 0, -2, 7], [9, -5, 0, 0], [0, 0, 6, 3], [-27, 7, 0, 0], [-12, 0, 0, 0], [25, 3, 0, 0], [-30, -8, 0, 0], [-3, 7, 0, 0], [-6, -8, 0, 0], [0, 0, -9, 7], [-30, -6, 0, 0], [0, 0, 8, -8], [0, 0, -4, 11], [-6, 10, 0, 0], [11, 9, 0, 0], [0, 0, 13, -7], [-24, 6, 0, 0], [-15, 3, 0, 0], [0, 0, -3, 1], [0, 0, -24, 5], [-3, -13, 0, 0], [0, 0, -20, 1], [24, 10, 0, 0], [0, 0, 30, -1], [21, -5, 0, 0], [3, 21, 0, 0], [0, 0, -18, 5], [0, 0, -10, -11], [0, 0, -12, 12], [0, 0, 1, 9], [0, 0, 20, -3], [-23, 15, 0, 0], [0, 0, 9, 1], [0, 0, -4, 4], [0, 0, 20, 1], [0, -20, 0, 0], [8, -6, 0, 0], [0, 0, -6, -5], [27, 5, 0, 0], [0, 0, 18, 9], [-24, 4, 0, 0], [-21, -15, 0, 0], [0, 0, 6, -15], [0, 0, 1, -7], [15, 5, 0, 0], [0, 0, -10, 7], [0, 0, -15, -7], [0, 0, -22, 7], [12, 8, 0, 0], [9, -17, 0, 0], [0, 0, -12, 8], [0, 0, 4, 7], [18, 8, 0, 0], [-45, -7, 0, 0], [0, 0, -9, -1], [3, -19, 0, 0], [0, 0, 24, -4], [0, 0, 13, -15], [0, -2, 0, 0], [-22, 18, 0, 0], [0, 0, 24, -3], [21, -1, 0, 0], [12, 24, 0, 0], [0, 0, 7, 7], [-3, -21, 0, 0], [-48, 2, 0, 0], [0, 0, 39, -1], [0, 0, -16, -16], [0, 0, -10, -3], [24, -10, 0, 0], [0, 0, 24, 3], [-9, 29, 0, 0], [0, 0, -20, -3], [0, 0, -3, 1], [-2, -18, 0, 0], [0, 0, 8, -4], [18, 18, 0, 0], [-31, 15, 0, 0], [0, 0, -6, 3], [0, 0, 23, -1], [0, 0, -22, -3], [0, 0, -8, 8], [-33, 1, 0, 0], [0, 0, 8, -9], [42, 10, 0, 0], [0, 0, 0, -20], [0, 0, -12, -1], [0, 0, -14, -13], [17, -9, 0, 0], [8, 12, 0, 0], [0, 0, 0, 8], [22, 12, 0, 0], [21, 11, 0, 0], [11, 9, 0, 0], [0, 0, 0, -1], [15, 17, 0, 0], [18, -14, 0, 0], [6, 8, 0, 0], [0, 0, 10, -13], [6, -24, 0, 0], [0, 0, 21, 1], [0, 0, -18, 3], [0, 0, -20, 4], [0, 0, 2, -1], [6, 4, 0, 0], [-3, -1, 0, 0], [0, 0, -36, -5], [0, 0, -8, -8], [-9, -31, 0, 0], [-18, 6, 0, 0], [51, -7, 0, 0], [-16, 30, 0, 0], [0, 0, -12, -17], [0, 0, -7, 17], [15, -19, 0, 0], [0, 0, 9, 9], [36, 8, 0, 0], [0, 0, 4, -12], [0, 18, 0, 0], [0, 0, 15, 7], [0, 0, 12, -13], [0, 0, -4, 11], [0, 0, -19, 9], [0, -34, 0, 0], [0, 0, -4, 12], [-3, -9, 0, 0], [0, 0, 2, -1], [6, 14, 0, 0], [0, 0, 0, 16], [0, 0, -29, -1], [-3, -13, 0, 0], [0, 0, 8, -13], [3, -15, 0, 0], [0, 0, 8, 0], [-30, 2, 0, 0], [3, 13, 0, 0], [0, 0, -12, 4], [24, 22, 0, 0], [0, 0, -24, 11], [-9, 5, 0, 0], [0, 0, 9, 9], [21, 19, 0, 0], [65, 3, 0, 0], [0, 0, -29, -1], [0, 0, -2, 1], [-6, -16, 0, 0], [0, 0, 15, 15], [0, 0, -4, 16], [-33, 5, 0, 0], [11, -3, 0, 0], [0, 0, -12, 4], [0, 0, -32, 9], [-6, 36, 0, 0], [0, 0, -3, 9], [2, -6, 0, 0], [0, 0, 0, 13], [-27, 3, 0, 0], [0, 0, -8, -11], [49, 15, 0, 0], [0, 0, 18, 11], [0, -12, 0, 0], [0, 0, -32, 8], [63, -7, 0, 0], [0, 0, -14, -7], [0, 0, 16, 17], [-24, 6, 0, 0], [41, 3, 0, 0], [0, 0, 24, -19], [6, -34, 0, 0], [9, -9, 0, 0], [0, 0, 30, -9], [0, 0, -10, -23], [-5, 9, 0, 0], [-2, 30, 0, 0], [0, 0, 8, 16], [10, -12, 0, 0], [0, 0, 5, 1], [15, -3, 0, 0], [0, 0, 2, 27], [-30, -6, 0, 0], [-3, -25, 0, 0], [0, 0, -36, 0], [0, 0, -13, 15], [0, 0, -45, -9], [0, 0, -28, -5], [27, 13, 0, 0], [0, 0, -24, -4], [-56, 6, 0, 0], [0, 0, -42, 9], [0, 0, 5, 17], [-42, 0, 0, 0], [0, 0, 0, 5], [21, -25, 0, 0], [6, -18, 0, 0], [-3, 31, 0, 0], [0, 0, -20, -9], [0, 0, 6, 5], [0, 0, -4, -8], [0, 0, 36, 7], [15, -19, 0, 0], [9, -13, 0, 0], [44, 0, 0, 0], [0, 0, 30, 3], [0, 0, -40, -12], [45, 19, 0, 0], [57, -13, 0, 0], [-32, 18, 0, 0], [21, -9, 0, 0], [0, 0, -10, 17], [0, 0, 39, -1], [0, 0, -26, 5], [0, 0, 0, 12], [0, 0, -24, -5], [72, 6, 0, 0], [-15, -13, 0, 0], [0, 0, 9, -15], [10, 18, 0, 0], [-27, 11, 0, 0], [0, 0, 24, -8], [-6, 24, 0, 0], [18, -16, 0, 0], [35, -27, 0, 0], [-16, -12, 0, 0], [0, 0, -12, 11], [0, 0, -32, 0], [-12, -16, 0, 0], [0, -2, 0, 0], [-15, 39, 0, 0], [33, 3, 0, 0], [0, 0, 12, 8], [0, 0, -12, -27], [0, 0, 34, -11], [0, 0, -24, -1], [0, 0, 16, -8], [15, -3, 0, 0], [-77, -3, 0, 0], [63, 9, 0, 0], [0, 0, 16, 1], [-6, 36, 0, 0], [6, 6, 0, 0], [0, 0, -40, 12], [0, 0, 16, 5], [24, 4, 0, 0], [0, 0, 18, 19], [0, 0, 43, 7], [0, 0, 30, -3], [15, -23, 0, 0], [0, 0, 20, 3], [11, -39, 0, 0], [0, 0, -12, 20], [0, 0, 24, 3], [9, 23, 0, 0], [36, -8, 0, 0], [0, 0, -18, -13], [0, 0, 4, 12], [0, 0, 16, -19], [-54, -6, 0, 0], [0, 0, -24, -24], [-51, -21, 0, 0], [-24, 10, 0, 0], [0, 0, -9, 15], [0, 0, 26, 17], [-18, 38, 0, 0], [0, 0, 0, -12], [39, -7, 0, 0], [25, -9, 0, 0], [-18, 14, 0, 0], [0, 0, 6, -29], [0, 0, 11, -25], [36, 8, 0, 0], [-9, 21, 0, 0], [-21, -7, 0, 0], [0, 0, -24, -4], [0, 0, 18, 5], [0, 0, -35, 1], [0, 0, 8, 17], [-27, -21, 0, 0], [0, 0, -12, -32], [22, -12, 0, 0], [24, -6, 0, 0], [56, 12, 0, 0], [0, 0, -36, -5], [72, -12, 0, 0], [0, 0, 5, 1], [-54, -20, 0, 0], [9, -25, 0, 0], [0, 0, 21, 1], [0, 0, 8, -36], [0, 0, -14, 7], [-23, 15, 0, 0], [0, 0, 24, 24], [-14, -24, 0, 0], [0, 0, 6, -33], [21, -33, 0, 0], [-45, -11, 0, 0], [0, 0, 15, 7], [0, 4, 0, 0], [0, 0, -32, 16], [12, 40, 0, 0], [27, -27, 0, 0], [0, 0, -6, -35], [0, 0, -68, -3], [-48, -2, 0, 0], [0, 0, 44, 12], [-12, 16, 0, 0], [0, 0, -12, -11], [0, 0, 11, -9], [0, 0, 46, -7], [-48, 2, 0, 0], [27, 29, 0, 0], [-9, 53, 0, 0], [42, -12, 0, 0], [0, 0, 30, -15], [-33, 1, 0, 0], [9, 47, 0, 0], [30, 14, 0, 0], [0, 0, 18, -9], [0, 0, -24, 0], [0, 0, -37, 7], [24, 10, 0, 0], [0, 0, -9, 23], [0, 0, -6, -15], [0, 0, -38, 9], [0, 0, -24, -4], [-33, -15, 0, 0], [-32, -12, 0, 0], [0, 0, 6, 27], [-27, 31, 0, 0], [30, -26, 0, 0], [-39, 27, 0, 0], [0, 0, 39, 15], [-20, 0, 0, 0], [15, 1, 0, 0], [3, -7, 0, 0], [0, 0, 36, -20], [0, 0, 10, 11], [0, -50, 0, 0], [-18, -2, 0, 0], [0, 0, 20, -15], [81, -5, 0, 0]]; 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_bb();". // 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_bb(: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_bb();". // 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_bb( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9216_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-6, 0, 1]>,<7,R![16, 0, -16, 0, 1]>,<11,R![6, 6, 1]>,<13,R![4, 0, -28, 0, 1]>,<17,R![-12, 0, 1]>,<19,R![6, -6, 1]>,<67,R![-26, -2, 1]>],Snew); return Vf; end function;