// Make newform 4598.2.a.bq in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4598_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_4598_2_a_bq();". // 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_4598_2_a_bq();". // 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, 2, -4, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-2, -2, 1, 0], [1, -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_4598_a();" function MakeCharacter_4598_a() N := 4598; order := 1; char_gens := [3269, 3631]; v := [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_4598_a_Hecke(Kf) return MakeCharacter_4598_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 := [[-1, 0, 0, 0], [-1, 0, 0, -1], [-1, 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0], [0, 2, 0, 0], [1, 0, -1, 0], [-1, 0, 0, 0], [-1, -3, 1, -1], [1, -4, 0, -1], [2, -1, 1, 3], [-1, -3, -1, 0], [0, 1, 3, 0], [-4, -1, -1, -1], [0, 1, 2, 1], [0, -2, -1, 1], [5, 4, 0, 3], [1, -5, 2, -4], [-6, -2, -2, -4], [-2, 0, -4, 1], [-1, 0, 3, -2], [2, -2, -2, 5], [7, 3, -2, 0], [2, -3, -3, 2], [-6, 7, 1, -1], [1, 2, -1, 5], [-4, 0, -4, 1], [11, 5, -1, 4], [-6, 1, 2, 0], [6, -1, 1, 4], [8, -2, 2, 2], [-6, 1, 3, -3], [-3, 2, 5, 2], [4, -2, -2, 4], [6, -8, -2, -4], [8, -3, -1, 4], [-5, -2, -3, -5], [-14, 1, 3, -3], [2, 9, 1, 9], [5, -2, 2, -3], [-8, 5, 2, -6], [-2, 2, 2, 2], [-2, 7, 2, -1], [0, -5, -7, -1], [-4, -1, -7, -3], [-5, 4, 1, 8], [-4, 10, 3, 3], [-4, -3, 7, -8], [-10, -2, 2, -6], [-2, 7, -5, 9], [-15, 7, 1, 1], [0, 2, 5, 0], [-4, 0, 0, 5], [-8, -4, -2, -4], [10, 0, -2, -2], [11, 0, 1, 10], [-8, 8, -1, 1], [9, -4, -3, 8], [-1, 7, 4, 2], [10, -5, 1, -4], [-4, 2, -6, -4], [0, -3, 0, 2], [-3, 9, -1, -2], [13, -4, 3, -10], [-4, 6, 1, 6], [-8, 0, 8, -8], [-20, -3, -2, 2], [-6, 2, -2, -1], [-7, -2, 1, -7], [-10, -3, -1, -7], [0, -8, -3, 0], [5, 2, 7, -2], [-10, 3, -4, 1], [-13, 0, 0, -5], [0, 7, 4, 4], [10, -14, 0, -6], [-3, -2, -1, 1], [2, -8, -4, -2], [-18, 2, -4, -1], [16, 6, 10, 8], [1, -11, 0, -2], [-4, 6, 1, -11], [10, -4, 4, 7], [-8, 5, 7, -7], [-12, -6, -2, -4], [-18, 1, -5, -3], [18, -14, -10, 0], [-8, 1, 4, 1], [-11, -2, -5, -3], [-3, 10, -5, 14], [2, 9, -1, 7], [8, -12, -7, -2], [-24, -1, 1, -7], [2, 3, -1, -13], [-3, 6, -7, 7], [7, -2, -10, 0], [-15, -5, -5, -12], [-12, 7, 1, -3], [-7, 15, -1, 12], [-6, 2, -6, 0], [12, -10, 5, -17], [-6, 4, 0, 0], [4, -2, -3, 5], [-30, 6, 0, -4], [-9, 1, -2, 10], [-7, 0, 1, -12], [-25, 2, 1, 5], [13, -1, -3, -7], [10, 2, -12, 6], [4, -2, 4, 16], [2, -1, -5, -8], [5, -12, 7, -3], [-12, 0, -9, -2], [26, -8, 4, 10], [-7, -7, 11, -11], [8, 0, 4, -3], [4, -8, -2, -18], [7, 5, 13, 7], [-2, -1, 9, 3], [-1, 11, 6, 7], [-1, -5, 7, -4], [10, -18, 4, -7], [-19, 12, 2, -7], [24, 4, 4, 4], [5, 14, -5, 13], [6, -2, 12, -2], [14, 6, 0, 12], [2, 4, -4, 20], [-14, -5, -8, -17], [-22, 1, -3, 11], [3, -12, -1, -13], [22, 3, -5, 9], [-4, -15, 1, -15], [-37, 0, -1, 7], [16, -9, -2, 1], [2, -7, 0, -11], [-2, 4, 13, -9], [1, 21, -1, 12], [-6, 8, 4, 8], [-1, 7, -11, 13], [27, -3, 3, -12], [-29, 11, 0, 2], [-25, -3, 1, 5], [-1, -1, -10, -1], [-4, 8, 0, 12], [-14, 5, 13, -4], [-28, 9, -7, -13], [26, -1, 3, 11], [17, 4, 5, 5], [8, -9, -9, -9], [-15, -13, -10, -17], [9, 0, -2, -2], [1, 7, -8, -2], [4, -8, 8, -11], [9, 2, -10, -7], [-4, 7, -5, -8], [-8, 12, -6, -4], [-1, 10, 18, 4], [21, -15, -9, 3], [-12, 4, 4, 12], [8, 13, 3, -5], [-30, -1, -3, -4], [-33, -4, -4, -16], [14, -1, -6, 6], [-6, 0, 0, -8], [8, 9, -13, 12], [-20, -10, 0, -7], [5, 9, 12, 10], [24, -14, -2, -13], [10, -8, 0, -4], [-12, 8, 2, -12], [-14, 16, 11, 1], [-5, -16, 3, -18], [1, -17, -7, -17], [42, -27, -11, -3], [12, -12, -3, -14], [31, -13, 0, 10], [26, -9, -15, 3], [12, 8, 2, 21], [-15, 9, 4, 0], [-23, 5, -7, 5], [17, -1, 9, -24], [12, -22, -5, -13], [-24, 3, -3, 10], [-15, -5, 5, 3], [25, -19, 2, 8], [2, 1, -14, 6], [8, 11, 4, 18], [2, 5, -17, 6], [-24, 8, -5, -4], [-20, -15, 13, -8], [3, -5, -2, 10], [2, -22, 6, -12], [16, -10, -2, -18], [-9, 5, 2, -18], [2, -9, 13, -3], [1, -12, 5, -12], [-13, 24, 7, 9], [-50, 1, 4, -3], [-12, -6, 12, -6], [22, 23, -10, 12], [-16, 4, 4, -10], [-6, -16, -4, -10], [18, 16, 6, 20], [-6, -17, 7, -9], [40, -22, -12, -2], [4, 5, 6, 11], [4, -10, 1, 1], [-4, -5, 10, -21], [-7, -14, 4, -21], [20, -9, 0, -7], [18, 11, 9, 9], [22, -3, 12, 13], [-14, 9, -8, -22], [-2, 18, -4, -4], [-26, -12, -10, -22], [32, -6, -3, 10], [4, 17, -11, 9], [-12, 24, 2, 11], [-21, -19, -12, -24], [13, 2, 8, 7], [4, -4, 20, -16], [10, 15, -13, 18], [-27, 14, 5, -2], [16, -24, -3, -25], [-19, -21, -2, -26], [8, 9, -15, 24], [-12, -5, 7, -23], [-14, 17, 15, -11], [23, 14, -11, 13], [-2, 15, -1, 19], [38, 2, 7, -13], [10, -5, 15, -28], [-12, 2, -2, 14], [-36, -9, 7, -3], [8, 6, -5, 10], [-19, -14, 9, -10], [-59, -2, -7, -3], [-33, 2, -3, -1], [18, -15, 9, -22], [-6, -5, 4, 0], [13, 24, -3, 9], [-15, -1, -5, -13], [-4, 16, 17, 7], [4, 7, 17, 10], [-15, 32, -2, 28], [-12, -1, 12, -11], [4, -10, 14, -4], [8, 12, 19, 13], [11, 10, 0, 22], [20, -17, -7, 11], [-18, 3, -8, -15], [5, -4, 11, -18], [-14, 9, -9, -14], [2, 2, -22, 18], [-13, -4, -5, -23], [-27, -11, -20, -20], [-3, -19, -17, -10], [-9, 9, -14, -9], [-25, -15, 5, -17], [-10, 33, 1, 25], [-2, -20, -5, -13], [14, 5, 1, 25], [19, 7, 8, 7], [-17, -8, 3, -2], [-2, -4, 14, -2], [-15, -11, -9, -24], [4, 22, -12, 26], [20, 22, -11, 1], [-29, -9, 2, 12], [36, -24, -5, 11], [-25, 0, -10, 17], [10, 2, 17, 4], [11, -6, 5, -14], [-24, 9, 3, 12], [-32, 27, 6, 11], [-48, 15, -1, 19], [-2, 9, -8, 12], [16, -3, 1, -20], [21, -9, 4, 0], [-10, -3, -13, 29], [38, -15, 0, 1], [13, 6, 15, -18], [-2, -1, 2, 10], [20, 14, -6, 6], [-3, 9, -9, 27], [4, 16, 2, 8], [0, -21, 4, -38], [1, -5, -21, 11], [-12, -7, 17, 2], [-45, 19, 6, 4], [-17, 20, -9, 15], [20, 2, 0, 19], [16, 12, -13, 7], [-47, -2, 4, -15], [-19, 7, 12, 12], [8, 23, 9, 28], [-26, 10, -10, 14], [-13, 9, -3, -10], [4, -10, -28, 16], [34, 15, -10, 21], [-9, 9, 10, -3], [-73, 6, 4, 0], [-20, 8, -6, 8], [-23, -15, 6, 0], [12, -8, -4, -7], [7, -6, 2, 11], [12, 6, -5, 15], [-3, -32, 3, -20], [-63, 20, 3, 11], [-27, -22, 2, -8], [-31, 32, 7, -3], [24, -9, -5, 20], [-12, -7, -16, 22], [30, 12, -17, 16], [16, -21, -15, -19], [15, -1, 2, -2], [-6, 13, -17, 19], [8, -24, 7, -4], [-19, 14, -1, -18], [21, -19, -11, 2], [5, -6, 0, -6], [7, -5, 0, -17], [8, 0, -6, -10], [-63, 39, 13, 1], [14, 39, -7, 40], [-32, 23, 8, 22], [-20, 7, -2, 0], [-18, -7, 7, -13], [-35, 2, -9, -3], [-5, 30, 7, 21], [43, 3, -3, 18], [-33, 8, -16, -16], [24, 1, -7, 21], [-9, 7, 2, -16], [32, 23, -5, 13], [-12, -24, 13, 3], [8, -17, 15, -19], [25, -18, -7, -16], [3, -12, 2, -24], [1, -43, -1, -40], [-11, -18, 6, -34], [18, -19, 12, 26], [-46, -4, 16, -2], [2, 24, -15, 31], [52, 8, -7, 5], [-18, 10, 12, 4], [24, -18, 4, -8], [16, 16, 24, 1], [6, 14, -8, 22], [-10, -18, -12, -30], [-46, -6, -2, -23], [-21, 14, -3, -13], [0, -4, 10, -1], [-38, 11, -8, -22], [-20, -20, 8, -8], [-19, 17, 15, 17], [17, -6, 7, -7], [-46, 13, -6, -7], [0, 18, 18, 3], [9, 28, 9, 25], [23, -19, -31, 9], [-4, -22, -8, -22], [19, -9, -15, 0], [-9, -8, -7, -11], [-2, -13, 7, 19], [-6, 5, -1, 9], [0, -4, -8, 16], [42, 0, 6, 17], [-38, -2, -12, -26], [-28, -16, -4, -14], [7, -6, -27, 16], [-22, -7, -12, -8], [-19, -20, 2, -12], [-5, 2, -7, 21], [10, -6, 12, -14], [23, -36, -1, -32], [-8, -16, -14, -28], [12, 3, 12, -9], [24, -22, -2, -21], [2, -14, 9, -41], [16, 6, 9, -12], [61, 8, -9, 2], [29, -23, 11, 1], [-42, 14, 17, -1], [8, 8, 30, -2], [-3, 26, -13, 22], [19, -20, 13, 15], [17, 8, -5, -3], [-46, 9, -3, 6], [-26, 18, -12, 18], [-19, 3, -7, 24], [-8, -36, 6, -10], [13, 20, 13, 1], [38, -31, -15, -5], [54, 5, -5, 8], [-12, -7, 9, 12], [4, -7, -5, 30], [27, -17, -6, 1], [-2, 19, 7, 13], [26, -6, -26, 8], [-31, 18, -6, 22], [-6, -10, 28, -24], [-20, 8, -20, 29], [-31, 22, 2, 5], [-37, -4, -11, -6], [50, -16, 12, 30], [14, 7, 9, 8], [-8, -13, -16, 4], [-1, -5, -8, -8], [44, 10, -6, -4], [-46, 21, -7, 25], [-10, -17, 10, -42], [9, 3, 16, -16], [31, 5, 5, 1], [61, 0, -1, -1], [20, -28, 4, -2], [-6, 42, 4, 21], [-27, -16, 9, -6], [0, 9, 13, -10], [1, -25, -17, -8], [41, 22, 2, 31], [18, -16, -28, 4], [-32, 20, 0, -6], [-12, 4, -22, 25], [33, -5, -3, 32], [33, -12, -6, 23], [13, 17, 21, 5], [43, -1, 20, 20], [2, -18, 11, -32]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4598_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4598_2_a_bq();". // 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_4598_2_a_bq(:prec:=4) chi := MakeCharacter_4598_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_4598_2_a_bq();". // 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_4598_2_a_bq( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4598_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, -2, -4, 2, 1]>,<5,R![-2, -8, -4, 2, 1]>,<7,R![-1, 1]>,<13,R![16, 16, -16, -4, 1]>],Snew); return Vf; end function;