// Make newform 8048.2.a.n in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8048_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_8048_2_a_n();". // 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_8048_2_a_n();". // 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, 1, 5, -3, -2, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-2, -1, 1, 0, 0], [2, 2, -4, -1, 1], [1, 4, -3, -2, 1]]; Rf_basisdens := [1, 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_8048_a();" function MakeCharacter_8048_a() N := 8048; order := 1; char_gens := [1007, 6037, 2017]; 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_8048_a_Hecke(Kf) return MakeCharacter_8048_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, -1, 0, 1, -1], [0, 0, 1, 1, 0], [0, 0, 1, -1, 0], [2, 0, -1, 0, 1], [2, -2, 0, -1, 0], [-4, -1, 0, 1, -1], [0, 1, 0, 1, -1], [1, 1, 1, 1, 2], [-2, -3, -3, 1, -3], [1, 2, 1, -1, 3], [-3, 1, 0, 3, 0], [-1, 2, 1, 2, 5], [-2, 1, 1, -1, 0], [0, -1, -5, 0, -1], [-5, 6, 1, -4, 3], [1, 4, 0, -2, 1], [-4, 2, -3, -1, 0], [-3, 0, -2, 0, 3], [7, -6, 1, 1, -4], [-3, -3, -5, -2, -6], [0, 1, -2, 4, -5], [3, 1, 2, -3, -2], [-3, 10, 1, -6, 2], [3, -2, 3, 0, -1], [-6, 6, -4, -6, 7], [2, -12, -3, 4, -3], [-2, 4, 3, -2, -2], [-6, 2, 4, 0, 3], [2, -6, 0, -6, -3], [-1, 3, -4, -4, -1], [2, 0, 0, -3, 5], [-1, -2, -1, 5, 3], [-1, 5, -4, -3, 8], [2, 2, -2, -3, 0], [-6, 1, -5, 3, 3], [1, 3, -2, -5, -5], [-11, -4, -5, -2, -8], [1, 3, 5, -6, 1], [4, 3, 0, -4, 5], [3, 0, 2, 0, -1], [7, -4, 4, 0, -5], [-5, 1, 1, -3, 1], [4, 6, 5, 3, 5], [-1, -8, 3, 2, 1], [-6, -9, -3, 9, -10], [4, -3, 3, -2, 4], [-3, 1, 2, 2, 8], [1, -8, 3, -4, 2], [-5, 4, -6, -1, 8], [-10, -1, -2, 0, -6], [3, -7, -2, -2, -11], [-2, 3, 3, 3, 2], [0, 12, 7, -5, 2], [1, -4, 0, -8, -4], [6, 8, 5, -2, 15], [-12, 7, -2, -3, -8], [-4, 14, 6, -1, 6], [6, -16, -3, 8, -2], [5, 14, 5, -12, 4], [-15, -5, 2, 6, -9], [3, 3, 9, 3, 11], [-15, 6, 0, -5, 2], [7, 0, -10, -1, -8], [2, 0, 6, -1, 4], [14, 10, 7, -5, 15], [-2, 6, 8, -3, 6], [-4, -1, -2, 13, -7], [2, -12, -2, 15, -3], [-5, 13, 8, -6, 0], [-8, 7, 9, 5, 1], [-10, -1, 1, 2, -2], [-9, 20, 7, 3, 7], [0, 4, 3, 0, 5], [-4, -1, -7, -4, -2], [2, 7, 2, 11, 6], [13, -15, -3, 6, 1], [7, -4, 4, 1, -7], [-9, -14, -7, 1, -10], [-16, 1, 3, 9, -3], [4, -19, -8, 0, -8], [0, 19, 7, -14, 3], [16, -3, 6, 2, 1], [-12, -4, -6, 2, -15], [0, 7, 0, -1, 0], [-12, -11, -7, 14, -13], [-16, 6, -2, 9, 8], [-5, 1, 11, 10, 7], [-8, -15, -2, 1, -13], [-21, -1, -11, -8, -5], [-9, -16, -3, 7, -6], [11, 1, 1, -9, -3], [14, -11, 4, -2, -5], [9, 15, -4, -4, 4], [14, -7, -2, -1, -11], [1, 0, 0, 0, 0], [-9, -21, -8, 8, -13], [0, 0, 3, 13, -9], [8, -7, 8, -4, 1], [7, 5, -5, 5, -9], [5, -3, 13, 8, 7], [-4, -5, -13, 3, -14], [3, 12, 4, -9, 6], [10, -6, 0, -1, -12], [21, 3, 4, -8, -1], [13, -1, -8, -9, 10], [-1, -5, -4, -13, 1], [6, -17, -2, 2, -2], [-2, 17, 9, -17, 10], [2, 14, -5, 4, 9], [3, -20, -1, 4, -16], [-8, 2, 8, 10, -4], [0, -5, -4, -7, -12], [9, -7, -14, 2, 0], [-2, -11, -6, 14, -8], [-12, -9, -5, 4, -3], [13, 6, -6, -9, 16], [-6, 6, 7, 7, 3], [-5, 6, 12, -11, 6], [-11, 8, 7, 5, -5], [-4, -13, -8, -4, -13], [-12, -20, 1, 5, -15], [-4, -9, -12, 13, -10], [22, -8, 5, 1, -5], [9, -16, 7, 10, 2], [11, -28, 1, 10, -10], [6, 5, 13, 7, 6], [-13, -17, -15, 11, -18], [-5, 9, -9, 1, -6], [-8, 2, -1, 3, 10], [-6, -2, -4, -4, 2], [-7, 19, -14, -9, 18], [1, -9, -4, 3, -10], [1, 32, -1, -6, 20], [-26, 3, -8, -16, -4], [9, -14, -10, 5, -10], [-6, 11, 5, -10, -9], [17, 0, -5, -5, -7], [18, -4, 11, 9, -5], [-13, 8, -1, 9, 6], [7, 21, 2, -12, 22], [-6, -19, -3, 26, -30], [14, 12, -11, -3, -5], [4, 0, 7, 2, 19], [-12, -7, -6, -5, -16], [-13, 18, 9, -5, 2], [8, 10, -14, 9, -6], [18, -7, 19, 13, 7], [-2, 20, 1, -13, 10], [-16, 1, -6, -4, -22], [7, -21, 7, 10, 3], [-2, 2, -6, -2, 3], [3, -21, -9, 11, -18], [-3, -12, 13, 6, -2], [-21, -2, -4, 2, -5], [23, -18, -2, -11, -10], [-16, 1, 7, 11, -15], [9, -5, -2, 1, -19], [14, -20, -10, -12, -10], [-8, 3, 0, 8, -9], [-13, -11, 19, -7, 12], [-9, 17, -19, -10, 19], [8, -9, 4, -9, 11], [-13, 6, 5, 19, -9], [-4, -14, -3, -15, -9], [-4, -11, 11, 4, -14], [4, 5, 9, 0, 15], [-8, 14, -6, 1, 18], [-10, -8, -8, 2, -4], [-8, 4, 25, -1, 4], [10, 0, -3, -4, -4], [4, -4, -1, 20, -5], [20, 1, -16, -1, 3], [-29, 0, 0, -5, 0], [6, 27, 2, -7, -2], [19, -8, -3, 5, 5], [33, -2, -4, 5, 2], [5, 17, -7, -6, 5], [13, -2, -2, -12, 17], [-30, 4, -7, 2, -6], [-5, 5, 8, 9, 0], [-24, 13, -16, 0, 12], [-1, -26, -5, 3, 1], [-5, -24, -7, 26, -28], [25, -24, 5, 7, -2], [-1, -8, -10, 11, 4], [-6, -7, 9, 1, -3], [9, -26, 15, 7, -12], [1, -7, 7, -11, -10], [-6, -5, 6, 12, -10], [-4, 7, 12, -23, 15], [14, 5, 13, 2, 3], [5, 14, 6, -12, -7], [-12, 24, -14, -13, 5], [1, -18, -5, 16, -31], [-2, 0, -18, -5, -2], [-9, -3, 8, 0, 11], [2, 20, -4, -25, 18], [-4, 23, -1, -14, 7], [18, 19, 16, -10, 10], [-29, 11, 9, 4, -1], [-17, 0, 0, 13, 4], [31, 22, 19, -8, 18], [-7, -16, 8, 1, 9], [-22, 3, -12, -8, 5], [-8, 8, -7, 4, 4], [14, 10, 21, 8, 19], [8, 2, 13, 21, 16], [4, 15, 15, -1, 2], [-3, 4, 14, 3, -14], [6, 15, -7, -17, 10], [20, -26, -4, -4, -7], [-37, 10, 10, 14, 7], [-2, -4, -2, 31, -5], [19, -36, -5, 17, -3], [0, -19, 8, 7, 16], [12, 8, 14, 4, 4], [37, 25, 13, -11, 20], [-13, 0, 1, 19, 0], [-15, -5, -1, -20, 5], [17, 4, 27, 3, 10], [10, -4, -10, -3, 0], [-15, 11, 7, -13, 28], [31, -19, 7, 19, 2], [2, -4, -11, 7, 3], [-4, 2, 3, -13, -17], [12, -1, 8, 17, 9], [26, 27, 7, 1, 25], [20, -6, 4, -1, 18], [13, -13, -4, -9, -2], [2, 5, 6, -35, 22], [-27, 12, 6, 4, 12], [7, -28, -24, -12, -17], [6, -24, -11, 9, -21], [-5, 0, 19, 5, 4], [-9, 25, 1, 7, 9], [-28, -1, -12, -19, -6], [13, 7, 1, -32, -3], [2, 5, 14, -5, -6], [10, 5, 4, -16, 12], [-6, -8, 3, 17, -5], [5, -35, -2, 23, -10], [2, 10, -18, 6, 2], [-3, -29, -19, 5, -29], [-20, 6, -23, -7, 9], [26, -5, -2, 4, 17], [-29, -14, -12, 18, -20], [4, 20, 27, 12, 19], [23, 12, 12, -16, 13], [3, 9, -21, -6, -1], [-11, 3, 4, 39, -1], [6, -36, -19, 19, -12], [-18, 25, -21, -17, 16], [28, -18, 8, 8, -26], [30, 21, 0, -8, -6], [-16, -2, 13, 14, -3], [22, -4, 20, 11, 1], [24, 14, 25, -9, 6], [31, -4, 2, 25, -4], [-15, -22, -2, 20, 9], [30, 16, 18, -7, 32], [-31, 3, 2, 10, -30], [-29, 9, 2, -5, -4], [12, -35, -8, 25, -14], [-37, -30, -12, 5, -20], [-15, 22, -3, -31, 35], [4, -16, 17, -5, -13], [34, -8, 17, -10, -26], [-5, -18, -22, 0, -14], [-5, 23, 6, -35, 23], [-27, 13, 8, 10, -6], [-18, -10, -15, 20, -12], [-8, -5, -6, -3, -2], [34, -3, -4, -23, 28], [-14, 17, -1, -27, 18], [-9, 6, 7, -13, 6], [20, -15, 7, 22, 8], [-9, 23, 18, -8, 1], [7, -3, -13, -27, 2], [-6, 1, -17, 4, 6], [-4, 43, 15, -32, 15], [46, 6, 4, -18, 1], [12, 8, 8, -2, -24], [1, 11, 0, 27, 7], [-30, -3, -7, -19, 6], [-36, -22, -19, 15, -25], [-8, 13, -9, -10, 5], [-9, 43, 15, -27, 15], [-19, -34, -28, 9, -22], [8, -34, -10, -5, -5], [7, -17, -11, -19, -5], [-27, -13, -4, 0, -4], [-7, -5, -13, 14, 10], [24, 6, -14, 3, 9], [20, -7, 1, 19, 2], [29, -18, -13, 13, 2], [-29, 19, -12, -24, 19], [12, -5, 3, 4, 19], [48, -30, 2, 21, -14], [-11, 44, 9, -20, 13], [-32, 26, -7, -12, 2], [7, -35, -22, 15, -21], [-39, 37, -5, -23, 11], [-16, 22, 27, 12, 8], [31, -11, 19, 7, -20], [-42, 0, -2, 4, -4], [-16, -23, 14, 0, -4], [31, 28, 25, -1, 29], [4, 19, -15, -3, 16], [27, 6, 12, -25, -6], [2, -2, -14, 20, -17], [9, -24, -9, -14, -15], [-1, -20, 4, 17, -25], [20, 22, 13, -14, 20], [-3, -37, 1, 31, -29], [-16, 32, 7, -10, 13], [9, -12, 0, 16, 9], [-5, -6, 14, -1, -22], [7, 34, 24, -30, 27], [32, 41, 30, -18, 27], [0, 3, 24, 18, 32], [-8, 3, -1, 1, -26], [-4, -13, -25, 6, -25], [-28, 7, 17, -11, 0], [-5, 9, 17, 7, 0], [14, -1, -11, 10, -24], [-12, 6, 7, -19, 12], [20, -39, -9, -24, -20], [-30, 52, 8, -4, 6], [20, -14, 2, -1, 11], [-13, 19, 11, -8, 24], [-1, 32, -7, 5, 17], [-20, -5, -2, -12, -19], [-31, -20, 5, 25, -40], [-13, -14, 4, 9, -18], [3, -1, -22, -18, 0], [47, -4, -12, -27, 28], [21, -5, 21, 10, 20], [-14, 6, 7, 18, -16], [17, -7, -2, -5, 40], [25, -16, -9, -19, -17], [38, -22, 10, 34, -3], [-10, 19, 11, 2, 13], [-36, 18, -6, -19, 5], [32, 4, 0, -12, -15], [1, -22, -35, -28, -26], [7, 0, -16, 25, -24], [12, -10, 24, 23, 17], [-23, 25, -11, -30, 33], [44, -12, 5, 8, -20], [-15, 0, -4, 40, -25], [-26, -15, 4, 30, -30], [-35, -4, -23, -4, -22], [9, 2, 4, -27, -9], [-6, -3, 21, 13, 46], [15, 22, 38, 4, 32], [23, -24, -5, 9, 3], [3, 3, -10, -42, 5], [8, -50, -24, 23, -7], [-8, 26, -4, -29, 0], [8, -13, -12, -1, -22], [21, 12, -16, -19, 16], [-28, 18, -6, -31, -7], [25, 0, 21, 3, 14], [-41, 16, 19, -2, -4], [-10, -12, 13, 21, 0], [0, -44, -8, 35, -29], [-21, 14, -20, -33, -21], [-4, 30, -3, -8, 5], [-11, -12, -3, -13, -22], [21, -4, 17, 21, 27], [1, -26, -5, 36, -18], [48, 3, -5, -9, 4], [9, 14, 23, 21, 23], [10, -15, -15, 4, 6], [-3, 21, -16, -10, 13], [2, 26, 14, 9, 38], [10, 8, -22, 15, -7], [0, -20, -21, 4, -29], [-63, 3, 6, 22, -7], [30, -30, 1, 9, 21], [16, -3, 7, 45, 0], [0, -17, 27, 12, -9], [-18, -12, -14, 4, -19], [-4, -39, -16, 18, -8], [0, 8, -11, -28, -11], [2, -36, -2, 35, -1], [-41, 37, -1, -20, 22], [42, -2, -10, -12, -27], [-5, -6, 34, 4, 0], [36, -52, -21, 4, -4], [2, -19, -23, -18, -14], [-16, 32, 16, -6, 11], [-1, -16, 12, 1, -13], [40, -20, 27, 11, -26], [-28, 14, -1, -5, -6], [19, -16, 11, -5, -7], [29, -9, 23, 17, 16], [-14, 30, -17, -12, 25], [-53, -1, 8, 19, -34], [-36, -3, 0, 9, -7], [-28, 52, 19, -16, 9], [-2, -18, -13, 5, -20], [29, -23, -13, -24, -13], [-29, -10, 9, -2, 19], [-26, 4, -3, 17, 15], [-27, -30, -16, -7, -27], [-24, -40, -24, 17, -45], [-16, 6, 15, 28, 32], [-21, -27, -19, 7, -53], [-17, 21, 6, -12, 4], [-40, 13, 30, -3, 12], [25, 27, -16, -4, 2], [14, 11, 12, 3, -18], [-22, -27, -24, -4, -38], [12, 1, 6, 22, 30], [-7, 43, -2, -20, -4], [-10, -41, -25, 6, -31], [-34, 38, -1, 3, 36], [30, 34, 21, -20, 61], [-23, 1, -29, 3, -4], [-46, -53, -35, 16, -43], [2, 3, 10, -6, -7], [15, -12, -18, 12, -34], [20, 39, 17, 1, 20], [19, 43, 41, -22, 23], [-9, 41, 20, -22, 26], [-16, 25, 15, -24, 30], [6, -35, 16, 21, -15], [17, 14, -2, 19, -6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8048_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8048_2_a_n();". // 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_8048_2_a_n(:prec:=5) chi := MakeCharacter_8048_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_8048_2_a_n();". // 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_8048_2_a_n( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8048_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, 4, 1, -6, 0, 1]>,<5,R![1, 7, -4, -9, 1, 1]>,<7,R![1, -7, -24, -11, 3, 1]>,<13,R![-69, 14, 81, -21, -5, 1]>],Snew); return Vf; end function;