// Make newform 8470.2.a.ct in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8470_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_8470_2_a_ct();". // 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_8470_2_a_ct();". // 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 := [11, 1, -8, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [3, -4, -2, 1], [-11, 2, 4, -1]]; Rf_basisdens := [1, 1, 2, 2]; 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_8470_a();" function MakeCharacter_8470_a() N := 8470; order := 1; char_gens := [6777, 6051, 7141]; 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_8470_a_Hecke(Kf) return MakeCharacter_8470_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], [0, 0, 0, 1], [-1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0], [-1, 1, -2, 0], [1, 0, 2, 1], [0, -1, -1, -2], [-2, 0, 0, 0], [3, -1, -1, 1], [4, 0, 6, 0], [-2, 2, -2, 2], [1, 1, 0, 3], [-1, 1, -2, 3], [-1, 5, -3, -1], [6, -2, 4, -2], [0, -1, -7, 0], [4, 2, -4, 0], [-4, 1, 1, 2], [1, -1, 6, -2], [-4, -4, -1, 0], [5, -1, 2, 0], [0, 0, 4, 3], [-5, -1, -6, 1], [8, 0, 11, 0], [14, 2, 2, -2], [7, -3, -1, -1], [-9, 5, -4, 1], [2, 4, 8, 0], [10, 3, 3, -4], [-6, 2, 4, 0], [10, -5, 7, 0], [-1, -1, -4, 1], [0, 0, 8, 4], [-3, -1, -10, -2], [-7, -3, -5, 5], [3, 1, -5, -3], [8, -5, -3, 0], [12, 0, 0, 0], [3, -1, 4, -6], [7, 0, 4, -3], [6, -8, 2, -2], [-5, -5, 4, -2], [6, -4, 10, -2], [6, -4, 6, -6], [4, 0, -2, -6], [12, 4, 8, -5], [-4, 6, -6, 0], [9, 4, -3, 0], [4, -4, 2, -4], [-7, 1, 2, -1], [5, 5, -12, 2], [0, -7, -3, -2], [-6, 6, -12, 0], [-6, 2, 3, -2], [18, 0, 6, -4], [-10, 4, -8, -6], [0, -2, 0, -8], [-12, 6, 0, 2], [-11, -1, 0, -7], [1, 7, -7, -3], [14, 4, 0, 0], [-10, 4, -1, 12], [22, -4, 0, -2], [12, -9, 3, -4], [-8, 2, 14, 0], [-6, 2, -12, -7], [6, -7, 13, 4], [6, -7, -7, -4], [12, -8, 2, 2], [9, -6, 3, -8], [27, -3, -2, -2], [-3, 7, -6, -6], [-8, 2, -2, 6], [-10, -4, -7, 4], [5, 5, 0, -10], [-13, 5, -6, 0], [-3, -7, 8, 2], [-16, 8, -21, -4], [16, -2, 12, 0], [9, -5, -4, 1], [-13, -3, -2, 4], [21, 1, 11, -7], [-1, 2, 21, 0], [8, 4, 0, -2], [-7, -5, -2, -7], [-1, 6, 1, -4], [5, -3, -6, 5], [8, -2, -10, -6], [-14, 10, 2, 2], [15, -5, 0, 2], [2, -2, 6, 14], [6, 2, -12, 4], [-13, -1, -24, -3], [4, 2, 13, 2], [-11, -1, -6, -2], [8, -4, 6, 6], [16, 3, 1, 0], [1, -6, 18, -9], [9, 9, 0, -2], [-16, 5, -23, -6], [0, 4, -6, -4], [-6, -4, 3, 0], [21, 4, 14, -7], [11, -7, 7, -5], [-3, 8, 6, 5], [23, -4, 7, 8], [-1, 0, -10, -5], [-5, -11, 5, 7], [11, 11, 0, -7], [3, -3, 1, -5], [-10, 0, -12, 4], [0, 5, -1, 8], [2, -5, 3, -8], [19, -3, 5, -1], [-16, -2, -18, 9], [8, 0, 12, 15], [7, -17, 5, -1], [-22, 6, 4, 4], [-9, 6, -11, 0], [4, 12, -14, 2], [22, -1, 15, -12], [-19, -1, -7, -1], [6, 2, -4, 8], [2, -13, 7, -8], [4, -10, 4, -14], [-25, -5, 8, -2], [-20, 16, -8, 0], [-11, -15, 4, 2], [19, -1, 12, -12], [20, -3, -1, -6], [12, -6, 16, 4], [-21, 5, -27, -9], [-4, -10, -4, 8], [18, 7, -11, -2], [8, -6, 2, -14], [5, 1, -20, 0], [5, -2, 3, 4], [-25, 1, -10, 16], [3, 0, 11, 0], [9, -5, -6, -9], [9, -17, 15, -5], [-8, 16, -12, -4], [30, -9, 19, 12], [-14, -2, -8, -10], [-34, 12, 6, 6], [5, 5, 12, -4], [23, -7, 32, 7], [-6, 7, -25, 4], [10, -14, 12, -2], [-16, 10, -16, -4], [-9, 3, 4, 15], [7, 9, -6, -5], [-1, 13, -5, -1], [-6, -3, 7, 14], [13, 5, 7, -7], [-3, 13, 1, -3], [6, 5, -11, 8], [-17, 6, -11, 0], [8, -4, -8, -6], [8, 5, 17, -6], [-15, 5, -10, 1], [-18, 8, -20, -6], [-16, 8, 18, 6], [-20, 6, 2, 14], [-19, 13, -1, 1], [11, -9, 24, 4], [-37, 1, -13, 1], [-24, 4, -19, 4], [-33, 1, 2, 0], [18, -20, 9, 4], [-16, 22, -10, 0], [19, -9, 4, -4], [-38, -3, -13, 6], [14, -8, 6, -4], [34, -4, 10, -10], [-8, 2, 3, -2], [-4, -14, -4, -6], [1, -3, -8, -2], [-26, 12, 0, 10], [-6, 0, -32, -6], [30, -6, 22, 10], [-30, 2, -16, 4], [33, 1, 11, 3], [-34, 0, -2, -4], [8, -8, 26, -4], [4, 4, 12, 2], [-13, -9, -20, -3], [11, -16, 11, 8], [-28, 10, -28, -12], [27, -2, 44, -5], [-32, -3, 7, 4], [-25, 6, -23, -8], [34, 2, 0, -2], [24, 0, 20, 8], [23, 7, 1, -15], [10, -4, 38, -3], [-48, -10, -20, 12], [21, 3, -6, 12], [-13, 17, -34, 2], [10, -8, 32, -4], [-24, 10, -4, -6], [3, -1, 20, -12], [-45, 3, -14, -1], [-9, 1, -10, -7], [-29, -3, -19, 17], [10, -16, -12, -6], [30, -1, 13, 6], [-15, 11, -18, 0], [31, 5, 10, -5], [-8, -7, 13, -14], [-16, -12, -10, 12], [32, 0, -14, 0], [-36, 4, -4, 11], [-8, -4, 4, 12], [43, 3, 18, -9], [-38, 8, -12, 6], [5, 5, 20, -1], [0, -6, -8, -20], [10, -4, -12, -6], [-10, 12, -28, 0], [-20, 2, 12, 10], [-37, -3, -2, 12], [22, -6, 32, 6], [-20, 4, 16, 16], [29, -17, -2, -20], [13, -18, 27, 4], [26, -4, 10, -10], [-6, 2, 16, 6], [-4, -8, 16, -1], [-2, 2, -8, 16], [-25, 1, -22, -3], [5, -3, -16, 0], [25, -17, 18, 4], [-36, 6, -25, 2], [7, 1, -19, 3], [-33, -5, 0, 13], [-4, 26, -36, 10], [-16, 10, -23, -14], [2, -6, 38, -8], [50, -11, 35, 18], [-23, -13, -12, 11], [39, -1, -1, 3], [-43, 1, 7, -1], [-7, 24, -15, 8], [22, -18, 24, 8], [-16, -4, -44, 0], [1, 7, -2, -11], [-5, -3, -21, -5], [2, 6, -6, -4], [-10, -22, 20, 2], [2, -4, -22, 2], [-4, 2, -22, -2], [38, -5, 1, 10], [-40, 0, -18, 6], [35, -22, -14, -11], [-31, 3, -34, -2], [9, -22, 14, 3], [-55, 3, -26, -4], [0, -19, -19, -4], [-24, 14, -2, -6], [-4, 20, -12, -12], [8, 6, 20, 14], [-31, -11, -9, 11], [9, 11, 6, 8], [13, 11, 24, 3], [26, -10, 22, -4], [18, -17, 19, 16], [-20, 3, -9, 14], [-18, -8, 4, 14], [-2, 14, -8, 2], [7, -15, -16, -9], [9, -6, -25, -16], [-13, 9, 17, -1], [3, -6, -31, -16], [-5, 7, -24, -8], [-8, 9, -33, 8], [9, 11, -21, -5], [-11, 10, -24, -3], [-22, 10, -40, 8], [-5, -15, 37, -5], [27, -5, 44, -4], [27, -13, 28, 8], [36, -4, 36, 0], [8, 17, 11, -6], [74, 4, 14, -6], [-22, 17, 25, 8], [0, -8, -4, -10], [28, -10, -4, -16], [12, -26, -2, 6], [1, -5, 5, -9], [-7, -3, -8, 23], [-37, -9, 8, 5], [-20, -1, 9, 8], [-36, 14, -28, -6], [-28, 12, -14, -6], [56, -6, 18, -8], [-17, 21, -5, 1], [11, -11, 26, -13], [-8, -2, 13, -14], [-9, -11, 4, -1], [22, -16, -4, -12], [0, 12, -20, 4], [-13, -19, -10, 7], [51, 0, 35, -12], [1, -7, -2, -1], [29, 11, 26, -5], [-34, 16, -22, 8], [9, 17, 11, 5], [-22, -14, 18, -16], [-11, -7, 12, 0], [19, -19, 19, -23], [3, 0, 23, -4], [-16, 1, 25, -6], [-20, -18, 12, 6], [-31, -3, -44, 9], [-17, 17, -20, 33], [19, -17, -3, 7], [-53, -3, -32, -5], [-22, -2, 9, 2], [39, 5, -8, -7], [-39, 5, -48, 9], [10, 8, 0, -8], [52, -10, 22, 0], [52, 1, -5, -4], [15, 7, 8, 13], [11, -7, 35, -1], [41, 5, 24, -2], [52, 6, 10, -18], [-24, -18, -32, 8], [-17, 7, -24, 10], [14, -6, 14, -8], [-18, 18, -18, 18], [12, 2, 17, -2], [-24, 9, -41, -2], [40, -8, 28, 4], [19, 1, 1, -31], [37, -8, 25, 8], [10, 11, -11, 14], [24, -2, 54, 0], [-32, 6, -26, -4], [1, 1, 26, 15], [16, -8, 20, 6], [45, 9, 13, -3], [34, 16, 22, -8], [29, -7, -12, -6], [14, 10, 4, 4], [-42, -2, 0, 12], [34, 4, -4, 4], [42, -18, 26, 0], [-14, 17, -9, -6], [35, -5, 12, -7], [-51, -1, -7, 11], [-17, 23, -29, 7], [-3, -15, -23, 3], [26, -18, 13, 2], [-2, 14, 6, -6], [-4, -9, -33, 2], [-4, 5, 25, -14], [2, 12, -30, 2], [-2, 18, 16, -10], [-6, 20, -30, 10], [10, 22, 2, 6], [-4, 25, -17, -6], [-28, -4, 16, 16], [-13, -1, -16, -29], [-33, 3, -31, -23], [24, -14, 12, -10], [4, -12, 49, -4], [-34, -10, -24, 9], [16, 8, -36, -4], [-18, 14, -24, -18], [15, 7, 16, 4], [10, 24, -34, 8], [-24, -18, 18, 0], [22, -17, 11, -24], [-11, -9, -26, 12], [17, -16, -3, 16], [-43, 13, 10, -3], [-48, -6, 14, 11], [18, 4, 18, -10], [-23, -3, 30, -1], [34, -16, 10, 8], [26, -20, -20, -20], [-65, -7, -44, 13], [21, 9, -7, -11], [11, -27, 6, 10], [-59, -1, -34, 8], [-14, 7, 37, 14], [-24, 4, 24, -4], [-23, 18, -9, -4], [1, -9, 34, 6], [3, -25, 4, 9], [85, -1, 5, -11], [-1, -23, 9, 15], [4, 14, 20, -12], [42, 6, 24, -22], [16, 11, 17, 0], [14, 6, -16, 1], [23, 7, -5, 17], [80, 10, 14, -16], [-15, 9, -8, -19], [30, -34, 2, -8], [17, -8, 9, -16], [20, -36, 18, 6], [6, -8, -4, 0], [0, -14, 16, 14], [-36, 9, -27, -12], [7, 21, 27, -9], [-42, -23, -7, 20], [60, -5, 15, 0], [2, 6, 18, 20], [36, 4, 24, -24], [33, 10, 10, 3], [1, -7, 18, -9], [-50, -2, 4, 22], [-66, -2, -20, -2], [-21, 5, -1, 11], [29, -18, 12, 5], [39, 27, 17, -21], [15, 5, 9, -15], [-23, 9, -12, -24], [6, 20, 2, 4], [-15, -1, -4, -11], [-24, 6, -10, -15], [10, 4, 24, -4], [-21, -7, -6, 15], [-19, 14, -4, 33], [23, -7, 36, -7], [6, -28, -20, 0], [62, -7, 19, -14], [13, 27, 1, 5], [3, 5, -41, -5], [7, 11, -24, -2], [3, -23, 11, -11], [22, 9, -17, 18], [37, 4, -19, -4], [-29, 15, 9, 21], [-69, 1, -22, 27], [-2, -6, -24, -16], [-16, -24, 8, 9], [-4, 20, -43, 8], [33, -7, 29, -17], [27, -3, 7, -9], [28, -5, -15, -4], [-26, 20, -18, 26], [-17, 17, -29, 19], [17, 29, -3, -19], [-20, 27, -21, 2]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8470_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8470_2_a_ct();". // 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_8470_2_a_ct(:prec:=4) chi := MakeCharacter_8470_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(3169) - 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_8470_2_a_ct();". // 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_8470_2_a_ct( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8470_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![11, 8, -7, -2, 1]>,<13,R![-4, 16, -13, -1, 1]>,<17,R![-19, 38, -17, -2, 1]>,<19,R![236, -18, -41, 3, 1]>],Snew); return Vf; end function;