// Make newform 8470.2.a.co in Magma, downloaded from the LMFDB on 29 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_co();". // 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_co();". // 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, 0, -7, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-4, 0, 1, 0], [0, -4, 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_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], [-1, 1, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 0], [4, -1, 1, 1], [3, 1, 0, 1], [-1, 2, -1, 0], [0, 0, 0, 2], [0, 1, -5, 1], [-6, -1, -3, 3], [2, 1, -5, 1], [6, 1, 0, 1], [3, -2, 1, -3], [0, -2, 4, -2], [3, -2, -4, -1], [-1, 0, 5, 2], [5, 1, 3, 2], [1, 1, 2, 1], [-2, 1, 3, 1], [11, -2, 7, 3], [-1, 0, -6, -1], [-1, -3, -6, 0], [-4, -1, -6, 3], [-3, 2, -1, -3], [-3, 0, -2, -1], [-7, 0, -8, 3], [-5, -2, -11, -1], [-5, 4, 0, 1], [2, -4, 3, -5], [5, -6, 4, 3], [7, 2, 9, 2], [7, 5, 2, 0], [0, 2, 8, -2], [6, 2, 4, -2], [-7, 1, -7, -2], [3, 1, 3, -4], [-7, 3, -10, -1], [1, 3, -3, 8], [1, 4, 2, 1], [-4, -7, 6, -6], [3, 4, 2, 7], [13, 5, 3, -4], [-2, 4, -4, -6], [12, 0, 4, -2], [11, 2, 2, -3], [6, -5, 12, 1], [1, 6, 0, 3], [-4, 0, -5, 1], [5, -1, 3, -10], [1, 5, 0, 2], [2, -7, -7, -5], [-6, 1, -8, -6], [-2, 4, -6, -4], [-7, 6, 1, -1], [22, -1, 3, -1], [-13, 2, 2, -3], [16, 3, 7, -9], [5, 5, -3, 8], [-12, -2, -7, -6], [2, 2, -10, -4], [13, 0, 2, 3], [-1, -3, -10, 2], [-8, 8, 6, 6], [-3, 9, -6, -3], [8, -1, 7, -5], [-8, 9, -4, 1], [-6, -6, -3, 5], [7, -3, 8, -1], [8, 2, 6, -6], [-1, 3, 6, 5], [14, -4, 2, 0], [-2, 9, 1, -1], [-5, -3, 11, -2], [-2, -9, -2, 7], [2, 9, -9, 1], [-3, 5, -3, 2], [5, -2, 20, 3], [12, -4, 3, -6], [10, 2, 20, 2], [14, 2, 11, 10], [-12, -8, 0, -6], [-8, -4, -10, -6], [-1, -1, 2, -5], [3, -8, -6, -5], [7, 6, 5, 3], [-3, -10, -13, -2], [-3, 3, 6, 10], [11, 0, 2, -5], [-3, -2, -8, -13], [0, 8, 10, 6], [0, 1, 1, 3], [-3, -5, 7, -12], [-4, 7, 12, 1], [-4, -9, -12, 3], [8, -8, 10, 2], [-8, -9, -1, 3], [10, -3, 4, -10], [-16, 2, -11, 1], [2, -5, -5, -9], [5, -11, -14, -9], [21, 10, 16, -7], [29, 1, 20, -2], [7, -2, -3, -4], [-6, 10, 0, 12], [-3, 7, -4, -1], [8, 0, -9, -1], [37, 5, 8, -5], [-21, 0, -16, 1], [2, -5, 4, 9], [20, 3, 3, -3], [19, 2, 18, -3], [18, 8, -5, 1], [-3, -6, 23, -2], [-24, 10, -8, -6], [12, 10, -1, 2], [19, 5, 10, 10], [-7, 6, 12, -1], [-3, -13, -23, -4], [22, -5, 12, 8], [13, 1, 15, 8], [14, -2, 17, 13], [-6, 9, -23, 7], [24, -6, 2, 2], [5, 16, 3, 2], [-23, -5, -7, -4], [-17, 6, -2, 9], [-7, -9, 11, -6], [0, -10, 26, -6], [-26, -4, -14, -2], [14, -7, -2, -10], [24, 0, 0, 6], [18, 3, 1, 5], [2, -3, -1, -13], [14, 7, -6, -6], [-12, -12, -2, 2], [-5, -16, -20, -1], [-18, -6, 3, -7], [-7, -3, -19, -4], [-15, -10, -13, 8], [2, 18, -5, 4], [27, 4, 0, 11], [-18, -1, -19, 9], [1, -11, -12, -3], [-8, -1, -11, 3], [8, -13, 11, -1], [-7, -9, -23, -8], [-17, -4, -13, -13], [-9, -10, 7, 4], [26, 5, 13, 7], [-4, 9, 3, -11], [-4, 9, 8, 17], [1, -8, 5, -1], [-7, 1, 7, -6], [-13, -7, -12, -9], [3, 7, -1, -2], [15, 10, 4, -7], [-16, -5, -2, 10], [-7, 19, 18, 7], [-12, 13, 15, 5], [5, -9, -4, -21], [-5, -3, -12, -20], [-23, -4, -20, 7], [8, 0, -6, 0], [-2, -16, 2, -2], [-7, -9, 15, 2], [-19, 1, 5, -8], [7, 1, 19, -4], [-10, -8, -15, -8], [7, -1, 3, -16], [-8, -3, 6, -1], [7, -12, 0, -5], [22, -3, -5, 7], [32, -4, -1, -11], [12, -3, -9, -7], [10, -8, -12, -14], [0, -19, 8, 5], [-4, -7, -13, 9], [10, 12, 12, -14], [-15, -13, -17, 8], [-5, 0, 0, 9], [-22, 14, -20, -4], [-5, 5, 11, 4], [4, 4, 22, 0], [14, -2, 4, -6], [-14, 3, 5, 5], [15, -9, -1, 2], [-41, 11, -6, -4], [7, 4, 11, -2], [-5, -9, -25, -10], [-27, 7, 14, 3], [13, 7, -14, 1], [-2, 1, 2, -4], [24, 11, 17, -9], [0, -2, 20, -10], [-4, 16, -16, 2], [-22, -8, -5, 8], [41, -4, 12, -3], [-34, 10, -16, -6], [9, -8, 6, -15], [17, 2, 14, 5], [-6, -2, 20, -2], [7, -13, -15, -14], [4, -1, -4, 11], [-14, -2, 9, -10], [-13, 9, 21, 8], [9, 7, 9, 0], [31, -17, 8, 5], [-6, 16, 26, 4], [30, 4, 15, -2], [-16, -14, 15, -5], [22, -6, 0, -20], [21, -7, 7, -10], [29, -15, 18, 4], [27, -3, 17, 10], [20, -9, 16, 5], [2, 13, 5, 1], [-14, 21, -2, 7], [19, 9, 9, -14], [27, 8, 6, -9], [-41, -1, -17, -14], [16, -2, 20, 0], [-4, -8, -14, -12], [16, 22, 8, -6], [-26, -2, 18, 2], [-44, 9, -19, -11], [29, 3, 18, -17], [-22, 16, -12, 0], [-4, 21, 21, 7], [0, -7, 30, 1], [-2, -18, -26, 0], [-36, -8, -7, -6], [-28, -20, 2, 0], [-30, -6, -24, 2], [-3, 1, -20, 4], [-18, -15, 17, -7], [24, 5, 10, -7], [46, -1, 27, -3], [8, -7, 14, 7], [-21, 0, -30, 15], [25, -11, 14, -7], [4, -10, -19, -4], [-8, 2, 22, 8], [15, 6, 42, 3], [19, 15, 16, 3], [27, 13, 9, -14], [17, -8, 24, -5], [22, -2, -1, -18], [16, -4, 16, 6], [-38, -15, -7, 11], [-2, 0, 18, 18], [-22, -6, -6, 6], [-1, -20, 2, 9], [0, 7, -26, 10], [-3, -8, -2, -23], [54, 3, 20, -12], [-1, -17, 37, -10], [-26, 10, -13, -1], [-2, 12, 24, 4], [7, 3, 16, -11], [1, 20, 6, -9], [-12, -4, 6, -4], [-11, 4, -20, 1], [29, 1, 41, 12], [-48, -2, -6, 0], [2, -14, -11, -6], [10, -27, 7, 5], [20, 5, 38, 12], [5, 1, 18, -1], [17, 8, -2, -7], [-4, -8, -2, 0], [-35, 0, 17, -5], [-19, 5, 30, 7], [-22, 11, 33, 5], [-21, 1, -20, -11], [-38, -6, -20, -12], [-29, -9, -28, 15], [-24, 4, -52, 0], [-11, -8, -1, -6], [14, -6, 2, 2], [22, -5, 19, 7], [-19, 6, 6, 13], [-11, -18, 28, -15], [-24, -10, 10, -2], [10, -14, 17, 1], [27, 2, 12, 13], [-20, -18, 9, 3], [6, -7, 17, 5], [-26, 1, 9, -1], [-38, -6, 16, -6], [4, -28, -18, -4], [13, -11, -2, -24], [31, 3, -20, 10], [23, 4, -1, 2], [13, -21, 23, 2], [3, 7, -27, 0], [8, -12, -18, 10], [-32, 14, 8, 18], [16, 20, -19, 8], [17, 11, -2, 0], [-1, -6, 35, -9], [38, -8, 8, 4], [3, -12, 2, -7], [-7, -18, -11, -7], [14, -1, -2, 4], [-19, 9, -28, -4], [25, 2, 5, -7], [31, 7, 15, 18], [2, -9, -27, -9], [-5, 18, 12, -13], [9, -2, -24, 9], [8, 10, 24, -2], [5, -20, 33, -8], [-29, -9, -10, -7], [16, 13, 21, 7], [-32, -5, -10, 1], [44, 6, 49, -10], [13, 2, 14, 11], [-23, -8, -23, 17], [-22, 6, -39, 0], [-66, 2, -15, 2], [59, 5, 28, -6], [21, -6, 16, 7], [0, 9, -3, 11], [-20, 14, -11, 7], [-38, 21, -24, -9], [-28, 4, -16, 0], [30, 4, -16, 0], [21, 15, 31, 10], [-28, -8, -16, -14], [-46, -5, 11, 3], [27, 16, 18, -15], [-34, 9, -19, -3], [5, -7, -24, -4], [20, -21, 16, 2], [-6, 0, -36, 0], [16, 8, -24, 16], [77, 7, -8, 3], [-14, 17, -24, 8], [-20, 4, -18, 26], [50, -2, 4, 2], [-7, -3, -14, -18], [16, -8, 14, -28], [-13, 4, -26, -17], [12, -14, -24, 4], [-36, 0, -62, 8], [-18, 10, -18, -4], [-28, -12, -38, -2], [-22, 15, -21, -23], [-41, -15, -25, 20], [11, -12, 17, -24], [-31, 9, 8, 12], [0, 23, 17, -11], [35, -1, 5, 8], [-53, -11, 15, -8], [-41, -4, -17, -13], [-22, 22, 30, 12], [-9, 4, -31, 6], [6, -2, -9, 13], [24, 12, 12, -10], [9, -15, -3, -22], [40, 10, 22, -22], [-27, -5, -33, -8], [-28, -21, -2, -8], [-40, 4, -62, 6], [-25, -7, -18, 0], [-4, -11, -1, -5], [-1, 5, -13, -22], [6, -14, -5, -16], [-26, -17, -10, 11], [-12, -28, 20, -2], [42, 9, -11, 7], [31, 7, 1, 0], [-16, -3, 7, 5], [21, -8, 38, 17], [-5, -40, -13, 8], [19, 11, 29, 18], [25, 5, 12, 3], [-28, -9, -18, -9], [3, -12, -13, 5], [0, -27, -15, -13], [11, -25, -16, -2], [3, 9, -9, 10], [-12, -16, -4, 8], [-8, 16, -1, 10], [19, -8, 24, -25], [-7, -8, 32, 5], [-29, -8, 22, 5], [-31, -5, 14, -5], [67, 10, 6, 7], [15, -40, 3, 6], [30, -18, 0, 14], [-46, -5, -40, 3], [34, -2, 18, 6], [12, 5, 3, 9], [44, 8, 14, 0], [-59, -5, 1, -18], [16, 7, -4, 14], [-12, -21, 12, -33], [-6, -22, -40, -2], [3, -11, -7, 12], [-19, -11, -14, 8], [-25, 6, -62, 1], [-19, -5, 28, -15], [36, 14, 14, -10], [-34, -12, -12, -16], [12, -16, 26, -2], [71, -10, 35, 12], [2, -30, 18, -2], [-43, -6, -35, -6], [27, 11, 2, -13], [22, -7, -11, 5], [-20, -24, -2, -18], [-34, 7, -16, 12], [9, 7, -48, 2], [-11, 5, 13, 4], [4, 10, 4, -2], [24, -14, 22, 22], [-45, 11, -20, -21], [22, 21, 9, 3], [46, -6, -26, -8], [18, -6, 24, -28], [29, -9, 25, 10], [6, 0, -31, 4], [-5, -8, -29, 1], [9, -8, 42, -17], [-19, 4, -17, 19], [1, 0, -17, 16], [-30, -20, -17, 20], [-27, 4, -20, 39], [-8, -7, -6, -24], [-20, -14, -10, 12], [-16, 26, 14, 12], [48, -14, 42, 6], [2, 32, 10, 14], [4, -1, 12, 2], [-7, -8, 3, -12], [31, 27, 31, -10], [-45, 18, -25, 3], [-39, 7, -13, -22], [-35, -5, -12, 24], [-14, 30, -17, -4], [28, -8, 6, 22], [38, 13, -7, -1], [-32, -5, 30, 4], [-14, -6, -28, -16], [10, 28, -4, 26], [-26, -6, -38, 14], [-10, 15, 2, 10]]; 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_co();". // 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_co(: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_co();". // 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_co( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8470_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![5, -10, -1, 4, 1]>,<13,R![-220, 0, 54, -14, 1]>,<17,R![-25, -30, 41, -12, 1]>,<19,R![145, -10, -29, 2, 1]>],Snew); return Vf; end function;