// Make newform 8470.2.a.cm 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_cm();". // 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_cm();". // 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 := [8, -7, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [-5, 1, 1], [5, 1, -1]]; Rf_basisdens := [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], [1, -1, 0], [1, 0, 0], [-1, 0, 0], [0, 0, 0], [-1, 0, -1], [0, -1, 0], [1, -1, 1], [3, 1, 0], [2, 0, 0], [-2, 1, -1], [4, -2, 0], [-4, -1, -1], [-2, -1, 0], [2, 1, -1], [2, 3, 0], [5, -3, -1], [0, -2, -1], [6, 1, 0], [6, -2, 1], [2, 1, 0], [-3, 1, -1], [1, 1, 0], [2, -1, 1], [0, 4, 1], [-3, 2, -1], [8, 1, -2], [-6, 0, -1], [-6, -1, -3], [1, -2, -3], [7, 3, 2], [-5, 3, 0], [1, 0, 3], [-3, -3, 1], [-8, -2, 2], [13, 1, 0], [3, 2, 3], [0, 4, -1], [-2, 3, 0], [-2, 0, 2], [-8, 3, -1], [-11, -2, -2], [3, -5, -4], [-5, 2, 3], [-4, -4, 1], [-2, -4, -4], [2, 4, 2], [-6, -1, -2], [-12, 2, 2], [12, 4, -1], [7, -4, 1], [13, 3, 2], [-6, 4, 2], [2, 0, -1], [6, -3, -2], [5, -3, 2], [1, 6, 2], [-4, 6, 0], [8, -3, 3], [1, 4, -2], [5, -1, 2], [1, 0, -5], [12, 2, 0], [-6, 7, 1], [-22, 6, 2], [-4, -1, 2], [4, -3, -5], [-5, 6, 1], [-12, 8, 1], [15, 4, -1], [2, 1, -2], [22, 4, -1], [18, -5, 0], [18, 0, -1], [0, 0, 4], [4, -5, -2], [-8, 9, 1], [-2, 0, -2], [-22, 4, 0], [2, -4, -2], [13, 5, -3], [8, -4, -4], [-13, 9, 1], [-2, -3, 5], [26, 3, 3], [-4, 6, -2], [9, 4, 0], [-9, 4, 5], [-20, 4, 1], [11, 7, -2], [-15, 9, 0], [4, -3, 1], [-7, 13, 0], [8, -11, 1], [-28, 2, -2], [-16, 1, 0], [13, 2, 4], [22, 2, -2], [-7, -5, 6], [-6, 6, 2], [30, 1, 0], [10, 5, -3], [1, -1, 4], [-1, -6, -6], [-4, 2, 4], [-14, 5, 4], [-5, -1, 6], [-18, -4, -5], [13, 3, 1], [-12, -3, -3], [-4, -4, -9], [-8, -5, 4], [-10, -4, 4], [-15, 11, 2], [-2, 6, 1], [3, 0, -6], [16, -12, -4], [20, 3, -2], [-6, -3, -2], [-16, 9, 5], [-11, 8, 5], [1, -2, 1], [-13, 12, 1], [-24, 0, 1], [-10, -4, 3], [-10, -1, 1], [4, 10, 4], [-20, 0, 6], [10, 4, 6], [-15, -6, -3], [4, -3, 1], [0, 8, 2], [-1, 9, -2], [6, -11, 0], [-22, -7, 3], [16, -2, -4], [5, 8, 3], [0, -4, -6], [11, 4, -1], [16, 6, 5], [4, 10, 2], [22, -8, 2], [20, 6, -4], [-26, -7, -2], [-11, -10, -1], [2, 9, 5], [-5, -2, 9], [-4, -12, -2], [-18, -4, -3], [12, -12, -2], [26, -3, 2], [4, 10, -2], [14, -8, -3], [-14, -1, 5], [-26, 2, -3], [3, -5, -5], [8, -4, -8], [-28, 4, -4], [10, 8, 4], [16, -12, -5], [12, -9, -9], [35, 2, -3], [4, -6, -4], [-13, -3, 3], [5, -14, -5], [4, -8, -8], [-19, 7, -1], [-15, 0, 3], [-20, -6, -7], [11, -8, -1], [-14, -9, -7], [-19, -8, 0], [12, 2, -2], [23, 0, -3], [-4, 0, 4], [-4, 4, 4], [32, 3, 1], [-2, -7, -5], [-14, 1, 8], [24, -2, -3], [7, -3, 0], [-10, 0, 1], [8, -13, -7], [0, -3, 11], [8, 0, 12], [31, -4, 3], [-36, 2, 4], [5, -15, 4], [-6, -2, -6], [-6, -1, 5], [-4, -4, -1], [-22, -10, 1], [10, 6, 4], [5, -12, -2], [18, -1, 8], [12, 13, -2], [-31, 12, 6], [30, 6, 3], [10, 10, 6], [-6, 14, 5], [2, 16, -2], [22, 4, -2], [-23, 16, 3], [18, -7, 5], [22, -12, 1], [-19, 0, 1], [-10, 6, 8], [22, 3, -7], [-21, 6, 8], [20, -4, 4], [18, 4, -6], [15, 4, -2], [11, -15, -8], [15, 7, 2], [-2, -5, -3], [-22, 0, 8], [-9, 9, -2], [-34, -5, -5], [11, -5, 4], [32, 6, 9], [-8, 3, -3], [6, 9, -5], [21, 4, -2], [15, -5, 6], [-37, 11, 2], [44, 1, 5], [-50, 4, -3], [18, -8, 1], [-26, 8, 1], [8, -16, -8], [-12, -13, 9], [-17, -5, -3], [-1, -5, 1], [-19, 8, 9], [-15, 5, 6], [-8, 3, 3], [14, 12, -2], [-4, -15, 8], [50, 2, 0], [-10, -1, 13], [18, -7, -4], [22, 6, 9], [-34, 2, -5], [-24, -9, 3], [-28, 11, -3], [14, -10, -6], [2, -7, 10], [7, 3, 5], [-14, -5, -9], [-41, 7, 0], [-4, 8, -1], [54, -2, 2], [-17, 5, 6], [22, 10, 2], [22, 4, 4], [-30, -11, 5], [16, -8, 0], [32, -16, 0], [-46, 6, 2], [6, -8, 0], [-16, 8, 2], [28, -9, -2], [-34, 4, 4], [13, -14, -3], [32, 4, -2], [-20, -2, -10], [30, -13, -3], [-36, -3, 1], [-8, -8, -7], [14, -9, -8], [-7, -10, 10], [2, -4, -6], [2, 6, -5], [23, 5, -9], [6, 15, -9], [20, 16, 1], [0, 10, 7], [8, -5, -1], [31, 0, -2], [-38, -6, 1], [4, -12, 7], [-29, -11, 5], [4, 12, 0], [-18, -8, 12], [-5, 9, 0], [-9, -5, -4], [7, 14, 1], [14, 8, -9], [-10, 18, -4], [4, -13, 3], [8, 16, 0], [-21, -5, -14], [35, -14, 1], [53, -1, -1], [-22, 0, -7], [-11, -16, 8], [28, -13, -3], [31, 12, -3], [26, 12, 3], [-12, -14, -10], [10, 10, -7], [46, 7, 4], [-2, 4, 15], [-10, -15, 10], [-64, 11, 1], [-3, -10, 5], [-8, 11, 10], [-17, -16, 2], [-29, 1, 9], [38, 1, -8], [-20, 8, -3], [8, -7, 11], [-16, 2, -7], [-26, 7, -7], [31, -23, -8], [-6, 17, -5], [19, 7, 12], [1, -13, 11], [-14, -3, -6], [4, 8, -7], [19, -3, -2], [46, -12, 0], [-17, -4, -10], [-3, 3, -16], [-5, 6, 11], [-26, 8, 4], [8, -7, 5], [-45, -3, -4], [27, 3, -2], [-12, 9, -9], [-12, -10, 6], [-10, 12, -3], [28, -11, -7], [-4, 4, -12], [-22, -18, 14], [3, -9, 10], [0, -13, -3], [-6, 2, -7], [-24, 9, -1], [-14, 12, 10], [8, 3, 0], [21, -26, -11], [-7, 18, -6], [-31, -1, 1], [50, 2, -2], [42, -25, -7], [-39, 20, 6], [20, 2, 6], [-14, 18, 2], [33, -6, 5], [-30, -4, -9], [-37, -14, 7], [20, 2, 8], [-46, 4, 3], [-2, -9, -9], [14, -25, 3], [-22, -6, 10], [61, -1, -6], [-1, 24, -5], [12, -12, -2], [18, -13, -6], [-10, 7, -1], [23, -1, 6], [54, -12, -4], [43, -9, 2], [-2, 2, -4], [-29, 4, 3], [1, 7, 8], [6, -2, 6], [30, -14, -6], [1, -7, -12], [-28, -12, 2], [20, 11, 21], [-60, -5, -3], [0, 17, 8], [-35, 1, -9], [39, 3, -7], [-42, -5, 10], [-36, -12, -2], [24, 4, 11], [19, -4, -5], [66, -8, -2], [8, -12, 8], [53, -8, -3], [-17, 1, -7], [2, 8, 15], [-22, 22, -1], [-38, 28, 6], [88, -2, 2], [46, 6, -7], [-20, 18, 7], [9, -16, -3], [14, 1, 3], [31, 11, 6], [1, 15, -1], [18, 12, -12], [-8, 10, 6], [34, -5, 3], [0, -15, 11], [-14, 14, 2], [-47, 0, 5], [-5, 4, 5], [-11, -11, -2], [-6, 1, 13], [-4, 12, 1], [-30, -2, -4], [-36, 8, 3], [23, 22, -3], [16, 10, 2], [17, 7, -6], [-16, -18, -3], [-36, 23, 8], [40, 4, 0], [-16, -12, -8], [15, -22, -11], [-23, -8, 0], [21, -9, -10], [-56, -1, -1], [-4, -9, 2], [-20, -13, 0], [36, 5, 13], [18, 18, 10], [-35, -7, -6], [-9, -5, -12], [42, -12, 4], [9, 8, -15], [-60, 10, -5], [-56, 2, -1], [-36, -4, -4], [12, 10, 6], [-24, -13, 7], [-14, 17, 1], [0, 11, -13], [34, -8, -6], [21, -14, -1], [38, 3, 1], [27, -2, 5], [-8, 12, -6], [8, -27, -7], [46, 7, -1], [28, -28, -12], [5, -2, 0], [22, -6, 6], [23, 3, 5], [-8, 8, 18], [33, 10, -9], [-12, 12, 6], [-10, 1, -8], [-84, -3, -3]]; 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_cm();". // 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_cm(:prec:=3) 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_cm();". // 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_cm( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8470_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![14, -9, -2, 1]>,<13,R![-50, -19, 4, 1]>,<17,R![4, -10, 1, 1]>,<19,R![-49, -37, -3, 1]>],Snew); return Vf; end function;