// Make newform 8470.2.a.cs 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_cs();". // 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_cs();". // 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_cs();". // 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_cs(: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_cs();". // 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_cs( : 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;