// Make newform 8008.2.a.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8008_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_8008_2_a_i();". // 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_8008_2_a_i();". // 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, -4, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-3, 0, 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_8008_a();" function MakeCharacter_8008_a() N := 8008; order := 1; char_gens := [6007, 4005, 3433, 4369, 4929]; v := [1, 1, 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_8008_a_Hecke(Kf) return MakeCharacter_8008_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], [1, 0, 1], [2, -1, 0], [-1, 0, 0], [-1, 0, 0], [1, 0, 0], [-5, 1, -2], [0, 2, -1], [-4, 3, 1], [2, -2, -2], [-6, 1, -1], [2, 1, -1], [-2, -6, 4], [3, 3, 0], [0, -4, 2], [-5, -1, -4], [2, 3, -3], [3, 1, 0], [-3, -2, 1], [-8, 1, -2], [-6, -4, 0], [0, 2, 3], [0, 4, 1], [5, 4, -1], [-2, 5, -7], [-6, 0, 1], [-4, -4, 4], [3, -7, -2], [8, 4, -1], [-14, -1, 2], [4, -4, -5], [4, -2, 6], [-8, 5, -5], [4, -2, 2], [4, 4, 1], [-2, -2, 4], [4, 1, 7], [-2, 0, 1], [-13, 3, 0], [-6, -6, -1], [-12, -3, -5], [4, 7, -1], [-4, 5, -1], [-3, -7, 6], [-5, 11, -4], [-6, 0, -9], [5, 5, 0], [2, 5, -9], [7, -1, 10], [10, -7, 0], [-6, -8, 6], [12, 0, 6], [-6, -4, 2], [-4, -11, 6], [6, -4, -6], [-4, 8, 1], [18, 4, -4], [-5, 9, -10], [-12, -4, -2], [1, -9, -4], [-8, 6, 6], [8, -4, -2], [3, -7, 4], [-22, -4, 3], [-10, 5, -9], [-6, -1, 1], [-6, 1, 0], [0, 12, -10], [10, -4, 1], [-16, 0, 6], [10, 7, 3], [-2, 12, 2], [1, -4, 7], [16, -4, 16], [-1, 2, -13], [-10, 9, 5], [-12, -11, 6], [9, 5, 2], [12, -10, -4], [-12, -10, 8], [15, -7, -4], [2, -16, 8], [14, -8, 8], [-4, 9, 9], [-6, -2, 4], [4, -15, 13], [-10, -15, 5], [2, -12, 9], [18, 10, -4], [-36, 3, -1], [-1, 0, -5], [14, -4, 5], [-8, -7, 5], [24, -2, 2], [6, 12, 3], [10, -6, -6], [-6, -5, 11], [-2, 19, -7], [-24, -6, 0], [26, 0, 8], [0, -8, 12], [-4, 6, 3], [-22, 8, -18], [-18, 4, 2], [18, 4, 3], [-21, 8, 5], [-6, 2, 4], [-34, 4, -14], [-6, 6, 12], [1, -5, -6], [4, -14, 2], [-2, 20, -4], [0, -14, 2], [-18, -7, -1], [7, -18, 13], [7, 14, 5], [8, 9, 5], [-28, -1, 3], [23, -6, 23], [-19, -1, -14], [30, -5, 10], [-4, 10, -16], [25, 9, 2], [11, -11, 2], [14, -5, -5], [-18, -2, 6], [-6, 17, 5], [-7, 16, -13], [-2, 21, 2], [22, -8, -6], [-2, 12, 2], [-2, 8, 0], [-30, 7, -13], [-4, -12, 13], [8, 2, 10], [-8, 20, 10], [5, 21, 2], [5, -7, -6], [20, -15, -3], [6, 24, -2], [12, 10, -3], [-3, -13, 14], [-8, 3, 11], [0, 6, 0], [12, 5, -5], [2, 1, -9], [-16, 18, -10], [4, 6, 15], [-28, -5, -1], [0, 0, -8], [-19, 11, -6], [-10, -5, 9], [14, 10, 2], [-24, 8, -4], [-20, 8, -26], [-22, 6, 8], [35, -5, -6], [5, 21, -12], [-4, -6, -7], [4, -4, -2], [-7, 14, -19], [-38, -10, 4], [18, -8, -8], [-12, -9, -18], [4, 5, 13], [-20, -5, 11], [16, 0, 0], [26, -12, 19], [-26, 6, -17], [22, -23, -9], [-2, 6, -20], [6, 16, -16], [-4, 2, 5], [-15, -25, 12], [13, -18, 13], [0, -7, 1], [-18, 12, -24], [-10, -3, 11], [-10, 12, 0], [34, -10, 3], [-8, 2, 7], [-22, -12, -8], [0, -19, -2], [-14, -14, -2], [3, -28, 23], [30, -4, 14], [18, -14, 22], [14, 30, -12], [2, 16, -3], [-40, 14, -2], [-6, 7, -15], [26, -10, 26], [18, -1, 5], [15, -9, 10], [8, 14, 10], [-26, -7, -1], [-11, 11, -4], [-15, 20, 7], [-20, 0, -4], [-12, -4, -20], [29, -11, -8], [-2, -14, -14], [34, -10, 11], [-24, 14, 7], [6, 1, 5], [-10, -9, -1], [8, -7, -11], [-16, -4, -8], [6, 14, -1], [-8, -26, 9], [-2, -34, 12], [-28, -1, -21], [22, 5, 3], [24, -12, 28], [14, -28, -5], [16, -5, -4], [-7, 3, -12], [-26, 16, -20], [-7, 11, -12], [-20, 19, 7], [2, 0, -22], [8, -4, 16], [-40, 9, -3], [-4, 5, -4], [48, 6, -4], [-8, 4, -10], [22, 12, -6], [-44, 8, -22], [8, 20, -20], [40, -12, 2], [11, 12, -19], [10, 12, -22], [-12, -42, 18], [20, 22, -6], [16, -15, 0], [-1, 5, 2], [-22, 21, -19], [12, 20, -19], [-24, 19, -9], [37, -2, 1], [16, -23, 13], [18, -10, 10], [-11, -5, -14], [8, 1, -11], [16, 28, -4], [28, -4, 0], [-12, -12, 4], [24, 24, -16], [32, -12, -2], [24, -2, 14], [12, 2, 1], [-6, 2, 16], [9, 10, 3], [-24, -10, -10], [-10, 6, -23], [-10, 8, 2], [5, 21, -8], [0, 6, -13], [-44, -11, -7], [24, 8, -12], [15, -1, -14], [-33, 15, -28], [36, 10, -10], [30, 6, 1], [19, -17, 4], [-33, -7, 0], [-16, 7, -17], [-5, 4, 1], [-46, -20, 10], [-22, -24, 8], [-4, 5, 17], [42, 13, 8], [20, -12, 29], [17, -19, -14], [7, -17, -4], [-20, 3, 17], [18, -27, 16], [17, 13, -8], [30, -24, 20], [28, 8, 0], [10, 24, -6], [8, -23, 24], [6, 26, 4], [2, 2, -26], [41, -5, 16], [-26, -9, -25], [-30, 14, -17], [-7, 27, -6], [14, 23, 13], [1, 11, 10], [-58, -3, 9], [-16, 4, -24], [-38, 2, -1], [-12, 26, -13], [20, 12, 8], [-38, 8, 4], [-1, -24, 21], [50, 5, 9], [-26, 2, -8], [-20, 12, 4], [-46, -16, 16], [-17, -15, 2], [-8, 0, 4], [20, 0, 8], [18, 13, -7], [14, 5, 15], [0, -1, 15], [-32, -3, -5], [-52, -6, -3], [0, 32, -6], [-32, 2, -12], [18, -24, 4], [36, 2, 26], [-26, -9, -11], [38, -24, 4], [28, 24, -11], [38, 5, 13], [2, -18, -16], [8, 8, -17], [-54, -2, -4], [-34, 16, -8], [18, -13, 5], [-49, 19, 2], [17, 10, -13], [-15, 17, -10], [54, 9, -5], [49, 4, -1], [6, 9, 13], [-20, -10, -15], [10, 22, 16], [25, 23, 10], [-36, -24, -1], [28, 10, 14], [0, -12, -3], [-34, 24, -16], [-10, 7, 7], [-16, 33, -13], [-32, 8, -2], [0, -5, 0], [-8, 20, -8], [-26, -5, -35], [-46, 11, -21], [46, -4, 36], [-8, -17, -19], [0, -29, -11], [-10, -12, 7], [44, -9, -9], [-56, -11, 1], [-12, -46, 14], [32, -33, -1], [-7, -35, 18], [26, 13, -15], [-54, -4, 10], [-22, 28, -18], [18, -12, 16], [14, 24, 8], [10, 15, -23], [12, -32, -8], [-8, 26, 18], [-36, -3, -2], [31, 21, -10], [4, -4, 29], [-36, 11, 15], [-24, 2, -14], [15, 14, 25], [38, -13, 4], [2, -30, 10], [16, -14, 4], [-2, -32, 20], [-40, 1, 15], [12, -26, -25], [-12, 6, 16], [-18, 2, -14], [30, -12, -4], [-35, 26, -23], [-6, -6, -6], [-56, 10, -17], [16, 4, 8], [11, 27, -4], [-8, -10, 15], [-26, 14, 3], [2, -3, -1], [10, -38, 0], [-26, -10, 23], [16, 4, 20], [-18, 1, 19], [44, -4, 20], [24, -7, 33], [66, 8, -4], [42, -15, -10], [38, 19, -3], [34, 23, 7], [17, 6, -15], [-16, -1, -12], [17, 28, -19], [20, 4, -20], [2, -16, 5], [18, -24, 36], [60, -12, 16], [-16, 10, -31], [-24, -22, 12], [-13, -16, -1], [10, -20, -14], [-35, -3, -16], [16, -26, -20], [46, -6, 4], [-6, -4, 20], [-14, -32, 7], [-68, -24, 12], [4, 23, 21], [-12, 6, 27], [50, -24, 12], [-30, 10, 8], [22, 30, 7], [-38, 12, -28], [-14, 23, 21], [50, 2, 10], [-2, 13, -2], [-20, -20, 25], [-28, -6, -16], [-40, 27, 8], [0, -4, 0], [-16, 26, -27], [-22, 34, 4], [25, -11, -8], [-54, -11, -12], [-12, -5, 3], [2, -20, 17], [-8, 27, -39], [2, 8, 16]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8008_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8008_2_a_i();". // 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_8008_2_a_i(:prec:=3) chi := MakeCharacter_8008_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_8008_2_a_i();". // 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_8008_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8008_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![7, -4, -2, 1]>,<5,R![1, 8, -6, 1]>,<17,R![-28, 37, 13, 1]>],Snew); return Vf; end function;