// Make newform 8280.2.a.bj in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8280_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_8280_2_a_bj();". // 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_8280_2_a_bj();". // 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, -9, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-6, 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_8280_a();" function MakeCharacter_8280_a() N := 8280; order := 1; char_gens := [2071, 4141, 4601, 1657, 3961]; 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_8280_a_Hecke(Kf) return MakeCharacter_8280_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], [0, 0, 0], [-1, 0, 0], [1, -1, 0], [-2, -1, 0], [0, 0, -1], [-3, 0, -1], [-4, 0, -1], [-1, 0, 0], [-5, 1, 1], [-3, 1, 0], [3, -3, -1], [-3, 0, 1], [8, 0, 0], [0, 0, -2], [1, -3, -1], [5, 1, 1], [4, -1, 2], [3, -3, -1], [7, 2, -1], [6, -4, -2], [0, -4, 0], [5, 1, -1], [4, -4, 0], [-2, 3, 0], [-9, 1, 1], [4, -1, 2], [-3, 3, 3], [8, -2, 1], [1, -3, 3], [0, 2, 0], [8, 2, -2], [4, -2, -3], [-3, -3, 1], [14, -4, -1], [-4, -5, 0], [-3, -1, 3], [8, -2, 1], [8, 0, 0], [0, 3, 0], [4, 0, -2], [2, 4, -1], [0, 2, -4], [4, -2, 0], [-2, -1, -2], [-14, 0, 0], [3, -1, 1], [6, 2, -2], [0, -4, 2], [0, 2, -2], [-8, -2, 0], [1, -5, 1], [-2, -6, -2], [20, 0, -1], [-12, 2, -4], [3, 0, -3], [17, 3, -1], [-9, -1, -2], [-6, -4, 0], [-6, -2, -4], [1, -3, -5], [-5, -5, -3], [2, 1, -2], [0, 4, -4], [3, 3, -4], [10, 2, -1], [3, 11, -1], [-8, -2, -1], [8, -9, 0], [9, -7, -1], [8, 0, 0], [0, 2, 2], [17, -1, -3], [-10, 8, 2], [24, -3, -2], [-9, -11, 3], [-2, -7, 0], [-8, 7, -4], [-6, -4, 2], [1, -1, 2], [10, -6, 2], [-12, 10, 1], [-4, 0, 6], [9, 5, 0], [-2, 8, 5], [-2, 8, -5], [-7, 3, -2], [9, 9, -1], [22, 0, 2], [6, 4, -4], [1, -7, 1], [0, -2, -6], [-8, -2, -4], [-13, 1, 5], [9, 3, 1], [5, 3, -2], [22, -10, -4], [-22, 8, 4], [24, -2, 2], [2, -10, 2], [6, -8, -3], [-3, 7, 1], [-1, 1, -1], [-6, 0, 0], [4, -5, -2], [0, -6, 2], [-10, -3, -6], [12, -10, 0], [-8, 7, 4], [-21, 5, -4], [32, 0, 4], [2, 8, -4], [-21, -10, 1], [-2, 2, -1], [12, 6, 4], [-30, 2, 2], [-21, -3, 1], [6, -8, -4], [8, 4, -3], [2, 6, -2], [-2, 1, -8], [0, 2, -2], [7, 3, 5], [10, 4, -3], [-16, 12, 2], [2, 0, 7], [28, -3, -2], [23, -1, -2], [-11, -6, 3], [-17, -1, -3], [-9, -7, 7], [18, -10, -1], [14, -8, -2], [13, -1, 9], [-9, 1, -6], [-40, 2, 2], [-14, -2, -6], [-1, -1, -5], [-21, -9, 3], [43, -4, -1], [-5, 13, 1], [-10, 0, 0], [4, 14, -2], [-5, 7, -5], [23, -5, -3], [18, 2, -4], [-12, 4, 5], [-30, 4, 0], [7, -3, -7], [22, 8, 2], [2, 1, 6], [-16, 0, -4], [18, -4, -3], [-38, 2, -2], [-33, -7, 1], [8, -4, 10], [8, 0, 0], [3, -1, 5], [-24, -2, -5], [-12, -3, 2], [-22, -8, 3], [10, -11, -4], [-8, 10, 8], [4, -9, 6], [-15, 1, -4], [17, 0, 7], [-13, 14, 7], [-4, -6, 2], [28, -6, 0], [16, -3, -4], [-8, 8, 4], [23, -7, -7], [-2, -2, 0], [23, 2, 5], [8, 3, 8], [-4, 0, -2], [40, -2, -2], [19, -11, 3], [1, -6, 7], [-12, 6, -1], [-22, -4, 2], [0, -9, 6], [-24, 0, -1], [-22, 6, 0], [3, 6, -3], [12, 18, -3], [0, 0, -9], [-35, -1, 1], [-21, 0, 7], [27, -6, -5], [-46, -4, 2], [2, -10, 2], [-2, -10, 10], [37, -5, -3], [-36, 6, 2], [-11, 13, 10], [-26, -2, -4], [-6, -12, 0], [-10, 12, 4], [26, 10, -4], [-24, 1, -2], [-14, -8, -7], [10, -11, -6], [-14, 6, 10], [4, 4, 4], [-18, -8, -4], [-12, 2, -2], [-18, 3, -2], [-9, -15, -4], [16, -16, 2], [18, 6, 0], [10, 0, 4], [46, 2, -3], [8, -4, 6], [-35, -9, 5], [-16, -20, 6], [-14, 4, -2], [-23, 13, 7], [21, -1, -4], [22, 4, -4], [31, 1, 1], [14, -16, 0], [-1, 0, 7], [-21, -3, 0], [-34, -2, -6], [21, 7, -5], [24, -2, -1], [19, -5, 12], [-2, -6, -5], [3, 13, 7], [8, 8, 1], [-42, 9, 4], [22, -2, 10], [-9, -18, 3], [47, -15, -7], [-36, 4, 0], [12, 4, 6], [10, 6, 0], [37, -1, 3], [47, -8, -1], [29, 7, -5], [3, -5, -3], [14, 4, -8], [26, -4, -10], [0, 4, -2], [18, -12, -3], [4, -4, -10], [-12, 8, 5], [6, -9, -8], [-15, -14, -5], [-10, 4, 2], [2, 0, -6], [-27, -5, 0], [2, -10, -2], [34, 0, -4], [-7, 5, -3], [-32, 1, 2], [0, 14, -12], [12, 4, 1], [2, -20, 4], [13, -8, 7], [-9, -13, 1], [13, -13, 1], [25, -5, -5], [24, 6, -6], [-28, 3, -2], [36, 5, 6], [-6, -4, 0], [21, -3, 1], [10, 2, -3], [3, -7, 1], [-29, -7, 3], [5, -23, 2], [-22, 6, 10], [-24, 6, 2], [-14, -8, 2], [0, 13, 4], [7, 5, 9], [-8, -6, -2], [14, -26, -5], [-2, 10, 10], [-4, 4, 4], [-12, -15, 2], [-4, -14, -6], [12, 6, 3], [-17, -14, -5], [39, -5, -7], [-2, 12, 10], [24, 9, 0], [42, 5, 2], [-10, 16, 8], [23, 7, -5], [-28, -2, 8], [66, 2, 1], [-40, 5, -8], [-4, -11, 10], [38, -10, 4], [-2, -15, 2], [-13, -9, -7], [-34, -1, -2], [-33, 3, -4], [17, 11, 3], [28, 2, -8], [-36, -4, -4], [14, 14, -1], [4, 6, -8], [26, -5, 10], [8, -10, 7], [44, -12, -6], [-12, 1, -12], [16, 16, -10], [0, -14, 11], [-24, -6, 4], [16, 5, 4], [25, -1, 0], [-8, 4, 2], [-15, -11, -5], [-9, -21, 6], [7, 11, 13], [-26, 3, 2], [20, 1, -2], [21, -12, -1], [-10, -4, -12], [-6, -2, -16], [-39, 8, -1], [-24, -8, 8], [-14, -6, -7], [-4, -2, -19], [34, -10, 7], [11, 17, -3], [-19, -11, 9], [-42, 1, 2], [-31, 3, 5], [-52, 2, 3], [-4, 0, 8], [26, -12, 9], [-30, 17, 0], [-22, 20, -4], [14, 11, 2], [-17, 6, -11], [-38, 16, 4], [-54, -9, 4], [2, 2, -6], [-20, 20, 2], [23, -1, 17], [-33, 13, 11], [-33, -13, 1], [0, -2, 9], [-26, -16, 4], [7, 5, 5], [-15, -5, 3], [-24, 10, 10], [-17, -1, 7], [73, -8, -5], [36, 22, -6], [-10, 14, -8], [22, -4, -2], [-44, -4, -3], [-42, 4, 0], [50, 4, 8], [-37, 23, 7], [-43, -11, -5], [-26, 4, -6], [48, -25, -10], [-6, -7, 2], [22, -16, -12], [28, -11, -12], [-25, -11, 3], [8, 12, -6], [-1, 11, 1], [-49, 19, 7], [14, -12, -8], [16, 11, 14], [20, 4, -12], [4, -8, -7], [-27, 11, 7], [30, -4, 6], [3, 6, 5], [24, 8, -2], [-2, 14, 2], [10, 17, -8], [-74, 7, 0], [-24, 2, -4], [-90, 0, 0], [-32, 0, 4], [-10, 10, 14], [1, 1, 12], [50, 6, -7], [15, 19, 1], [-25, -7, -4], [22, 4, -4], [24, -16, 7], [-37, -11, 7], [16, 4, 8], [-38, 0, 8], [-3, -7, -12], [-2, 20, 8], [0, 12, -12], [-2, -10, -13], [-11, 17, 7], [9, -15, -11], [18, 24, -6], [65, 7, 1], [-19, -11, 0], [-17, 1, -5], [5, -20, -7], [-15, 11, 15], [0, 8, -5], [-56, -8, 2], [-12, -4, -4], [20, 6, 12], [14, 11, -14], [44, -8, 8], [16, -1, 2], [75, -13, -6], [62, 0, 8], [54, -16, -3], [-44, -4, -2], [13, -16, 15], [6, 6, 15], [26, -23, -16], [-22, -14, 4], [-58, 6, 0], [24, 8, 2], [12, 20, 6], [36, 6, 2], [-68, 20, 8], [13, -1, 0], [-4, -14, 0], [-13, -16, 7], [16, 6, 12], [18, 29, -12], [56, 0, 2], [5, -8, -5], [8, -30, 2], [-4, -14, 12], [21, 6, -3], [30, 18, -8], [-14, 19, -10], [-26, 0, 2], [10, -6, 14], [74, -10, -6], [-7, 11, 19], [8, -12, -8], [23, -10, -1], [-24, -16, 0], [18, -2, -10], [52, -26, -8], [33, 24, -7], [-34, 25, 12], [-6, 4, -19], [-58, 2, -8], [38, -25, -6], [-16, -16, 13], [-40, -2, 3], [43, 11, -5], [-9, 7, 1], [4, 6, -16], [23, -1, 11], [19, -26, 1], [45, -9, -5], [18, -18, -14], [-26, -2, -11], [5, 7, -9], [4, -1, -4], [-28, 0, 2], [44, -4, -2], [-21, -3, -15], [18, 10, -12], [-10, -5, 8], [-56, -16, 4], [-50, -2, 2], [-22, 12, -1], [32, 4, -13], [-37, 11, 15], [33, 5, -5], [45, -11, -11], [76, 0, -2], [-88, 0, -2], [34, 15, -10], [-4, -20, 16], [16, 6, 22], [15, 16, 13]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8280_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8280_2_a_bj();". // 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_8280_2_a_bj(:prec:=3) chi := MakeCharacter_8280_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(3457) - 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_8280_2_a_bj();". // 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_8280_2_a_bj( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8280_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![1, -8, -2, 1]>,<11,R![-14, 7, 7, 1]>,<13,R![-50, -23, 1, 1]>,<17,R![-83, 10, 10, 1]>],Snew); return Vf; end function;