// Make newform 3360.2.a.bj in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3360_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_3360_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_3360_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 := [1, -3, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [3, 4, -2], [-5, 0, 2]]; 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_3360_a();" function MakeCharacter_3360_a() N := 3360; order := 1; char_gens := [1471, 421, 1121, 2017, 1921]; 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_3360_a_Hecke(Kf) return MakeCharacter_3360_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, 0], [1, 0, 0], [1, 0, 0], [1, 0, -1], [1, 1, 0], [0, -1, -1], [3, 0, 1], [0, 1, -1], [0, -1, -1], [3, -1, 2], [0, 1, 1], [0, 0, 2], [2, 1, 1], [0, -1, 1], [-3, 0, 1], [8, 0, 0], [-2, -1, 1], [0, -1, -3], [1, -1, 0], [3, -1, 0], [4, 0, 0], [2, 0, 2], [0, 2, 0], [1, 1, 0], [-10, 0, 0], [0, 0, 0], [4, -1, 1], [-6, -2, -2], [3, 0, 3], [6, 2, 0], [6, 2, 0], [-7, -2, -1], [11, 0, 1], [-10, 1, -1], [6, 0, -2], [-5, -1, 0], [-10, -1, -1], [-10, 1, 1], [0, 0, -2], [17, 0, -1], [-16, -1, -1], [-1, 1, 0], [6, 2, 2], [-5, 2, 1], [15, -1, -2], [-4, -2, 2], [-2, -4, -2], [6, 4, 2], [-2, 3, 5], [-5, -2, -3], [1, -1, 0], [4, 0, -2], [-4, -2, -2], [10, 1, 3], [6, -3, 5], [-10, 2, 6], [5, 1, -2], [-20, -1, -1], [0, 0, 2], [-8, 0, -4], [-16, -2, 0], [-6, -2, -4], [-4, 0, 4], [17, -1, 2], [-7, -4, 1], [-20, 2, -2], [10, 2, -2], [2, 1, 1], [-2, -3, 3], [-4, 1, -7], [7, 3, -2], [18, -2, 0], [2, -1, -7], [10, 4, -2], [-14, 1, 5], [-26, -1, 1], [-11, 3, -6], [22, -2, -2], [6, 6, -2], [10, 2, 4], [10, 0, -4], [21, -1, -4], [-11, 3, 2], [11, 1, 0], [-18, 1, 5], [2, 2, -2], [12, -2, 0], [-18, 2, -2], [-2, 4, -2], [18, 2, 0], [8, 4, 4], [-14, 0, -2], [-7, 4, 3], [-22, -4, -2], [-22, 3, -5], [6, -2, -6], [-8, -2, 4], [-2, -2, 0], [14, 2, 6], [-16, -5, 1], [-7, 0, 5], [-16, -4, 0], [2, 2, 6], [-34, 0, -2], [-15, -1, 2], [-24, -4, 0], [4, 3, 7], [11, -3, 0], [0, -2, -4], [-10, 0, -6], [4, -3, 1], [7, 2, -3], [11, -2, 3], [0, -6, 2], [-4, 2, -4], [-14, 4, -2], [-10, -1, 3], [7, 4, 3], [3, 2, 3], [-22, 3, -7], [0, -4, -2], [4, 4, -2], [-18, 3, -5], [-11, 4, -5], [4, -1, -5], [34, 2, 2], [-22, 0, -2], [-6, 2, 4], [29, -3, 0], [-14, 2, 0], [-34, 1, 1], [-4, -4, -4], [-10, -1, 5], [0, 0, -6], [8, 4, 6], [4, 2, -8], [-18, 4, 2], [0, 0, -2], [-4, -2, 8], [3, -4, 5], [12, 5, 5], [-26, 0, -6], [-4, -3, 3], [16, -3, -7], [-30, 0, -2], [-23, 3, -2], [-10, -3, 3], [21, -2, 1], [-8, 3, 5], [-8, -1, -5], [-16, -2, -4], [0, 5, -1], [-16, 5, 3], [-24, -3, -1], [-13, 5, 0], [32, 0, -4], [-8, 0, -6], [-41, 1, 2], [-10, -8, 0], [-2, 1, -11], [-21, 0, -5], [-12, -6, 2], [4, -4, 8], [15, 2, 5], [-8, -7, -1], [24, 0, 4], [-3, 3, 2], [-2, 2, 2], [-4, 4, 6], [21, 0, 3], [-28, 1, 1], [5, 1, -6], [-14, 2, -2], [29, 7, 0], [-28, 0, -2], [28, -2, -6], [-18, -5, -3], [-34, -6, 0], [36, 3, -1], [0, 0, -8], [0, 2, -2], [26, -5, 5], [-4, -5, -1], [26, -3, 1], [-34, -4, -4], [-40, 1, 1], [-6, 2, 0], [10, 2, -2], [-38, 0, -2], [1, 1, -8], [-12, 3, 5], [-10, 4, 2], [18, 6, 6], [42, -3, 5], [-8, 7, 7], [14, -2, 6], [6, 1, 3], [-2, -7, 7], [2, 3, -5], [4, 3, 7], [19, 7, 2], [-29, -7, -2], [-4, -2, 8], [-4, 4, 0], [-4, 6, 4], [-11, 7, 0], [12, 1, -1], [8, 0, -6], [-3, 2, 13], [8, -4, -10], [10, -2, 14], [12, 8, -4], [10, -4, -10], [2, -8, -2], [24, 4, -2], [-6, -4, 10], [20, 12, 2], [46, 1, 1], [11, -4, -9], [18, -6, -6], [43, 1, 0], [4, 2, 4], [24, 6, 2], [-4, -2, -6], [26, 0, 4], [-2, -7, -1], [3, 3, -6], [-24, -2, 2], [1, -8, -1], [30, -3, -1], [-1, 0, -3], [-6, -6, 0], [8, 4, -10], [-14, -6, 0], [24, -5, 5], [6, -10, 6], [-1, -2, -7], [33, 4, -5], [-14, -6, -4], [24, 5, -1], [15, 2, 3], [-16, 4, -4], [22, 0, -8], [-38, -1, -3], [-2, -6, 0], [46, -2, 4], [-22, -4, -6], [-16, 4, -8], [4, -9, 5], [-8, 3, 7], [0, 4, 14], [8, 1, -9], [22, 0, 4], [40, 2, -4], [21, -2, 5], [-30, -2, -2], [28, -4, -4], [28, -6, -8], [-35, 3, 2], [18, 8, -2], [6, 1, -3], [14, -7, -1], [-25, -5, 0], [20, -3, -11], [-19, 4, -5], [36, 3, 7], [4, 6, 0], [36, 1, 7], [15, 0, -9], [-24, 1, -3], [-42, -1, -1], [41, 5, 4], [28, 6, 2], [-7, -7, 8], [-14, 4, -6], [12, -9, 1], [54, 6, 2], [26, 0, 0], [4, -8, -4], [12, 1, -5], [-4, 0, 16], [8, -1, 1], [20, -1, 11], [42, -2, -4], [11, -1, 6], [-16, 6, 12], [29, 0, 9], [27, -1, 10], [20, 4, -6], [-12, -9, 3], [-6, 6, 12], [-53, -2, -3], [10, 4, -6], [54, 5, -1], [-10, 10, -2], [-7, -1, -12], [-26, -6, -2], [48, 2, -2], [2, 0, -6], [15, 5, -2], [7, 8, -1], [-10, -12, 2], [10, -3, 5], [-2, 4, 2], [-34, 0, 4], [0, -3, 15], [2, 5, -9], [21, 3, -8], [-18, 3, 9], [20, -7, -13], [-44, -1, 3], [46, -6, 10], [8, 1, -13], [-16, -1, 1], [4, -6, -4], [-26, -8, 2], [43, -3, 0], [17, -7, 0], [-6, -4, -4], [13, -6, 5], [-16, 8, 10], [6, 2, -10], [12, -4, 0], [9, 8, -3], [-8, 4, -10], [-62, 0, 2], [40, 6, 6], [24, 5, 11], [9, 6, 7], [18, -10, 6], [3, -4, -1], [53, -5, -4], [-52, 6, -6], [-24, 2, 10], [-48, -2, -2], [10, -10, -10], [6, -5, 5], [48, 2, 8], [-42, 4, -2], [10, -5, -11], [27, -6, 9], [6, -2, -14], [2, 2, 8], [-33, -8, -1], [-5, 0, -19], [-30, 7, 13], [8, -5, -7], [6, 10, 0], [-8, -2, 8], [-5, 6, 11], [-76, 2, 4], [12, -7, 9], [12, -4, 16], [-2, -4, -4], [16, 13, 1], [-58, 4, -10], [-14, 4, 2], [55, 8, -1], [2, -1, 7], [26, -5, -11], [-24, 6, -4], [14, -9, 3], [7, 2, -9], [-18, -4, 2], [-8, 2, 0], [26, 2, 10], [-28, -8, -4], [-30, -4, -4], [-15, 6, 17], [-19, 6, 17], [6, 3, -1], [-50, -1, -7], [-51, -3, -6], [-56, -1, -5], [14, 2, -8], [-33, 1, -8], [5, -1, -18], [-16, 6, -4], [-29, -5, -4], [-54, -4, -8], [-45, -8, -5], [2, 0, -10], [-7, 0, -11], [15, 8, 5], [14, 5, 5], [28, -6, -18], [-55, -3, -4], [48, 7, 5], [-28, -11, -1], [-22, 10, -2], [18, 4, 0], [-1, -2, 11], [-14, -6, 8], [41, 3, 4], [38, -4, -8], [41, -3, -2], [36, -6, 4], [38, -6, 4], [-36, 1, -7], [-20, 3, 7], [-37, 4, -1], [8, -8, 8], [-14, 3, 9], [2, 4, 16], [-59, 5, 2], [8, 7, 3], [38, -6, -14], [40, -2, -2], [20, 4, 8], [23, 5, 6], [-7, -2, -1], [-50, 1, -3], [-36, -4, 4], [-18, 2, -14], [6, -6, -2], [-39, -5, 4], [14, 12, -2], [44, 5, 5], [28, -3, 7], [18, -5, -15], [-43, 3, 10], [10, 1, -3], [-32, -4, -4], [17, -9, 10], [-8, 6, -8], [-24, 7, 5], [34, 4, 4], [21, 2, -11], [-30, 4, -6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3360_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3360_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_3360_2_a_bj(:prec:=3) chi := MakeCharacter_3360_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_3360_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_3360_2_a_bj( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3360_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![32, -16, -4, 1]>,<13,R![104, -36, -2, 1]>,<17,R![40, -52, -2, 1]>,<19,R![32, 0, -8, 1]>,<23,R![-128, -64, 0, 1]>],Snew); return Vf; end function;