// Make newform 6024.2.a.j in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6024_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_6024_2_a_j();". // 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_6024_2_a_j();". // 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], [-2, 0, 1], [2, 2, -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_6024_a();" function MakeCharacter_6024_a() N := 6024; order := 1; char_gens := [4519, 3013, 2009, 4273]; v := [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_6024_a_Hecke(Kf) return MakeCharacter_6024_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], [0, 0, 1], [-3, 0, 0], [0, 1, 1], [-2, 1, -1], [2, -2, -1], [1, 0, -1], [0, 0, -1], [-1, -1, -2], [-1, -2, 0], [-3, 1, 2], [-2, 3, 0], [5, -2, 1], [4, -3, -1], [-7, 2, 1], [-5, -2, -1], [1, -2, -1], [1, 5, -3], [-11, 0, -1], [-7, -4, 2], [-1, 1, 1], [0, -2, -1], [-2, 3, -3], [-3, 6, 1], [-4, 5, -2], [-2, -1, 3], [14, 1, 1], [0, -3, -1], [2, 2, 1], [-1, 8, 1], [-2, -7, 2], [-1, -6, -1], [7, -3, 4], [0, 4, 2], [-6, 7, 3], [-18, 0, -2], [-6, 7, 1], [-8, 1, 1], [-10, 2, -1], [2, 5, -2], [-2, -9, 1], [-9, 2, -3], [-15, -2, -3], [-16, 1, 0], [2, -6, 0], [4, -1, 1], [1, 7, -2], [8, -6, -5], [-2, -3, -1], [20, -2, 0], [-5, 3, 6], [-7, 10, 4], [1, 0, 0], [-1, -6, 1], [-2, -8, 2], [-4, -11, 3], [9, -5, 1], [0, 0, 2], [7, -5, 4], [3, 3, 4], [3, 2, 1], [-7, -3, -6], [8, 1, 4], [-4, 3, 3], [14, -1, 1], [1, 3, -5], [-7, 1, 3], [-2, 1, 9], [10, 0, 4], [7, 7, 2], [-16, 14, 3], [10, -12, -2], [-9, -4, 2], [-27, -2, 3], [-3, 15, -2], [0, -10, -4], [-11, 1, 2], [11, 5, 0], [5, -13, -2], [-14, 6, 2], [0, -6, 2], [14, -4, 2], [32, 1, 1], [18, 2, -4], [-12, 7, -4], [6, 1, -5], [-31, -3, 2], [3, 1, -8], [0, -11, 5], [13, -10, -7], [-7, -3, 4], [-3, 6, -3], [20, 4, -7], [16, -13, -5], [-6, -9, -2], [-2, 4, 4], [-10, -2, -2], [19, -6, -2], [-21, 2, -2], [13, 3, 9], [16, 6, -6], [-7, 5, -4], [-2, 9, 4], [-1, 12, -2], [-7, -10, 6], [2, 10, 3], [-6, 13, 10], [-1, 4, 3], [8, -2, 0], [-1, -4, -10], [4, 1, 3], [-14, -3, 3], [-5, -9, -9], [9, -7, 4], [17, -17, -6], [-4, 0, 4], [11, 3, 0], [2, -7, -3], [24, 4, -4], [-11, 14, -5], [25, -2, -1], [-22, -2, -9], [6, 7, 0], [2, -5, -1], [-22, 1, 3], [-1, 7, 3], [10, -8, -6], [-11, 3, -7], [-29, -6, 3], [-12, -4, -4], [-6, 0, 10], [-8, 5, -1], [3, 12, -2], [-3, 5, 12], [-18, 7, -3], [16, -16, -5], [7, 5, 16], [-26, 7, 1], [12, 12, 0], [8, -11, 3], [-14, -20, 9], [1, -5, 4], [0, 6, -2], [-11, 0, -1], [-16, -6, -3], [-15, 12, 4], [-1, 13, -6], [5, 12, -6], [-12, 15, 12], [4, 3, 3], [-2, -12, 12], [-4, 4, 16], [-2, 0, -2], [27, -15, -1], [28, -3, -5], [-6, 9, -1], [15, 4, 3], [-34, -7, -1], [-16, 3, -13], [-2, -21, 6], [2, 7, 9], [-21, 16, 8], [2, 14, -12], [2, 10, -10], [28, 5, 7], [-1, 7, 0], [2, 6, 20], [-19, 5, -5], [14, 2, 14], [-12, -2, 14], [-27, 9, 9], [-32, 2, 7], [34, 3, 1], [-27, -1, -9], [-24, -11, 10], [-14, 1, 11], [6, -5, 13], [-4, 14, 4], [23, -13, -3], [23, -16, -8], [22, 5, 7], [5, 7, -13], [4, -17, -8], [11, -5, 4], [6, 4, -5], [19, -6, 6], [13, -19, 3], [-44, 7, 3], [4, 4, -12], [19, 2, 14], [44, -3, 3], [-17, -17, 8], [-4, 26, -2], [5, -1, 2], [2, -4, -5], [-1, 7, -17], [55, 4, -2], [-5, 1, 16], [14, 9, 17], [8, -16, 2], [-21, 6, 12], [5, 4, 16], [-2, 0, 16], [-22, -5, -6], [-24, 0, -13], [6, -13, -13], [12, -14, -18], [-24, 4, 2], [1, 1, -1], [-24, -13, 1], [13, 7, -2], [5, -12, -14], [12, 10, -9], [8, -15, 8], [21, 13, -11], [13, 7, 8], [-4, -1, 8], [24, 2, -2], [36, -1, 12], [-22, 3, -1], [-33, 1, 3], [36, 0, -3], [-3, -7, 14], [64, -6, -2], [19, 13, -1], [-26, 21, 1], [-3, -5, -12], [-11, 1, -19], [16, -5, 8], [21, -11, -15], [-16, 21, -5], [6, 20, 0], [26, 13, 9], [10, 1, -17], [-4, -4, -9], [-22, -10, 6], [19, -5, 0], [-32, -14, 0], [-6, 17, -4], [-2, 20, 10], [9, -25, 1], [-3, -6, 19], [21, 20, -3], [39, 0, 7], [-50, 1, 1], [-11, -25, 2], [10, -5, 6], [-10, -11, -3], [1, -10, 3], [37, -2, -4], [0, -15, -15], [0, -5, -14], [12, -9, -9], [22, -29, -7], [-18, -1, -9], [-7, -15, -11], [27, 12, 12], [-14, 6, -9], [-5, 8, -11], [38, -10, -6], [22, -8, 10], [11, 14, -19], [16, -10, 6], [-22, -18, 12], [-16, 11, 3], [-45, -3, 0], [-33, 14, 11], [1, -5, 1], [34, -19, -9], [-28, 28, 12], [-58, 13, 5], [9, 9, 2], [-38, 15, 1], [33, -19, -13], [-39, -10, 5], [-41, 3, 4], [-3, -5, -14], [-7, 2, -13], [-3, 15, -2], [-26, -12, -14], [-20, -3, -1], [0, -1, 1], [38, -27, -5], [-1, 5, -11], [8, -6, -19], [-25, 17, 0], [0, -17, -3], [28, 4, 12], [23, 10, 3], [-22, 9, 7], [-4, 14, 12], [-26, 5, -6], [-44, 18, 11], [-1, -29, 10], [-28, 1, 16], [0, 0, 8], [58, -6, 2], [28, -16, -17], [35, -6, 11], [-8, -4, -4], [-30, 10, -1], [-9, 0, -11], [-20, -10, -2], [23, -14, -12], [-61, 16, 2], [7, 20, 7], [-43, 20, 8], [38, 12, 3], [7, 7, 7], [-5, 4, -21], [68, 2, 4], [30, -19, 6], [-2, 7, -29], [-18, -13, -14], [41, 7, -9], [4, -6, 3], [18, 8, 6], [-6, 1, -13], [-28, -7, 4], [25, 0, -4], [-28, 11, 13], [30, -9, 15], [16, 18, -18], [-1, 20, -1], [-7, 17, 9], [-20, 4, 18], [11, 2, 4], [6, 4, -16], [26, 5, -10], [24, -9, 1], [-25, -9, 12], [-21, 23, -6], [45, 8, 7], [13, 18, 12], [3, 8, -3], [9, 17, -2], [41, 0, 17], [-13, 9, 15], [63, -16, -4], [27, -17, -12], [4, -8, 6], [-27, -5, 12], [14, -19, -11], [14, -25, 6], [-46, -3, 13], [-3, 36, -7], [8, -11, 7], [-2, 22, 0], [9, -9, -10], [15, -33, 7], [-14, 11, 5], [38, -6, -7], [-41, -4, 12], [0, -12, -4], [-34, 31, 7], [-52, 6, 7], [-22, 8, 14], [3, 19, -14], [-8, 34, -1], [4, -25, -17], [-20, -13, 21], [-18, 16, -16], [-29, 4, 11], [15, 9, -11], [-17, 1, 9], [-38, -2, -6], [5, -5, -10], [6, -25, -7], [36, 24, -15], [-4, 3, -7], [9, -8, 27], [16, 4, 6], [1, -14, -20], [-10, 1, -21], [-30, -10, -8], [-60, 0, 3], [-26, 9, 11], [-15, 17, -16], [-57, 13, 12], [-5, 4, 7], [2, -15, -6], [4, 13, -1], [-32, 19, 8], [-54, 5, 9], [14, 20, -10], [-29, 19, 9], [-46, -5, -1], [1, 0, -12], [8, -13, -5], [24, -33, -9], [-15, 22, -11], [-30, 17, -14], [43, 3, 15], [0, 22, -6], [17, 11, 2], [5, -24, -12], [46, -5, 16], [19, 16, 2], [-17, 10, -5], [-13, 24, -15], [24, -23, -6], [-15, 10, -15], [24, 11, -24], [-22, -9, 21], [-41, 2, -5], [-38, 15, -15], [-28, 0, 18], [-50, 0, -6], [60, 7, -6], [-49, -19, 4], [16, -7, -5], [55, -24, -5], [54, -6, -6], [-2, 14, 8], [-6, -3, -14], [30, -3, -3], [13, -19, -10], [-59, 8, -3], [-26, -37, 18], [-13, 6, 13], [-27, -3, -5], [-12, -14, 0], [-20, -14, 8], [35, -25, -11], [-18, -23, 10], [-54, 19, 9], [0, 6, 11], [0, 11, 31], [-10, -2, -14]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6024_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6024_2_a_j();". // 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_6024_2_a_j(:prec:=3) chi := MakeCharacter_6024_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_6024_2_a_j();". // 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_6024_2_a_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6024_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![13, -9, -1, 1]>,<7,R![3, 1]>],Snew); return Vf; end function;