// Make newform 6048.2.a.bq in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6048_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_6048_2_a_bq();". // 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_6048_2_a_bq();". // 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 := [9, -6, -10, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [9, -1, -3, 1], [9, -7, -3, 1], [0, -7, -1, 1]]; Rf_basisdens := [1, 3, 3, 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_6048_a();" function MakeCharacter_6048_a() N := 6048; order := 1; char_gens := [4159, 3781, 3809, 2593]; 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_6048_a_Hecke(Kf) return MakeCharacter_6048_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, 0, 0], [1, 0, 1, 0], [-1, 0, 0, 0], [-1, -1, 0, 0], [-1, 1, 0, -1], [1, 0, -1, 1], [0, 0, -1, -1], [-2, 0, 0, 1], [-1, 1, 0, -1], [-3, -1, -1, 0], [-2, 1, -2, 1], [0, -1, -1, 1], [-1, 1, -2, 1], [-1, 1, 0, 1], [4, -2, 2, 0], [-5, -1, 2, 1], [-2, 0, 2, -2], [-1, -1, -2, -1], [-3, 0, 1, -2], [-1, -1, -4, 1], [1, 3, -2, -1], [-5, 1, 0, -1], [6, 1, -1, -1], [4, 0, 4, 0], [1, 0, 3, 1], [-1, -3, -1, 2], [-11, 0, -1, -1], [-3, -1, -1, 0], [-14, 0, -2, 0], [-1, -1, 2, -1], [-7, 1, 2, -1], [-1, 1, 0, 3], [0, 2, 2, 0], [-4, 2, 2, -2], [4, 0, 2, -2], [-7, 1, -6, 1], [5, 3, 0, 1], [-14, -2, -4, 0], [-2, 0, 0, 1], [-3, 0, -1, 3], [-6, -4, 0, 0], [0, -2, 4, 1], [10, 2, 2, 0], [0, 2, 2, -2], [-8, 1, 2, -3], [0, 2, 4, -2], [4, -1, 2, -1], [-4, 2, 2, -2], [-14, 0, -2, 2], [12, 2, 2, 2], [-7, -2, 1, 3], [-5, 1, -6, 1], [2, 0, 6, 2], [-1, 3, 0, -2], [-8, -4, 2, 3], [4, 2, -4, -3], [0, -2, 2, 4], [-2, 1, 6, 1], [-11, 1, 2, -1], [6, -2, 0, 2], [5, 2, 5, 1], [10, 0, 1, 1], [-7, -1, -8, 1], [21, -1, 0, 1], [7, -3, -4, 1], [-20, -2, -2, 0], [1, 2, 0, -2], [-8, 5, 5, -5], [0, 2, 0, -2], [-12, -1, -1, 3], [-9, 2, -1, 1], [8, -1, 4, 1], [-17, -1, -3, -2], [3, 1, 2, -3], [-8, 2, -2, -2], [-10, -4, 2, -2], [4, 0, -2, 6], [5, -1, 0, 5], [13, 1, -4, -5], [4, -4, 8, 2], [-14, -1, 4, 1], [2, 0, 0, -1], [-1, -5, 2, -1], [8, -4, 4, 0], [-7, 1, -8, 0], [-2, -4, 2, 4], [5, 0, -8, 0], [6, 6, -6, -3], [4, 2, 0, -2], [-13, 5, -4, -3], [8, 0, 0, -4], [-17, 3, 0, -3], [4, -3, 1, 3], [-3, -3, 8, 3], [-3, 1, -6, 1], [-3, -2, -7, 1], [11, -3, 6, -4], [11, 2, -2, 4], [-10, -1, 0, -3], [-18, -2, -4, -2], [10, 0, 2, 0], [3, -3, 4, 3], [10, 2, 8, -4], [12, 2, -4, -2], [6, -2, 2, 4], [-6, 0, 6, 0], [33, -1, -2, 2], [-4, 4, -6, 1], [13, 1, -2, 1], [-24, 2, -6, 0], [-17, 1, 5, -4], [-25, -1, 2, 1], [5, 6, 2, -4], [-17, -1, 0, 1], [2, 2, 4, -4], [-2, -8, 5, 5], [-23, 1, -6, -1], [-19, -7, -2, -1], [-13, -5, 0, 0], [-23, -1, -4, 1], [2, -2, -6, 4], [13, -4, 5, -2], [13, -7, 12, 0], [2, 4, 6, 2], [-11, 1, -10, 5], [-19, 1, -3, -4], [9, -3, 2, 7], [18, 2, -4, 2], [9, 3, 2, -5], [26, 2, -2, 0], [17, 2, 3, -2], [9, -3, 12, -5], [-8, 6, 4, -2], [3, -6, 11, 3], [15, 5, -4, -5], [-1, -4, 3, 4], [-2, 4, -10, 2], [-11, -2, 3, -10], [1, -3, -6, 5], [-13, 2, -4, -6], [1, 3, -8, 3], [-18, 0, -4, -4], [20, 9, 1, 1], [1, 1, 4, 3], [3, 1, 0, -1], [-8, 0, 0, 4], [-6, -1, -3, 3], [22, -4, -3, 1], [5, 2, 17, -1], [12, 0, -6, 2], [20, 3, 3, -1], [2, 4, -2, -2], [-1, -1, 6, -5], [10, 2, 14, -4], [1, 2, -11, 3], [4, -8, 4, 8], [15, 6, -5, 3], [-2, 0, 2, -10], [-5, -2, 9, 6], [24, 5, 1, 5], [-9, 1, 4, -5], [4, -8, 10, 2], [15, 5, -4, 7], [-14, 2, 4, 2], [25, -1, -4, 3], [-3, -3, 6, 1], [2, 0, -4, -4], [0, -4, 5, 1], [-2, 0, -8, 5], [7, 8, 5, -3], [-9, 1, -6, 1], [13, 2, 1, -1], [2, 0, -7, 1], [-12, 1, 2, -3], [7, 1, -10, 0], [-14, 0, 2, 2], [24, 4, 16, -2], [-17, 7, 7, -8], [2, -2, 8, 2], [12, 0, -6, -2], [14, -2, 12, -4], [-14, -1, -6, -1], [13, -2, -11, 1], [21, 0, 5, 0], [-22, -4, 0, -7], [12, -2, 4, -2], [-8, 0, 12, -4], [-12, 2, -3, 3], [0, 2, 14, -4], [-13, -3, 8, -5], [-15, -2, -3, -9], [0, -4, 10, -6], [37, 4, -5, -3], [9, -1, -10, -6], [-30, -5, -3, 1], [-12, -6, -9, 5], [4, 2, -4, 2], [21, -1, -2, 4], [-12, -4, -8, -6], [-21, 1, -4, 7], [24, -3, -8, 7], [3, -3, 2, -11], [-6, -4, 14, 2], [25, -3, 6, -3], [-9, 6, -9, 3], [-34, 2, 4, -6], [4, 3, -9, 5], [-19, 1, -6, 7], [14, 6, -8, -6], [-17, -2, -4, 6], [11, -2, 1, 6], [5, 1, 8, -9], [7, -1, -6, 9], [-2, 8, 2, -2], [2, -10, 8, -2], [1, -5, 0, 9], [-13, -1, 4, -4], [2, -2, -2, 5], [-26, 8, 2, -6], [8, 0, 10, -6], [22, 4, 6, -6], [1, -5, 12, -7], [35, 5, -6, -3], [-16, 0, -8, 2], [-28, -3, -4, 3], [-3, -7, -4, -4], [27, 2, -3, 2], [11, 1, 3, 2], [40, 1, 1, -5], [-7, 7, -13, -4], [40, 2, -5, 1], [23, 7, 2, -1], [-2, -8, 6, -4], [-26, -6, -1, 5], [41, -3, 6, 1], [-6, 2, 10, 0], [25, 5, -6, -3], [7, -2, 7, -7], [33, 5, -2, 1], [-12, 7, 7, -5], [1, -6, 6, 4], [-22, -3, -10, 5], [28, -3, 2, 1], [21, -3, 0, 4], [1, 11, -8, -1], [7, 3, -3, -6], [23, -1, 2, -5], [-7, 3, -16, 1], [21, 4, -7, -4], [34, 0, -4, 4], [19, 2, 3, -1], [9, -8, 7, 7], [-18, 2, 10, -4], [1, -4, 1, 2], [-30, -1, -3, -1], [-18, 1, -20, 7], [24, -10, 7, 1], [-9, -2, -9, 7], [19, -7, 6, 5], [11, -5, 12, 5], [-23, 3, 6, 4], [-3, 5, 6, -3], [25, 5, -8, -1], [-27, -1, -6, 2], [3, -6, 4, 2], [-19, -13, 0, 5], [31, 2, -1, -1], [28, 4, -2, -6], [-31, 5, -10, 1], [29, 9, 0, 3], [14, -6, -4, -2], [37, 1, 14, -3], [13, 3, 8, 1], [-37, -5, -4, -7], [-35, -7, 7, 6], [-24, -1, -1, -11], [34, -8, 8, 0], [-12, -4, 2, -2], [15, 1, -4, -1], [-11, 9, 2, -5], [-2, 0, -4, 8], [-34, -2, 8, 4], [29, -7, -4, 7], [3, 2, -10, 4], [7, 8, -9, 0], [-49, 6, -4, -2], [-45, -5, -6, -3], [-9, 3, -1, -4], [48, -7, 13, -3], [-12, 4, 0, -14], [-49, 3, 6, -7], [45, 3, 0, -3], [14, 8, -2, -4], [21, -3, 13, 0], [-18, 6, 10, -3], [-16, 1, -4, -1], [-5, 0, -15, -5], [-17, 7, 6, -13], [-23, 12, -18, 2], [-32, -4, -2, 2], [19, -1, 10, -1], [12, -8, 10, 2], [-4, 3, -9, 13], [6, 2, -2, 16], [-37, -10, -10, 0], [-37, 1, 6, -6], [-29, -7, -6, -3], [-27, -6, -9, 6], [-16, -9, 0, -3], [20, 10, -6, -6], [-33, 3, -2, 1], [8, 2, -6, 0], [-5, -9, 14, 3], [-1, -10, -3, 10], [-12, 4, 12, 0], [-19, -3, -2, 5], [32, -4, -14, 3], [-32, -2, 2, 4], [-11, -9, 8, 7], [-22, -4, 6, 10], [-6, 6, 1, -5], [-16, -8, -10, -1], [35, 1, 2, 5], [2, 0, 6, -10], [-12, 10, -2, 0], [-20, 6, -8, 10], [-12, -6, -1, -3], [-13, 2, -5, -3], [-28, 6, -10, -6], [-15, 5, 3, -2], [20, 10, -18, -10], [25, 1, -3, 12], [41, 9, 2, -3], [15, -5, -6, 3], [8, -14, 2, 8], [-1, -7, -11, 4], [21, 1, 4, 8], [17, -3, -12, 15], [-13, 9, 5, -4], [-19, 9, -3, -12], [-9, -3, 0, 7], [39, -4, 9, -9], [27, 3, 2, -1], [-21, -9, 2, -7], [-7, 10, -7, -9], [-7, -3, 4, 11], [-25, 3, 6, 3], [3, -11, 16, -1], [29, 0, -9, 13], [22, -4, 3, 7], [-40, -4, 2, -10], [-28, -12, 4, 0], [-16, 1, 0, 3], [3, 3, -7, 6], [4, 5, 9, 3], [-27, -5, -4, 7], [-48, -2, -6, 4], [19, -1, 18, -7], [-55, -10, 1, 3], [14, 8, -4, 12], [7, -7, 2, -6], [-27, -4, -1, -5], [-1, -1, -8, 0], [48, -2, 6, 0], [14, 4, 3, -1], [27, 0, 7, -6], [-20, 4, 2, -2], [24, -2, 13, -9], [-37, 3, -12, 2], [-6, 2, 7, -11], [-21, -6, 3, 9], [-23, -3, -18, 7], [-1, 1, 21, -4], [24, -1, 6, 3], [-31, 11, 16, -17], [11, 8, 5, -3], [10, 2, -2, 4], [-33, 3, 4, -12], [-45, 1, 0, -5], [12, 0, 2, 3], [-12, 6, -2, -10], [-51, -13, -4, 1], [5, 15, 0, -1], [-45, -6, -2, 4], [-57, 5, -4, 1], [20, -2, 4, -6], [-10, 2, 0, -10], [-30, -4, 10, -6], [-38, 8, -14, 2], [26, 2, -6, 8], [33, 0, 1, 4], [26, -3, -1, -1], [42, 12, 2, -2], [19, 14, -9, -7], [33, 2, -8, -10], [8, 7, 12, 1], [24, -1, 11, -1], [23, -5, 4, 5], [-23, -7, -18, 5], [7, -5, -4, 1], [7, -3, 0, 7], [22, 6, -18, -4], [60, 5, -3, 5], [-9, 12, -1, -14], [-38, -11, -6, 1], [26, -1, -8, 5], [15, -9, 6, -1], [25, -6, -4, 10], [12, 6, 22, -8], [-18, -10, -4, -2], [13, -7, -14, -1], [29, -8, 15, -9], [-32, -8, 4, -4], [-56, -8, -2, -2], [5, 4, 9, -8], [-35, 8, 13, -16], [-4, -10, 14, 0], [-23, 6, 9, -7], [-12, -2, -18, 2], [18, -8, -6, 6], [8, 10, 6, 4], [-54, -14, -2, -1], [-49, 3, -6, 5], [-12, 0, 10, -2], [-1, -2, 7, -1], [11, -6, 23, 5], [-10, -16, 10, 6], [-15, -8, -8, 8], [21, 1, 2, 1]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6048_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6048_2_a_bq();". // 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_6048_2_a_bq(:prec:=4) chi := MakeCharacter_6048_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_6048_2_a_bq();". // 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_6048_2_a_bq( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6048_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![27, 10, -12, -2, 1]>,<11,R![-73, -114, -32, 2, 1]>,<13,R![-32, -176, -40, 4, 1]>,<17,R![48, 80, -32, -4, 1]>],Snew); return Vf; end function;