// Make newform 2736.2.a.be in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2736_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_2736_2_a_be();". // 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_2736_2_a_be();". // 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 := [10, -8, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-6, 1, 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_2736_a();" function MakeCharacter_2736_a() N := 2736; order := 1; char_gens := [1711, 2053, 1217, 1009]; 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_2736_a_Hecke(Kf) return MakeCharacter_2736_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, -1, 0], [1, -1, -1], [3, 1, 0], [-2, 2, -2], [-3, 1, -2], [1, 0, 0], [2, -2, 0], [-2, 0, 2], [6, 0, 0], [4, 0, 0], [-4, -2, 2], [3, 1, 1], [3, -1, -2], [-2, -4, 2], [-4, 4, -4], [1, -3, 1], [0, 0, 4], [0, 0, 4], [5, 1, -3], [8, -2, 2], [6, 0, -2], [-2, 0, -2], [6, 4, 0], [-4, 2, -2], [2, 4, -4], [2, 2, 4], [0, -4, 0], [2, 4, 2], [-2, 0, -4], [-9, 1, 4], [-9, 3, 2], [5, -1, -1], [-3, 3, 0], [8, -2, 2], [2, 0, 0], [0, -4, 4], [-10, -2, 0], [-2, 4, 2], [10, -2, -4], [-2, -2, 6], [-3, 1, 2], [8, -2, 2], [-20, 2, -2], [3, 1, 1], [-8, 4, 0], [-4, 6, -2], [-10, -2, 0], [-1, -1, 3], [-5, 7, -6], [1, -7, 6], [-4, 6, -2], [-19, 3, -4], [4, 2, -2], [17, 1, -2], [-6, 0, 2], [-4, 8, 0], [-13, -5, -1], [-2, -4, -6], [15, 5, -3], [8, -2, 6], [-8, 4, 0], [-9, 3, -2], [2, 8, 0], [8, -2, 2], [4, -8, 4], [2, -4, 0], [7, 5, 0], [-11, 5, -7], [0, -4, -4], [-7, -3, -6], [-8, -4, 4], [4, 8, 0], [-2, -2, -2], [18, -6, 4], [3, 9, -4], [-11, -3, 1], [20, 6, -2], [2, 0, 8], [10, -4, -2], [-14, 6, 6], [-22, 2, 0], [-2, 0, 4], [-2, 4, 8], [-5, 9, 0], [-8, -2, -2], [3, -9, -5], [-19, 3, 0], [9, 3, -5], [1, -1, -12], [10, -6, 4], [-12, -2, 2], [14, 4, -6], [3, -7, 1], [-6, -2, 8], [-6, 4, -6], [-8, -6, -2], [-14, -2, 6], [-1, -5, 7], [-18, 6, -6], [-13, 1, 4], [-12, -8, 4], [18, 0, -2], [-8, 4, -4], [1, 5, 9], [5, -1, 0], [0, 0, -8], [16, -8, 4], [28, -2, 2], [26, 0, -4], [-1, -5, -1], [3, 3, -10], [28, 0, 0], [-3, -9, 7], [-18, 0, 2], [-5, 9, 1], [23, -1, 2], [-17, -11, 4], [18, -6, -4], [8, -4, 4], [12, -6, -2], [6, 4, 6], [6, -6, 12], [-11, -1, 7], [-12, 10, 6], [-18, 0, 8], [23, 3, -2], [17, -1, 7], [-6, 0, 0], [13, -9, 7], [-4, 12, -4], [-8, -2, -2], [-23, -7, 5], [5, 17, -2], [9, -7, 5], [24, 2, -2], [-4, -8, 0], [20, -2, -2], [-5, -13, 6], [-18, 6, 10], [15, 1, 0], [9, -13, 3], [4, 16, 0], [-8, -12, 4], [26, -6, -4], [10, 12, 4], [-16, 2, 6], [-3, 15, -9], [0, 8, 8], [-2, 6, 6], [-5, 3, -10], [-11, 7, 7], [4, -4, 12], [-10, -6, 10], [-34, 6, 4], [16, 12, -4], [12, 0, -4], [-11, -15, 5], [-10, -16, 6], [34, -8, 10], [-40, 6, -6], [16, -8, 8], [-40, 0, 0], [2, 4, -2], [12, 4, 8], [0, 14, -10], [-9, 3, -17], [-4, -2, -14], [21, -5, 0], [4, -8, -4], [-10, -14, 2], [-2, -2, -4], [34, 0, 8], [10, 0, 4], [23, -5, 2], [4, 0, -8], [-3, -5, 8], [-4, 10, -6], [15, 7, -5], [29, 7, -1], [6, -6, 12], [10, -14, 2], [-8, 6, 2], [3, 7, -2], [-25, -3, 4], [-34, 6, -10], [-40, -4, 0], [14, 4, -8], [-26, 10, -16], [-34, 0, 4], [-13, 13, -4], [30, 6, -2], [28, -2, 10], [-6, 4, -2], [2, 12, -6], [29, 1, 5], [18, 8, 0], [-16, 16, -8], [-7, 13, 2], [-28, 10, -2], [28, -10, -2], [24, -8, -4], [-22, 12, 0], [10, -12, -2], [-11, 15, -12], [-11, 3, 19], [12, 0, -12], [-37, -5, 6], [18, -2, -10], [26, -12, 12], [36, 6, -6], [3, -15, 9], [28, -4, 8], [14, 14, 0], [-16, -2, -10], [-32, 8, 16], [36, -10, 18], [-46, 2, 4], [-17, -19, 4], [-4, 8, -12], [8, 18, 2], [2, -12, 18], [35, 1, 1], [12, 0, 0], [5, -7, -3], [20, -14, 10], [4, -20, 12], [-28, -2, -6], [39, -3, -8], [49, -7, 13], [10, -10, -10], [38, -4, -4], [16, -14, 14], [39, -11, 5], [-13, 11, 10], [5, 1, 5], [-13, 9, -12], [26, 8, 2], [-4, -28, 4], [2, -6, 12], [41, -5, 11], [4, 12, 4], [16, 0, 4], [6, -4, -10], [-35, 13, -10], [19, -7, 17], [-16, -4, 8], [-8, 8, -16], [7, 7, -14], [-10, 12, 12], [-3, -15, -10], [2, -6, -4], [-20, 2, 14], [-7, -5, -12], [18, -4, -10], [-31, -3, -7], [-6, 18, -2], [8, -14, 10], [-2, 4, 12], [20, -2, -18], [40, -4, 8], [27, -1, 11], [-18, 18, -6], [-21, 19, 6], [-18, -10, -2], [10, 8, 10], [25, 5, -2], [-28, -4, 0], [-35, -5, 8], [42, -10, 14], [-28, 0, 8], [15, 11, -1], [9, -17, 15], [-16, -10, 2], [-16, 12, -4], [1, 15, 4], [2, -2, -2], [16, -6, -10], [-19, 7, 0], [68, -4, -4], [40, -8, -8], [5, 5, -10], [40, -8, 4], [-1, -11, -3], [-17, 7, 18], [-31, -3, 9], [-50, 8, -6], [-3, 27, -13], [-54, 0, -2], [-20, 22, -14], [9, -1, -4], [-2, -12, 22], [20, 24, 0], [-24, 0, 12], [44, -10, 2], [-4, -6, -14], [19, 5, 8], [-6, -10, 8], [7, 5, -19], [-21, -5, -1], [-18, 8, -2], [19, -3, -3], [46, -10, -4], [-9, 13, -3], [26, -16, 8], [-8, -16, 8], [70, -2, 2], [-1, -13, 14], [34, 4, 12], [25, 5, 10], [4, -6, 6], [48, 2, -6], [-48, 0, -8], [30, 22, -4], [14, -24, 8], [7, 17, -4], [-6, -22, -4], [19, 7, 3], [-37, 11, -10], [-8, -16, 8], [-43, 13, 9], [18, -4, -10], [-18, 20, -12], [-16, -16, 0], [-20, -10, 18], [10, -6, -2], [0, 4, 20], [-44, 12, -16], [-45, 9, -12], [-3, 5, -7], [-26, -12, -6], [8, 4, -4], [3, 1, -8], [3, -7, -23], [13, 3, -4], [2, 10, 10], [-18, 16, -14], [21, -19, 9], [0, 0, -8], [-32, 12, -16], [-33, -13, 2], [-42, 12, 14], [-36, -6, 14], [8, -14, 14], [46, -2, 8], [14, 4, 4], [-30, 14, 2], [-70, 2, -8], [3, -3, -8], [10, 6, 18], [-44, 14, -2], [17, 11, -20], [2, 4, -20], [20, -16, -20], [26, 12, 10], [34, -10, -16], [-18, 0, 6], [-25, -29, 2], [-2, -2, -8], [9, -15, 21], [20, 0, 4], [0, 8, 0], [-8, 4, 16], [-37, -7, 17], [14, 12, -8], [20, 10, 14], [34, -20, 20], [-2, -8, -8], [-2, 24, 6], [54, -6, -10], [19, -21, 18], [-16, -10, 6], [3, 9, -15], [13, -23, 5], [-14, -6, 8], [30, -14, 24], [-25, -1, 3], [37, -3, 2], [-52, 6, 6], [22, -12, -6], [-37, 3, -10], [-47, -5, -5], [23, 3, -14], [-14, -6, -10], [-10, -26, 8], [-8, -32, 8], [-10, -8, -8], [7, -19, 21], [14, -18, 8], [8, 6, 10], [-10, 24, -26], [-18, -28, 10], [-13, 1, 9], [-36, 4, -4], [-26, -12, 4], [28, 10, -2], [50, 8, -10], [-42, 2, -10], [-4, 6, 10], [-8, 0, 12], [-13, -5, 6], [-30, 8, -20], [34, -8, -14], [-48, 14, -22], [-31, 7, -9], [-27, 17, -19], [-20, 10, -18], [-14, 14, -6], [-21, 5, -4], [-6, 8, 0], [-11, 19, 8], [2, 14, -8], [-7, -13, -5], [-38, 16, -24], [36, -14, 14], [22, -2, 8], [40, 10, -2], [52, 12, 0], [-62, 10, 4], [24, -26, 26], [-58, -10, 2], [7, -13, 6], [18, -14, -16], [-8, 6, -18], [2, -16, 6], [22, -26, 8], [1, -19, 22], [-23, -29, 3], [-25, -17, -6]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2736_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2736_2_a_be();". // 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_2736_2_a_be(:prec:=3) chi := MakeCharacter_2736_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_2736_2_a_be();". // 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_2736_2_a_be( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2736_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-2, -7, -2, 1]>,<7,R![56, -23, -2, 1]>,<11,R![-2, 25, -10, 1]>,<13,R![-160, -44, 4, 1]>],Snew); return Vf; end function;