// Make newform 3240.2.a.j in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3240_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_3240_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_3240_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 := [-6, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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_3240_a();" function MakeCharacter_3240_a() N := 3240; order := 1; char_gens := [2431, 1621, 3161, 1297]; 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_3240_a_Hecke(Kf) return MakeCharacter_3240_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], [-1, 0], [1, 1], [0, 0], [0, 0], [-2, 0], [-2, -2], [-5, -1], [3, -2], [6, -2], [-6, 0], [-5, -2], [2, 4], [-7, -1], [-2, 4], [-6, 2], [-3, 0], [-1, -5], [0, 4], [4, 4], [-2, -2], [-3, -1], [-7, 4], [2, 0], [-2, 0], [10, 0], [-5, -3], [-1, -4], [0, -4], [-7, 3], [-2, 2], [0, 8], [0, 8], [-21, 0], [12, 0], [-14, -4], [2, -4], [1, 3], [-12, 0], [-4, -4], [5, -6], [-10, -2], [-12, 4], [4, -8], [18, 2], [12, 0], [3, 1], [-18, 4], [-9, 2], [-10, -4], [12, -4], [11, -6], [-2, 2], [18, 4], [18, 0], [3, 8], [26, 2], [-14, 0], [-13, 2], [-11, 3], [-2, 8], [-17, 5], [-2, -2], [4, -8], [-16, -4], [-2, -2], [-20, -4], [-6, -8], [-17, 6], [0, 0], [14, 6], [22, 0], [2, 8], [22, 2], [2, 0], [3, -8], [-22, 0], [6, 8], [10, -8], [20, 0], [6, 8], [10, 2], [-2, -4], [-8, -4], [9, -7], [2, 8], [-2, 8], [17, -2], [22, -4], [-2, 4], [6, 10], [6, -8], [-4, -4], [-28, 0], [3, 9], [-9, 2], [19, -4], [-27, -3], [17, -2], [-29, 5], [34, 4], [-3, -3], [10, 0], [16, 4], [-26, 0], [15, -3], [-2, -12], [-10, -6], [14, 0], [-15, 7], [-10, -8], [4, -16], [-10, -6], [4, -8], [-9, -2], [-3, -7], [9, -1], [18, 4], [20, -4], [10, 0], [-16, 0], [24, -4], [-6, 12], [34, 2], [1, 12], [-5, 6], [6, -14], [-19, -5], [-10, -8], [-38, 2], [15, -9], [36, -4], [12, 8], [35, 0], [-15, 4], [16, -4], [-30, -8], [-8, 0], [-30, 0], [18, -2], [-7, 4], [-1, -3], [7, 19], [11, 16], [-10, 6], [28, -4], [20, -8], [0, 16], [55, 1], [-20, 8], [11, -2], [-23, 9], [-18, 8], [-29, 3], [-14, 10], [10, 6], [18, 8], [8, -20], [13, -6], [-29, 7], [-24, 4], [17, 13], [46, -6], [16, -12], [-9, -1], [14, 6], [6, -12], [-19, -12], [-6, 4], [28, 12], [29, 6], [-28, 0], [-8, 0], [-28, -12], [6, 16], [-36, 8], [-30, 8], [3, -1], [35, -12], [-2, 0], [2, 10], [-38, 8], [24, 12], [39, 1], [-49, -2], [20, 8], [-7, 3], [-11, -2], [6, -14], [4, 8], [-15, 1], [-26, -2], [21, 12], [-1, -5], [20, 0], [7, 2], [16, 4], [30, -16], [-5, -1], [2, -8], [20, -4], [-32, 0], [-19, -4], [-4, 0], [12, 4], [-16, 16], [13, 7], [-27, 16], [-20, -8], [26, -16], [-10, -8], [-41, -7], [14, -4], [20, 8], [31, 6], [25, -3], [-26, -16], [-11, 23], [-36, -12], [5, -4], [32, -12], [-51, -8], [-37, 7], [-29, -9], [22, 0], [-6, -28], [-12, 4], [-11, -9], [6, -26], [50, -4], [-16, 8], [8, -4], [-41, 10], [-3, 1], [-15, -3], [1, -10], [-48, -4], [-2, -18], [-20, 8], [-15, 11], [-4, -4], [-21, 13], [-1, 14], [-50, -4], [6, -18], [1, 13], [42, 14], [-30, -10], [-13, 3], [22, 0], [39, 4], [-26, -16], [-7, 22], [4, 4], [0, 8], [-14, -24], [-17, -5], [18, 0], [-10, -8], [43, 5], [-1, -9], [-14, -16], [2, -24], [-28, 8], [-14, 18], [26, -8], [-41, -8], [-47, 9], [14, 0], [-10, -8], [15, -7], [10, -16], [46, 6], [60, 8], [27, -19], [-45, 3], [-6, -16], [-21, -14], [0, 28], [18, 16], [40, 16], [-2, 16], [-11, 14], [7, 5], [-60, 8], [-48, -12], [20, 8], [52, -4], [-51, 2], [-2, -16], [-30, -8], [-42, 16], [10, -6], [10, -12], [11, 24], [-32, 4], [0, -12], [-18, 18], [50, 0], [46, 4], [-4, -28], [22, -18], [18, -28], [-54, 14], [-52, 0], [-49, -7], [-21, 10], [-14, 2], [-16, 20], [-10, 4], [3, -6], [-10, -16], [13, 11], [-58, -8], [49, -4], [10, -10], [-12, -4], [-42, 20], [66, 0], [-28, -16], [12, 4], [-42, -8], [42, -8], [22, -24], [15, 30], [-14, 18], [-7, -17], [-75, 3], [-22, 16], [-11, -8], [26, 12], [4, -24], [-46, 12], [-2, 6], [-25, 7], [39, -4], [60, -4], [-17, -10], [-6, 12], [-44, -8], [36, 4], [-53, 12], [14, 18], [-74, 0], [16, 12], [-34, 8], [19, 1], [-2, 18], [60, -8], [2, 2], [28, -20], [33, -12], [-13, -25], [-73, 6], [-2, 4], [36, 0], [56, -8], [-8, -8], [67, -3], [-4, 8], [-9, 28], [-18, -24], [-4, -20], [-26, 20], [2, 8], [0, 12], [47, 9], [-10, 0], [66, 6], [8, -16], [19, 19], [43, -4], [-40, 16], [-20, 28], [24, 4], [48, 0], [24, -16], [45, -2], [14, 12], [9, -26], [-56, -8], [-3, -9], [-12, 12], [38, -6], [2, -8], [2, 6], [-24, -16], [25, 7], [41, -7], [-51, -18], [4, 4], [22, 6], [22, -20], [-38, 14], [-18, 12], [22, -30], [26, 8], [50, -6], [22, -32], [-17, 16], [30, 16], [-6, -36], [62, 4], [63, 0], [-20, -12], [52, 12], [-21, -24], [-23, -5], [54, -14], [-36, 16], [36, 24], [1, 33], [-44, 12], [56, -12], [-35, 14], [-72, -12], [2, 20], [0, 12], [39, -7], [23, 4], [-8, 28], [-18, -16], [26, -30], [-2, 8], [-36, 20], [-23, -11], [-78, 0], [-32, 4], [18, -34]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3240_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3240_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_3240_2_a_j(:prec:=2) chi := MakeCharacter_3240_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_3240_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_3240_2_a_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3240_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-5, -2, 1]>,<11,R![0, 1]>,<17,R![2, 1]>],Snew); return Vf; end function;