// Make newform 2842.2.a.r in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2842_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_2842_2_a_r();". // 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_2842_2_a_r();". // 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 := [2, 1, -5, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [-3, 0, 1, 0], [1, -5, -1, 1], [3, -3, -2, 1]]; Rf_basisdens := [1, 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_2842_a();" function MakeCharacter_2842_a() N := 2842; order := 1; char_gens := [1277, 785]; v := [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_2842_a_Hecke(Kf) return MakeCharacter_2842_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 := [[1, 0, 0, 0], [0, 0, -1, 0], [0, -1, 0, 0], [0, 0, 0, 0], [2, 0, -1, 0], [2, 1, 0, 2], [0, 1, 1, 0], [-1, -1, 1, -1], [1, -1, 1, -1], [1, 0, 0, 0], [1, 0, 3, -1], [-3, 1, 1, 1], [0, 3, -1, 0], [-1, 1, 0, -1], [3, 2, 1, 1], [1, 1, 2, -1], [-5, -2, 2, -3], [9, 1, -1, -1], [0, -4, -2, -4], [7, 1, -3, 5], [6, 1, 1, -2], [-3, 1, 4, -3], [3, 4, 0, 5], [0, 3, 1, 0], [0, -3, 1, 0], [3, -1, -1, 1], [2, 0, -4, -2], [6, 2, 0, 2], [5, 1, -2, 3], [2, 2, 4, 0], [8, -2, -2, 0], [3, -1, -3, 3], [2, -2, -2, -4], [-9, 0, -2, 1], [3, 1, -4, 1], [1, -5, -3, -1], [7, -3, -3, -3], [0, 2, 5, 2], [0, -4, -4, 4], [3, -4, 4, -5], [10, -2, -2, -2], [2, 1, 0, 2], [2, 0, 4, 2], [6, -2, 4, -4], [2, -2, 6, -4], [0, -2, 2, 0], [-9, -3, 4, -5], [-6, 0, 0, 2], [5, -4, -2, 3], [-1, 1, 7, 1], [7, -5, 0, -3], [-2, 2, 6, -6], [1, -7, -4, -1], [-8, -2, 1, -4], [-7, -1, 6, -1], [-1, 7, 4, -1], [-9, -1, -5, 1], [3, 4, 1, -3], [2, 6, -6, 4], [8, 4, -7, 4], [5, -4, -6, -1], [7, 1, -1, -3], [-6, -2, -1, -2], [-16, 3, -3, 2], [-7, 1, -4, -1], [-8, -6, -2, 2], [-7, 3, 2, -3], [-10, 0, 0, 4], [-12, 2, 0, 4], [2, -1, -8, -2], [-8, -2, 2, -2], [15, 3, 2, -1], [8, -1, -3, 2], [-15, -5, -6, 3], [0, 4, -8, 8], [10, -2, 4, -2], [-7, 1, -1, 1], [20, 3, -6, 12], [3, 7, 4, -3], [8, -3, 5, 0], [5, -2, 2, -5], [-7, 1, -7, 1], [0, 8, -8, 8], [2, -3, -1, 2], [-8, 4, 2, 4], [11, -5, -1, -3], [10, 4, 2, -4], [-5, 1, -7, 3], [15, 3, -9, 1], [-5, -7, 1, 1], [14, 0, -7, -2], [-1, -6, 7, -3], [12, 0, -4, -4], [-11, -5, -4, 1], [-4, 8, 6, 4], [23, 4, 5, 1], [-12, 9, 2, 0], [-5, 1, 2, 5], [-11, 2, 0, 7], [2, 8, -6, 4], [-6, -8, -2, -2], [0, 2, -2, -2], [-16, -2, 7, 8], [6, 4, 12, -4], [4, -2, -8, 8], [8, 3, -3, 8], [11, 6, 8, -3], [-9, 3, -8, 1], [3, 3, 2, -1], [-10, 5, 5, 10], [5, -2, 7, -9], [5, -1, 0, -5], [4, 2, -4, 10], [-6, 8, 1, 2], [-2, 2, 14, 2], [4, 10, -4, 2], [-15, -6, 6, -5], [-8, -10, -6, -8], [8, 6, 4, 2], [14, 0, -3, 0], [-5, 8, 0, 3], [6, -6, 9, -6], [13, 7, 3, -1], [-6, -6, -6, -2], [-25, -4, 12, -11], [5, -9, 2, -9], [-9, -5, 10, -11], [-8, -6, -4, 4], [-4, 7, -7, 6], [29, 5, -1, 3], [-11, 3, 0, -3], [-16, -4, 4, 4], [4, 0, 12, 0], [3, -1, -11, 7], [-16, 6, 6, 2], [0, 1, 1, -8], [11, 1, -3, -3], [-5, 2, 10, -3], [-5, -7, 17, -7], [-2, -6, 2, 0], [1, 8, 2, -5], [-15, -7, -4, -5], [-2, -6, 2, -14], [-2, 0, -13, 0], [11, -5, -5, -7], [9, 10, 5, 3], [-9, -5, 5, -3], [17, -11, 4, -9], [-2, -2, -3, 2], [1, 3, 9, -1], [-23, -5, 0, 7], [20, 1, -5, 12], [12, 0, -6, 0], [35, 6, 1, 1], [1, 9, -3, -1], [-15, 5, 0, 5], [5, -1, -3, 3], [2, -4, -4, -8], [-32, 6, -2, 6], [-22, -11, -6, -6], [-9, -7, 8, -9], [0, 0, 1, 4], [-25, 5, 6, -9], [-31, -3, -1, -3], [-6, -10, 13, -10], [1, -6, -7, -13], [-6, -2, 4, -14], [23, 7, 9, -7], [20, 4, -1, -8], [-3, -11, -9, -9], [-24, 4, 4, 8], [-14, -5, -12, -2], [0, -8, 14, -4], [-6, -6, 16, -8], [-8, -4, -6, 8], [-4, 6, 2, 10], [-6, -6, 12, -6], [-15, -7, -11, 5], [-15, 4, 1, -9], [-5, -6, -14, 7], [21, 11, -2, 5], [10, -2, -9, -6], [8, 10, -6, 14], [9, -13, 6, -1], [-6, 10, 6, 6], [14, -3, 4, -10], [-11, 1, -13, 9], [-31, -5, 9, -7], [-30, 2, 10, -4], [8, 2, 2, 8], [5, 15, -2, 11], [-9, 1, 10, 7], [-21, -7, 12, -1], [-25, -1, -15, 5], [19, 3, -5, -11], [34, 6, 6, 0], [2, -12, 16, -8], [26, -6, -6, -4], [19, -7, 0, -11], [8, 12, -4, 0], [14, -4, -10, 16], [12, 8, -8, 0], [21, -5, -3, -9], [12, 7, 5, -12], [-11, -3, -5, -7], [10, -5, -10, 14], [-9, 0, 5, -15], [-4, -10, 0, -8], [6, -10, -13, -6], [20, -2, -15, 16], [-6, 12, 8, 8], [-5, -4, -16, 3], [15, 1, 0, 3], [-26, -6, -5, 2], [18, 9, -21, 12], [-9, -1, -22, 9], [-11, 3, 3, 7], [8, -17, 7, -12], [-20, -4, 2, -8], [5, -11, 3, -11], [-17, -13, 1, -13], [-8, -6, 24, -8], [2, 12, 10, 12], [-13, -1, -10, -5], [-27, 6, -4, 7], [-24, -2, 0, -6], [26, 9, -7, 18], [16, 6, -2, 8], [1, 10, -3, 19], [42, -4, 0, 2], [10, -12, -4, -12], [29, 4, -8, -13], [-30, 6, 6, 14], [34, 8, -10, 4], [-47, -6, 4, -1], [-6, 5, -9, 12], [10, -1, -7, 6], [-7, 5, -13, 13], [0, -16, -12, -12], [-1, 12, 7, 9], [-22, 20, 1, 12], [-33, -4, 0, -11], [22, 6, 0, -2], [-9, -1, -7, -1], [-20, 2, 10, -6], [-38, -6, 10, -14], [-32, 10, -10, 4], [17, -2, 8, -9], [-30, -4, -2, 6], [-9, 1, 16, -1], [-7, 1, -17, 5], [-29, -1, -8, -3], [21, 3, -10, 9], [18, 11, 3, 10], [23, -9, -7, 9], [24, 0, -10, 4], [-14, 8, -12, 4], [-24, 2, -7, 12], [-1, -8, -8, -1], [-23, -3, 20, -1], [46, 4, 4, -6], [26, 8, 3, 0], [39, 6, -10, -13], [-3, -1, 7, 3], [-36, -9, 15, -24], [-10, -12, -7, 2], [39, -9, 5, -1], [24, -3, -5, 0], [38, 6, -18, 14], [-5, -13, 12, -3], [2, 15, 4, -10], [16, -2, 0, 12], [-23, 7, 2, -1], [-26, -8, -4, 10], [32, -11, 11, -8], [-3, -10, -7, -9], [7, 3, -17, 13], [-23, -3, 16, -13], [14, 2, -6, -16], [-13, -4, 4, 5], [0, 16, -6, 12], [37, 7, -11, 11], [-34, -8, 16, -22], [7, -14, 2, -5], [-15, -5, 9, 1], [16, 6, 6, -8], [-2, 8, 4, -12], [4, 6, -14, 6], [-12, 12, 2, 4], [-13, -11, 20, -19], [-11, -3, 16, -17], [-39, -2, -4, 7], [9, -3, -16, -5], [28, 0, -13, 8], [-17, -9, -1, -21], [16, -9, -2, 0], [35, -2, 5, -7], [32, 1, 10, -12], [-31, 4, 16, 7], [-7, 3, -3, 5], [18, -9, -11, 6], [28, 6, -22, 2], [-40, -12, -4, 0], [9, -9, -2, -19], [-15, -3, -3, -5], [49, -15, 0, -9], [9, -9, -2, -11], [16, -3, 8, 4], [3, 13, -9, 21], [-29, 5, 4, -7], [-20, 2, 20, -12], [12, 18, 8, 14], [34, 12, 12, 8], [18, 4, 2, 6], [-8, -4, -6, -4], [-9, 7, 4, 1], [-9, -13, 9, 3], [-20, -8, -4, 4], [5, 3, -2, -5], [-46, 2, 8, -8], [-58, -10, -15, 2], [14, -6, -7, -22], [-3, -8, 4, -7], [-19, -13, 2, -7], [19, 15, 0, 5], [-30, 13, -1, 2], [-14, 16, 12, 2], [-7, -2, 0, -5], [15, 13, 4, -5], [-11, 3, 8, 11], [-7, -7, 3, -3], [17, -3, -8, -13], [-14, -16, -2, 2], [-3, 9, 7, 5], [-39, 7, -12, 5], [-17, 2, 2, -15], [0, 18, 10, -2], [62, -7, 5, -6], [-32, -3, 11, -12], [-30, -6, 10, 2], [1, 1, 5, 9], [54, 8, 4, 8], [29, 9, -25, 15], [-17, 1, -13, 17], [0, -24, -12, -26], [-3, 3, -2, 17], [-17, 11, -13, 9], [-9, 23, -1, 13], [-9, 4, 17, -23], [13, 3, -11, 7], [33, 8, -12, 15], [-9, 5, 1, 19], [-20, -10, -18, 14], [40, 6, -14, -8], [-17, -17, -5, 3], [12, -3, 3, 12], [-25, 0, -17, 1], [33, 2, -16, -9], [-36, -2, 12, -6], [-9, 1, -5, -15], [-33, -1, 10, 5], [35, -3, -16, -19], [35, -5, 2, -5], [-10, -6, 4, 18], [18, -4, -15, 2], [26, 8, 10, 12], [-25, 1, 9, 13], [-10, -18, -6, -14], [-10, 10, -28, 12], [-40, -2, -7, -2], [-35, 24, -12, 15], [-47, 7, 1, -1], [30, 0, 14, 0], [-38, -8, 12, 2], [-11, 19, 0, 11], [33, 1, 3, 5], [3, -11, -16, 15], [-28, -5, 5, 16], [-1, -3, -2, -7], [11, -1, 2, 5], [1, -7, 9, -9], [-27, 13, 16, 23], [-21, -13, -12, 3], [-8, 2, -10, 2], [7, 0, 6, 13], [-8, 6, -6, -12], [48, 0, -16, 0], [-22, -5, 10, -2], [14, -8, -9, 8], [12, -7, 11, -14], [-8, -2, -12, 14], [10, 1, -14, 2], [19, -15, 14, -5], [-62, 8, 3, -6], [31, -13, -4, -13], [57, -3, 8, -1], [-4, -22, -12, -24], [10, 9, -5, 18], [-44, -16, -18, -8], [16, 6, 4, -14], [1, 6, 6, 1], [2, -4, 10, -12], [18, -20, 0, -14], [5, 17, 16, 11], [-8, 3, -6, 8], [-4, 8, 20, 8], [-43, 7, -12, 11], [-8, 8, -10, 10], [-2, 4, 25, -14], [-19, -18, -2, 15], [-34, 2, -14, -4], [1, -17, -12, -17], [2, 0, -8, 6], [-34, 10, 2, 6], [-8, 0, 6, -14], [-27, 13, 9, 11], [-5, 13, 8, -13], [-6, 2, 12, -10], [-22, 21, 3, 22], [25, -10, 11, -5], [36, -16, 4, -14], [-15, 21, -7, 3], [5, 13, 4, 5], [-38, -2, 9, 14], [-17, -7, 14, 1], [-8, 31, -4, 24], [-12, -10, -16, 16], [-24, -16, 4, -2], [7, 26, 20, 25], [5, 14, -9, 23]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2842_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2842_2_a_r();". // 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_2842_2_a_r(:prec:=4) chi := MakeCharacter_2842_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_2842_2_a_r();". // 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_2842_2_a_r( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2842_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![8, -4, -10, 1, 1]>,<5,R![4, 24, -14, -1, 1]>],Snew); return Vf; end function;