// Make newform 1421.2.a.j in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1421_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_1421_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_1421_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 := [-2, 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_1421_a();" function MakeCharacter_1421_a() N := 1421; 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_1421_a_Hecke(Kf) return MakeCharacter_1421_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, 1], [-1, 1], [1, 0], [0, 0], [1, 1], [1, -2], [2, 2], [-6, 0], [-2, -4], [1, 0], [-3, 5], [-4, 0], [-4, -6], [5, 1], [-1, -3], [1, -6], [-2, -4], [2, -2], [0, -4], [-6, 2], [-4, 0], [-1, 1], [-2, 4], [4, -6], [4, 6], [8, 4], [2, -2], [-12, 2], [7, -4], [-2, 8], [-10, -4], [-10, 8], [12, 0], [-14, 0], [-5, 2], [0, 10], [0, 6], [11, 5], [6, 2], [-18, -4], [2, 6], [3, 8], [14, -8], [-8, -2], [2, 0], [-8, -6], [-1, 13], [6, 2], [-6, -10], [12, 6], [7, -8], [-14, 4], [7, -8], [-13, 5], [21, 2], [7, 3], [-6, 18], [-1, 11], [-6, 8], [15, -12], [-6, 4], [-2, 4], [7, 7], [-14, -8], [-7, 2], [-6, 18], [-1, 1], [2, -14], [-6, -6], [-9, -10], [-10, 12], [11, 5], [-18, 0], [-15, 8], [10, 12], [12, 6], [20, 12], [-25, -4], [-13, 4], [2, -12], [18, 6], [6, -22], [14, -4], [-8, 16], [6, 4], [-30, -4], [-18, 12], [18, 12], [-14, 0], [-26, 0], [3, -25], [3, 7], [-20, -6], [-17, 3], [2, -12], [-13, -9], [19, 6], [15, -10], [-16, -8], [-16, -4], [16, -14], [-6, 8], [5, 3], [-34, 4], [-8, -16], [10, -14], [-2, 4], [11, -6], [-17, -19], [20, 2], [-5, -9], [-9, 0], [12, 8], [-35, -1], [-34, 2], [-2, -14], [-24, 6], [-34, 4], [-16, 10], [13, -1], [-22, 8], [1, 16], [22, 0], [4, -12], [-48, 0], [-9, 22], [15, -10], [6, -10], [10, 8], [-4, -32], [-3, -5], [18, -4], [14, -8], [34, 6], [-6, 28], [18, -22], [28, 6], [6, -34], [2, 38], [8, -20], [-8, 2], [7, 6], [-26, 20], [23, -7], [10, -14], [15, 5], [6, -12], [9, 2], [-7, 9], [-14, -22], [-23, 10], [-14, 0], [-4, -30], [-27, -7], [-6, -20], [-31, -11], [-6, -10], [-30, -8], [-6, -16], [17, -28], [21, 13], [-13, 16], [-31, 3], [10, -4], [39, 2], [-5, -19], [-10, 2], [0, 20], [19, -6], [4, 8], [-20, 22], [17, 8], [-2, 10], [2, -8], [26, -4], [22, 12], [0, 32], [-14, 10], [33, -17], [26, 20], [43, -3], [-7, 7], [-2, 8], [-19, -30], [-28, 24], [-21, -8], [26, 14], [-2, -4], [18, -2], [-28, 6], [3, 24], [5, -13], [21, 13], [28, 16], [-34, -20], [-2, -4], [44, -8], [-34, 4], [43, -16], [48, 4], [44, 6], [-26, 24], [-32, -18], [-4, 14], [10, -20], [27, 8], [-7, -33], [58, 2], [1, -26], [-31, -23], [24, 14], [-22, 16], [49, -3], [39, -13], [-38, -16], [13, 18], [0, -4], [-20, 4], [48, 8], [-38, -4], [-26, 2], [12, 18], [-2, -4], [3, 21], [52, 8], [18, -4], [0, 16], [34, -16], [-17, -27], [-24, -30], [-12, -8], [0, 20], [-2, -20], [-16, -28], [-10, 6], [-18, -28], [-22, 18], [4, 26], [28, -10], [-7, -7], [23, 27], [-26, -30], [-46, -14], [-36, 8], [18, -8], [50, -14], [18, -8], [10, 28], [-34, 22], [75, 1], [50, 2], [-17, 22], [-3, -19], [0, -28], [-48, 18], [-24, -16], [-16, 14], [27, 29], [54, 4], [31, -4], [-6, 8], [-55, 9], [-22, 36], [22, 26], [-18, 14], [-21, -33], [-32, 12], [6, 2], [18, -4], [-9, 28], [7, -26], [-18, -38], [51, -12], [-18, -28], [-36, 16], [9, 33], [-10, 12], [-55, -20], [4, -16], [6, 32], [-2, 10], [26, 14], [20, -40], [-18, -48], [-14, 16], [10, 28], [-22, 26], [32, -18], [0, 34], [-57, -12], [-3, -12], [-62, 4], [-19, -12], [-29, 9], [-18, 40], [-15, -34], [-45, -27], [-21, 14], [0, -26], [22, 12], [2, 26], [46, -16], [42, 24], [41, -15], [2, -12], [47, -6], [7, -39], [-19, 10], [36, -10], [5, 24], [34, 32], [-16, -14], [-10, -16], [-34, 8], [32, 10], [57, 4], [-6, -44], [-22, -2], [-3, 6], [26, -34], [15, -3], [-49, 6], [-14, 0], [21, -31], [55, 12], [-24, -6], [-10, -30], [-48, 0], [15, -31], [-53, -16], [-36, -20], [27, -2], [-56, 0], [22, -24], [-37, -21], [-20, 12], [-10, -32], [26, -8], [6, -40], [60, 8], [-62, -6], [-22, -36], [-2, 46], [-36, 20], [10, 28], [33, 29], [-46, -14], [-54, 16], [37, 31], [-34, 0], [-30, 20], [-22, -44], [-6, 16], [10, -28], [-6, -38], [42, -14], [81, 1], [12, -18], [-18, 14], [2, 42], [33, -20], [7, -40], [35, 45], [14, -50], [27, -13], [-60, -14], [-36, -40], [86, 6], [18, 24], [-25, -3], [-42, 2], [24, -42], [2, 0], [26, 52], [-21, -36], [-18, 28], [41, -15], [46, -30], [15, 42], [-43, 2], [-20, 4], [31, -30], [37, -21], [-2, -14], [-28, -42], [20, 50], [22, -48], [41, -6], [-1, 29], [58, -4], [-54, 0], [-5, -2], [-47, -1], [-19, -51], [-29, 43], [-1, 54], [40, -32], [-64, 16], [26, -6], [18, -12], [50, -28], [-26, -12], [10, 28], [27, 32], [-51, -18], [50, -28], [15, 8], [-82, -14], [-1, 15], [60, -20], [-30, 48], [-7, 6], [-32, 26], [-22, -48], [32, -18], [-6, 0], [-43, 1], [18, 54], [40, 4], [25, 15], [-6, 64], [-54, 12], [-15, -15], [-51, 25], [-15, -12], [-31, -16], [14, 20], [-36, -36], [18, -16], [15, -17]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1421_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1421_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_1421_2_a_j(:prec:=2) chi := MakeCharacter_1421_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_1421_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_1421_2_a_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1421_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-1, 2, 1]>,<3,R![-1, 2, 1]>,<5,R![-1, 1]>],Snew); return Vf; end function;