// Make newform 4761.2.a.w in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4761_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_4761_2_a_w();". // 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_4761_2_a_w();". // 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 := [-1, -1, 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_4761_a();" function MakeCharacter_4761_a() N := 4761; order := 1; char_gens := [2117, 1063]; 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_4761_a_Hecke(Kf) return MakeCharacter_4761_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, 1], [0, 0], [0, -2], [-2, 2], [-4, 2], [3, 0], [2, 2], [2, 0], [0, 0], [3, 0], [3, -6], [0, -2], [1, -4], [0, 0], [1, -2], [-2, -4], [-4, 4], [2, -8], [4, 2], [-11, 2], [9, 4], [6, -8], [-10, -2], [-8, 4], [-14, 6], [-2, 4], [-2, -10], [6, -12], [0, 0], [10, 2], [-11, -6], [-15, 6], [-12, 16], [-7, 6], [14, -16], [3, -2], [4, -12], [-7, -2], [-4, -4], [-18, 8], [3, 6], [-8, 14], [-20, 10], [5, -8], [-1, -4], [16, 6], [-16, 12], [4, 0], [-6, 10], [12, 0], [9, 4], [-15, -2], [12, -18], [6, 6], [5, -4], [-2, -8], [3, -8], [8, 0], [13, -4], [-10, -2], [-24, 6], [-4, -4], [12, 4], [-7, 10], [-12, 20], [-18, 12], [-11, 14], [-16, 12], [0, -16], [17, -12], [3, 20], [-10, 16], [-2, -10], [-4, 6], [-12, 20], [12, 8], [28, 4], [-17, 12], [-8, 10], [9, -20], [-12, -12], [14, 6], [-20, -4], [-24, -10], [-15, 6], [-27, 18], [10, -8], [-6, -18], [-1, -4], [-20, 0], [-18, -8], [18, -22], [-11, 6], [-17, -14], [23, 6], [-22, 8], [11, -28], [24, -12], [-30, 18], [-27, 12], [-11, 30], [0, 12], [-28, 8], [-16, 10], [18, 6], [13, -16], [21, -6], [-2, -8], [-24, 16], [37, -16], [24, -4], [-2, 6], [-10, -4], [-12, 12], [0, 20], [0, 28], [16, -22], [-3, 6], [21, -28], [-2, 14], [18, -8], [3, 0], [18, 0], [-13, 22], [12, 8], [-10, -10], [26, -26], [8, 8], [24, -6], [30, -2], [33, 10], [30, -18], [12, 20], [12, 22], [-29, 28], [12, -18], [-8, -4], [-32, -12], [-22, 20], [-22, -16], [-33, -14], [34, -8], [-21, 30], [4, 4], [18, -36], [-30, 18], [-18, -12], [1, 4], [-13, 6], [3, -30], [-34, 4], [38, -10], [4, 0], [-7, 26], [18, -36], [14, -28], [-30, 18], [29, 8], [-28, 10], [-2, -14], [17, 10], [18, -4], [9, -30], [14, -4], [-32, -14], [-38, 4], [24, 0], [2, -24], [-6, 12], [29, -8], [0, -10], [-15, 32], [10, 26], [0, 0], [45, 2], [-54, 6], [9, 30], [-27, 48], [30, -24], [-26, 18], [33, -18], [4, -40], [21, 20], [18, -24], [24, -30], [6, 6], [14, 28], [34, 12], [-3, 0], [24, -48], [51, -12], [8, -20], [-16, -18], [2, -16], [60, -2], [32, -28], [-32, 38], [-28, 36], [4, 28], [-25, -22], [-28, -16], [-16, 24], [25, -12], [0, 6], [-24, 12], [7, -8], [24, -42], [-3, -2], [-23, 52], [-51, 18], [35, -36], [-47, 4], [-12, 20], [-2, -4], [-4, -16], [-8, -24], [3, -42], [-9, 0], [24, -30], [13, 4], [-1, 12], [12, -6], [31, -44], [-6, -26], [21, 10], [3, 12], [-28, 26], [-1, 14], [6, 8], [8, 32], [2, 0], [-66, 10], [-12, 54], [-6, 8], [36, 8], [46, -26], [4, 36], [-24, 22], [13, 2], [31, -10], [-24, 56], [-9, 18], [-41, 18], [43, 0], [6, -8], [-39, -10], [23, -42], [6, 28], [-18, 0], [32, -26], [6, 14], [-48, 22], [-46, 28], [6, 18], [-18, 48], [-25, 22], [-28, 18], [12, -18], [35, 4], [33, 12], [8, -44], [-36, 48], [-38, 32], [-42, 30], [38, -8], [-42, -12], [0, 2], [4, -8], [2, 40], [47, -4], [41, 16], [-2, -12], [-32, 24], [-2, -30], [13, 24], [-57, 6], [-40, -4], [-18, -16], [-4, 42], [28, -52], [-17, 34], [52, -10], [32, -70], [30, -44], [20, -44], [12, 12], [38, -8], [22, -32], [-20, 12], [2, -44], [-14, -8], [74, -10], [62, -16], [2, -20], [21, 4], [-58, 20], [2, 6], [-27, 0], [63, -34], [-15, -26], [-62, 6], [-46, 8], [-62, 30], [-13, -22], [-40, 12], [-41, 36], [-44, 16], [-16, 4], [0, 46], [10, 8], [33, 6], [30, 24], [6, 34], [37, 18], [18, 36], [-2, 14], [-13, -34], [33, -22], [-40, 24], [-47, 16], [-36, -20], [46, 2], [57, -4], [28, -60], [-36, 18], [-36, 12], [-28, 6], [33, 6], [-48, 18], [-46, 46], [-3, -12], [69, -12], [-11, 38], [15, -6], [30, 20], [-1, 26], [12, 42], [-18, 6], [-9, -12], [24, -36], [-25, 52], [8, -8], [-38, -20], [-30, -24], [-60, -6], [8, 20], [-33, 48], [20, 16], [-42, 16], [-56, 2], [-53, 2], [-43, 84], [-26, 8], [-38, 12], [46, -46], [-78, 24], [6, -6], [44, -34], [7, 20], [0, -28], [-44, 6], [51, -16], [-57, -6], [12, -2], [-47, 38], [21, -44], [-75, 20], [-8, 18], [-42, -24], [23, 34], [-12, 0], [69, -42], [-34, 70], [44, -44], [39, 0], [-9, 10], [-56, -10], [6, -28], [12, -8], [39, -64], [-20, 14], [34, 28], [-24, 48], [-39, 48], [2, 22], [-45, -18], [-7, -2], [10, -24], [64, -30], [-18, 22], [-2, 38], [-54, 0], [-15, -18], [12, -48], [-78, -6], [32, -38], [-32, 60], [-50, -10], [2, -42], [-14, 44], [51, -60], [-45, 12], [-66, 6], [74, -14], [17, 52], [31, -22], [-34, 12], [43, 28], [-50, -6], [48, 32], [35, 24], [-7, 16], [-8, 10], [-6, 32], [26, -36], [19, -4], [-12, 32], [8, -36], [12, -60], [48, -28], [90, 6], [-80, 0], [-12, -40], [20, -40], [1, 24], [29, -64], [-30, -8], [-6, 20], [-33, 14], [-75, 18]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4761_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4761_2_a_w();". // 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_4761_2_a_w(:prec:=2) chi := MakeCharacter_4761_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_4761_2_a_w();". // 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_4761_2_a_w( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4761_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-1, -1, 1]>,<5,R![-4, 2, 1]>,<7,R![-4, 2, 1]>],Snew); return Vf; end function;