// Make newform 2541.2.a.y in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2541_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_2541_2_a_y();". // 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_2541_2_a_y();". // 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_2541_a();" function MakeCharacter_2541_a() N := 2541; order := 1; char_gens := [848, 1816, 2059]; v := [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_2541_a_Hecke(Kf) return MakeCharacter_2541_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], [-1, 0], [1, -2], [1, 0], [0, 0], [-3, 2], [2, 0], [5, -4], [-2, 0], [5, 0], [4, 4], [5, -4], [-4, 4], [-2, -4], [3, 4], [0, 4], [-5, 8], [6, 4], [-7, 10], [4, -4], [-1, -6], [6, -4], [14, -4], [6, -4], [-10, 12], [16, -4], [-4, -4], [-3, 14], [-4, -4], [-12, 4], [10, -12], [0, 12], [10, -12], [4, -8], [21, 0], [10, 0], [10, -4], [-7, 2], [14, -12], [-14, 4], [6, -20], [0, 8], [22, -4], [8, -4], [2, -16], [14, -16], [-6, 20], [2, -16], [10, -4], [14, 0], [2, 8], [3, -18], [-3, -10], [-11, 0], [9, -14], [21, -6], [-6, 12], [-19, 16], [-2, 20], [5, 4], [1, -12], [0, 0], [4, 16], [0, 0], [2, 8], [-16, 8], [-16, 0], [-34, 4], [8, -8], [-11, 18], [11, -10], [16, 8], [-16, 0], [6, 0], [15, -2], [12, -8], [-32, 4], [14, 8], [-26, 0], [-14, 4], [17, 0], [-5, -20], [17, 2], [26, -12], [1, 16], [14, 0], [-6, 4], [-26, 20], [6, 8], [-3, 6], [23, -8], [-10, -16], [-20, 16], [17, -18], [23, -2], [-16, 16], [14, 12], [-19, 18], [-21, 12], [8, -20], [26, 0], [-23, 24], [-4, 16], [-2, 0], [6, 24], [20, 4], [17, -32], [30, -16], [-2, 4], [-9, -6], [-15, -8], [-24, 0], [-12, 24], [-20, 16], [-12, 24], [-4, 12], [14, -32], [5, 28], [24, 8], [5, -18], [20, -28], [12, 8], [24, -4], [32, 8], [24, -24], [-38, 16], [-41, 4], [-11, 20], [-10, 4], [6, -20], [26, 16], [5, -22], [9, -10], [-35, 12], [18, 4], [1, -10], [-3, -10], [-13, 36], [-41, 2], [17, 20], [47, -4], [-37, 24], [9, -26], [-23, -2], [-8, -8], [-29, 36], [-10, -12], [-34, 12], [22, -8], [0, 16], [-28, 0], [37, 2], [-9, -10], [4, 8], [4, 32], [4, -36], [16, 4], [3, 6], [-6, -20], [46, -12], [-44, 0], [-11, 12], [-32, 28], [25, -8], [10, 4], [-40, 0], [5, -10], [14, -44], [-26, 52], [-7, 6], [49, -10], [-10, -28], [-12, 24], [-8, 12], [50, 0], [-18, -12], [-30, 8], [-26, 12], [-9, 32], [27, -18], [-35, 14], [14, -12], [6, -24], [26, 0], [-12, 0], [-15, -18], [-4, -24], [-2, 0], [40, -24], [6, 20], [-24, 0], [27, 2], [-39, 26], [27, -38], [-43, 22], [23, 6], [-50, 4], [-22, -8], [-34, -16], [24, 12], [-2, -16], [3, -16], [2, 24], [11, 10], [-44, 0], [14, 12], [36, -12], [35, -30], [18, -40], [-16, -4], [-64, 4], [23, -30], [-45, 18], [4, -24], [-24, -20], [26, -20], [-28, 36], [16, 20], [0, 20], [-36, 24], [52, -8], [15, 24], [-4, -12], [-13, -22], [50, 0], [12, 36], [23, 22], [2, 12], [-45, 0], [-5, -30], [-2, 0], [43, -20], [-22, 40], [17, -20], [24, 4], [-28, -20], [6, -36], [45, -48], [10, 0], [-37, 52], [-14, 24], [-4, 16], [-12, 0], [22, 0], [-32, 48], [18, -32], [20, 16], [-45, -8], [64, -12], [5, -22], [-46, 4], [46, -52], [24, 24], [52, 0], [-16, 20], [-24, -8], [21, 4], [-37, 20], [-1, 10], [-37, 34], [52, 0], [-8, 24], [-58, -4], [19, 14], [-13, -10], [-14, -36], [52, 0], [-59, -6], [-38, 44], [14, 40], [-36, 48], [-35, 50], [-36, -4], [-20, 0], [3, -6], [52, -24], [14, -36], [34, 16], [-24, -24], [-22, 44], [-8, 36], [36, -16], [-10, -24], [15, -18], [-3, 12], [-46, 20], [-9, -8], [-30, -8], [-6, 56], [30, -44], [24, -60], [16, 0], [-19, 20], [-44, 48], [-30, 24], [6, -12], [28, -16], [-50, -4], [18, -24], [9, -20], [-55, -2], [23, 16], [-52, -16], [-8, -24], [-27, 50], [-17, 0], [-16, 16], [-48, 40], [24, -32], [32, -36], [-32, -28], [-16, 0], [1, 4], [32, -32], [60, 0], [34, 4], [8, 32], [-40, -8], [-30, 24], [-30, 16], [17, -4], [27, -24], [14, -12], [-44, 8], [-33, 12], [-18, 12], [32, 8], [12, 8], [31, 14], [-7, -48], [38, -28], [-54, -8], [-87, 8], [20, 28], [14, -4], [55, -48], [-45, 36], [-60, 24], [-4, 52], [8, -24], [14, -32], [54, -52], [42, -76], [3, -54], [-24, -28], [65, -34], [12, -44], [-28, 0], [-38, 8], [42, -68], [12, 16], [-83, 0], [32, 20], [17, 0], [-48, 0], [24, -24], [-13, 36], [-26, 32], [49, -12], [-20, -20], [-24, -4], [26, -28], [-16, 28], [-27, 54], [-22, -16], [63, 0], [-60, -4], [-82, 24], [-60, -4], [42, 4], [-54, -20], [63, -26], [-17, 24], [-8, -32], [-25, 28], [-79, -8], [-57, 6], [3, 26], [22, -32], [-18, -28], [-21, -22], [-18, 0], [36, -28], [33, -60], [-60, 32], [-29, -20], [-79, 22], [24, -16], [60, -12], [-43, -4], [65, 8], [-42, 76], [52, -48], [14, 8], [10, 24], [42, -24], [13, 40], [91, -2], [-72, 24], [81, -32], [-31, 42], [22, -48], [28, -52], [-50, 36], [34, 4], [-9, 0], [-5, -52], [-46, 16], [-4, 8], [4, -8], [-42, 12], [-6, -24], [30, 40], [14, 32], [-54, -12], [-7, 14], [9, -6], [16, -16], [-66, 20], [-34, 40], [2, 4], [-15, -22], [-30, -4], [28, 4], [64, -64], [-55, 30], [24, -64], [9, 28], [-22, -4], [10, 4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2541_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2541_2_a_y();". // 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_2541_2_a_y(:prec:=2) chi := MakeCharacter_2541_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_2541_2_a_y();". // 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_2541_2_a_y( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2541_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-1, -1, 1]>,<5,R![-5, 0, 1]>,<13,R![-1, 4, 1]>],Snew); return Vf; end function;