// Make newform 2541.2.a.ba in Magma, downloaded from the LMFDB on 28 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_ba();". // 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_ba();". // 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, -4], [3, 0], [2, 4], [-3, 0], [0, 0], [-3, 4], [8, -4], [-2, 8], [-3, 0], [-4, 8], [5, -4], [10, -4], [1, -6], [8, -4], [9, 2], [6, -12], [6, 0], [2, -4], [2, 0], [-4, -8], [0, 4], [13, -2], [0, 0], [0, -4], [2, 0], [-4, 4], [-10, 8], [12, -8], [5, 8], [-2, 12], [6, -12], [17, -6], [6, 8], [14, -4], [-10, -4], [16, -12], [-10, 8], [12, 0], [-14, 8], [-6, -4], [18, -12], [6, 0], [-22, 4], [-18, -4], [-6, 16], [3, 6], [3, 6], [3, -12], [23, -14], [-3, 2], [-26, 4], [3, -4], [-6, -4], [-3, 12], [7, -16], [-16, 16], [4, 8], [8, -16], [-14, -4], [24, 4], [24, 0], [-14, 24], [-16, 8], [-13, -6], [-3, 22], [-32, 8], [16, -24], [22, -12], [-1, -2], [-12, 16], [-4, 20], [6, 4], [22, -20], [22, -12], [31, -4], [3, -12], [17, -22], [-10, 12], [23, -20], [18, -16], [22, 4], [26, 0], [14, 8], [-3, -18], [9, 4], [-22, -4], [-12, 16], [-15, 22], [-33, 6], [0, -12], [-6, 20], [11, 2], [-3, 0], [-4, 12], [-2, 24], [-39, 0], [4, -4], [-34, 16], [10, 8], [-24, 0], [-33, 20], [22, -32], [-18, 12], [-7, 10], [15, -36], [-24, 4], [-4, -16], [-12, 36], [-12, 0], [0, -24], [18, -12], [-21, 16], [-32, -8], [29, -2], [-8, 24], [-12, 28], [-4, 20], [-16, 4], [-32, 0], [-6, 8], [31, -12], [27, -32], [6, 24], [18, 4], [6, 28], [-19, 2], [-15, -2], [5, 12], [-26, 20], [15, -18], [-21, 6], [-27, 0], [-31, 10], [1, -20], [17, 24], [-5, -8], [1, -10], [9, -18], [0, -24], [-19, 8], [42, -12], [-22, -4], [-14, 32], [40, -20], [36, -24], [35, -38], [31, 6], [-12, 12], [-12, 0], [24, -28], [-36, 8], [13, -26], [-10, -12], [10, -20], [-36, 0], [21, -20], [-36, 4], [-17, -4], [6, 32], [16, 16], [-51, -2], [-30, 12], [-22, 4], [-9, -10], [17, -26], [26, -20], [-4, -16], [12, 24], [-10, 0], [-46, 20], [38, 4], [-14, 4], [9, 12], [-19, -18], [-3, -34], [14, 8], [14, -8], [-30, 24], [-4, -16], [-41, -2], [36, -48], [2, 4], [32, -12], [30, 16], [-56, 0], [19, -22], [1, -6], [13, 26], [-3, -18], [-23, 38], [18, -24], [-30, 0], [-2, -20], [52, -8], [-34, 56], [-59, 12], [-6, -4], [37, -6], [-20, -8], [-6, 36], [0, -12], [19, -46], [-6, -8], [12, 24], [-36, 4], [-15, 26], [-29, 50], [-12, 24], [36, -12], [-2, -28], [16, -4], [-12, 40], [12, 0], [36, -12], [12, 36], [49, -12], [-24, 32], [-37, 42], [22, 16], [-24, -12], [1, -26], [34, -20], [-3, 36], [43, 10], [-2, -40], [29, -48], [30, -40], [57, -20], [-4, -8], [40, 20], [2, 24], [-19, 16], [30, -20], [-43, -8], [-34, -4], [-20, 8], [28, 20], [22, 24], [0, -32], [-34, 16], [44, -24], [-27, -4], [-28, -12], [21, -38], [-34, 52], [18, 24], [16, -16], [20, -24], [12, -36], [32, 8], [-11, 12], [13, -24], [-31, 26], [-11, -6], [-36, 16], [0, 0], [34, -60], [-19, 22], [-27, 54], [26, 16], [20, -40], [-37, 2], [58, -12], [-58, 8], [-20, 0], [29, 18], [-24, -8], [12, 44], [-27, -6], [36, -12], [-2, -20], [-14, 8], [-48, -12], [42, 0], [-36, 8], [-20, -40], [-14, 52], [-39, 6], [83, 0], [-30, -24], [-1, -8], [-22, 8], [-10, -20], [-70, 8], [-28, 40], [48, -24], [5, 4], [-36, 44], [-22, 16], [-22, -8], [-60, 16], [-54, -4], [62, 4], [-9, 32], [15, -42], [-41, 8], [-36, 12], [-24, 68], [-35, 2], [7, 0], [-24, 52], [48, -36], [32, 4], [-60, 0], [-44, -4], [64, -40], [-39, 28], [8, 40], [-4, -8], [38, 4], [-24, 56], [56, -8], [-62, -4], [-14, -16], [-1, -48], [27, -24], [58, -20], [-28, 0], [-57, 12], [50, -44], [-32, -24], [-20, 60], [-49, 22], [-15, 48], [-30, -28], [-38, 16], [31, 12], [72, 0], [-2, 0], [41, -28], [-37, -20], [-12, -20], [16, -16], [-64, 0], [-2, 24], [26, -8], [14, -40], [27, -14], [-12, 0], [17, -58], [72, -36], [-12, 32], [14, 8], [6, -52], [-36, 32], [-3, -24], [28, -4], [15, 36], [0, 4], [8, 52], [-51, 0], [-30, 8], [-39, -12], [56, -64], [60, -36], [30, -36], [-36, 28], [-3, 38], [70, -24], [-9, 48], [-16, 32], [-38, 0], [40, -52], [34, 8], [18, -8], [31, -74], [-57, 0], [-8, -24], [-33, 76], [-33, 12], [39, 6], [-43, 34], [-2, 16], [2, 28], [45, -6], [78, 12], [-48, 12], [-23, 52], [20, -72], [29, -16], [1, -34], [-8, -40], [48, -16], [-61, -16], [-87, 0], [-14, -4], [-12, -12], [10, -36], [-26, 32], [26, -36], [19, -60], [-11, -2], [24, -48], [-79, 8], [-25, 10], [6, 12], [-56, 32], [18, -12], [-42, -8], [-39, 68], [51, -60], [-30, 0], [44, -56], [20, -8], [34, -12], [-30, -16], [42, -40], [-78, 12], [30, -4], [-33, 6], [9, 26], [80, -20], [-62, 44], [-6, 20], [22, 40], [7, -54], [-22, 8], [24, 8], [0, 48], [39, 6], [48, -68], [-63, -20], [34, 36], [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_ba();". // 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_ba(: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_ba();". // 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_ba( : 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![-1, 4, 1]>,<13,R![11, -8, 1]>],Snew); return Vf; end function;