// Make newform 2541.2.a.bb 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_bb();". // 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_bb();". // 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, -1], [1, 0], [0, 0], [-1, 0], [-3, 2], [0, 0], [-2, 2], [3, -6], [-9, 2], [2, -8], [2, -5], [-1, 0], [-3, 7], [7, -1], [-6, -3], [-7, -2], [-8, 2], [-5, 4], [6, -9], [9, -3], [-6, 0], [-7, -1], [-17, 0], [2, 10], [-9, 6], [17, 2], [-2, 9], [-3, 14], [-3, 7], [1, 7], [-8, -8], [-12, 9], [-10, 5], [-18, 5], [-6, -7], [-2, -13], [10, 1], [-14, 1], [-1, 2], [1, 2], [11, 2], [10, 8], [-8, 2], [-2, 9], [9, 6], [-9, 6], [6, -21], [-8, 6], [5, -2], [21, 3], [-21, 6], [13, -22], [22, -8], [-3, -6], [19, -8], [1, 12], [-18, 12], [15, -1], [11, 6], [-12, 2], [13, -10], [10, -11], [-9, 11], [13, -20], [-11, 6], [5, 6], [-18, 12], [17, -29], [-23, 4], [-19, 23], [15, 6], [24, -10], [-9, 3], [-11, -5], [12, -14], [9, -27], [16, 7], [-27, -1], [-11, 7], [12, 15], [-31, 11], [-10, -12], [3, 4], [-28, 14], [7, -14], [3, -15], [17, -10], [-18, 29], [27, -8], [11, -17], [16, -1], [0, -6], [-5, 0], [-15, 28], [27, -24], [-13, 10], [3, -28], [17, -5], [9, -22], [2, 2], [-29, 26], [23, -26], [7, 0], [-15, -9], [26, -16], [-14, 16], [21, -17], [0, 24], [-18, -3], [19, -25], [-7, 5], [24, -18], [-3, 10], [15, -21], [-16, 0], [11, 14], [-3, -6], [27, -24], [-3, 0], [-15, -12], [8, -25], [-15, 28], [9, -9], [27, 0], [-42, 14], [-8, 1], [24, 3], [-7, -18], [26, -27], [29, 5], [-27, 3], [44, -2], [-3, -5], [42, -4], [18, 12], [-31, 3], [-17, 25], [18, -21], [-12, 18], [12, 10], [-4, 36], [15, 16], [21, -27], [13, -26], [16, 11], [-53, 2], [-27, 24], [19, -20], [30, -19], [-7, 8], [-25, 13], [-19, 4], [6, -21], [-37, -7], [-24, -12], [4, 32], [-7, 0], [-48, 10], [0, 1], [26, 11], [40, 6], [-21, 6], [33, 0], [-25, 8], [12, -35], [-6, -12], [-13, 21], [35, -32], [-17, 4], [-23, 5], [-2, -32], [-19, -24], [-15, -25], [24, 7], [-9, 27], [0, 9], [13, -28], [-26, -3], [40, -4], [4, 31], [9, 10], [30, -4], [-19, -19], [-8, -14], [-27, 30], [30, -52], [-11, 32], [7, 20], [8, 12], [38, -13], [-24, -8], [-24, 37], [-5, -4], [-22, 37], [11, -18], [-55, 3], [-53, 7], [-12, -23], [-22, 29], [-3, 10], [-18, -18], [42, -24], [-58, 1], [-12, 0], [-29, 33], [0, -12], [18, -21], [-45, 34], [2, 22], [-48, 0], [20, 3], [-33, -8], [37, -4], [28, 3], [29, -42], [-42, 18], [-29, 44], [-45, -2], [12, 0], [55, -30], [42, -39], [21, -29], [-27, 25], [-8, -9], [33, -3], [-42, 19], [16, 29], [-28, 10], [2, -21], [-15, -25], [-43, 15], [42, 10], [-4, 6], [31, -41], [-31, -8], [-29, 11], [-10, -40], [-24, 12], [-18, 9], [-7, 13], [-13, -21], [3, 39], [-41, 10], [-5, 35], [25, -9], [-33, 50], [-26, 42], [54, 0], [55, -9], [-3, -10], [-12, 0], [39, 2], [-37, 22], [-27, -16], [-3, 0], [-21, 48], [78, 0], [19, -27], [-21, -15], [-42, -20], [-27, -16], [-65, 3], [51, -21], [-53, 41], [26, -22], [-30, 44], [26, -9], [15, -2], [0, 19], [-3, -48], [69, 0], [34, -28], [-66, 8], [11, 1], [15, -24], [36, 18], [-48, 5], [10, 34], [26, -19], [15, -46], [33, -40], [12, 10], [0, 21], [-28, -5], [-20, -12], [15, 6], [11, -47], [-48, 21], [49, -44], [-12, 10], [26, 11], [30, -61], [37, -6], [-17, -31], [48, 18], [27, -6], [-21, 57], [77, -8], [4, 0], [5, -19], [-72, 9], [-28, 14], [27, -45], [-5, 6], [19, -67], [15, -25], [-56, 32], [27, 14], [17, -16], [-16, 2], [57, -30], [-23, 24], [11, -16], [28, -1], [-35, -20], [-32, -17], [36, 11], [54, -33], [36, 27], [-33, 22], [-9, -43], [-33, 59], [-58, -16], [-53, 35], [-33, 6], [45, -57], [3, -30], [18, -28], [-37, 48], [64, 8], [12, -24], [11, -34], [-48, 60], [1, 34], [49, -28], [-24, 16], [-11, 21], [24, -32], [49, 0], [-23, 67], [-27, -16], [21, -18], [12, 39], [-6, 42], [-30, 9], [-39, -6], [-53, 40], [-7, 35], [-39, 72], [-30, -9], [65, -11], [-58, 34], [-15, 35], [-63, 39], [21, 18], [-42, 63], [48, -15], [-21, -20], [-5, -24], [-11, 11], [-53, -8], [10, -20], [37, 32], [20, -12], [-39, 24], [4, -15], [5, -6], [-66, 16], [32, -39], [-21, 15], [37, -4], [83, 3], [12, -38], [-45, 30], [10, 44], [55, -22], [-10, -25], [-39, 24], [41, 2], [12, 49], [5, -24], [14, -7], [21, 3], [-40, 13], [-24, 32], [27, 22], [-24, 31], [-32, 40], [-44, 48], [16, 1], [-7, -26], [3, 15], [27, 0], [-13, 19], [72, -24], [55, -50], [-10, -36], [3, 18], [-9, -42], [78, -8], [-30, 36], [12, 2], [18, 39], [39, -39], [3, 15], [42, -30], [-26, -15], [-63, 51], [-20, -12], [-77, 25], [65, -22], [-33, -20], [-3, 42], [-5, -16], [38, -1], [-9, 54], [1, -31], [31, -19], [-1, 52], [16, 9], [1, -26], [-27, -21], [-2, 12], [16, -1], [-9, 46], [15, 15], [-26, 11], [39, -78]]; 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_bb();". // 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_bb(: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_bb();". // 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_bb( : 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, 3, 1]>,<13,R![1, 1]>],Snew); return Vf; end function;