// Make newform 2541.2.a.w 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_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_2541_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_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 := [[1, -2], [-1, 0], [0, 1], [1, 0], [0, 0], [2, -2], [3, -3], [1, -3], [5, 1], [-6, 0], [0, 5], [4, -1], [8, 1], [6, -6], [2, -4], [8, 2], [2, -2], [0, 0], [0, 0], [0, 8], [10, -6], [-10, 0], [14, -4], [5, -1], [-6, 0], [0, 11], [-3, -7], [9, -11], [-12, 9], [-4, 2], [4, 6], [-8, 14], [2, -10], [15, 3], [0, 8], [-18, -4], [-12, 18], [-10, 0], [-20, 2], [-13, 1], [-8, 11], [-10, 16], [8, 5], [-20, 3], [20, -4], [15, 3], [-6, 20], [19, -1], [10, -4], [10, -10], [-12, 6], [-11, 13], [12, 0], [24, -6], [9, -25], [9, -3], [-6, 12], [-12, 17], [3, 5], [-22, 8], [-8, 15], [-4, 23], [-12, 9], [12, -14], [-14, -10], [8, 2], [10, -12], [24, -5], [-4, 17], [-18, 6], [-18, 0], [12, 9], [12, 15], [12, -7], [24, 4], [-6, 24], [14, -2], [12, -30], [0, -12], [-16, 10], [-2, 2], [3, -11], [12, -5], [-30, 2], [0, -3], [21, 1], [-32, 16], [14, -12], [-6, 0], [-16, 12], [20, -10], [4, 8], [24, -6], [-9, -17], [10, -18], [-12, 4], [-24, 5], [-8, 7], [15, 3], [-12, 7], [-34, 4], [10, -20], [-12, -4], [-2, -22], [24, 14], [-32, 6], [6, 12], [12, -17], [-23, 1], [6, 4], [1, 21], [-9, 21], [2, 26], [-20, 27], [-28, 28], [14, 2], [1, -15], [34, -4], [4, -20], [-24, 3], [-14, 30], [-30, 24], [22, -20], [20, -11], [37, -19], [26, -22], [0, 13], [18, -34], [-17, 3], [-2, 26], [30, -18], [-12, 7], [24, 0], [5, -9], [-6, -12], [6, 8], [6, -4], [-24, 3], [36, -9], [6, -2], [16, 12], [-6, 8], [-10, -24], [7, -37], [-8, 36], [-18, 36], [-38, 6], [-22, 20], [12, 0], [-8, 7], [10, -4], [7, -7], [6, -22], [14, -22], [20, 6], [4, 8], [-28, 4], [33, 15], [18, -26], [-11, -17], [12, -3], [2, -16], [-16, -20], [36, -30], [-26, -20], [-6, 8], [-32, 2], [-16, 46], [22, -4], [3, 9], [-24, 11], [36, -12], [20, 5], [23, -15], [-12, -12], [4, -1], [-36, 4], [-2, 28], [28, 20], [36, -12], [-34, 22], [-8, 10], [-43, -9], [0, -21], [47, -1], [15, 13], [-21, 5], [0, -17], [-28, 15], [24, -12], [-48, 6], [-24, 45], [-36, 6], [36, -21], [15, -9], [38, -28], [-12, 33], [-36, 45], [-30, 16], [20, -20], [28, -40], [31, -1], [-14, 6], [22, -34], [18, 12], [8, -25], [44, -36], [12, -16], [-22, 14], [9, 9], [13, 15], [-39, 35], [30, -42], [-34, -20], [-10, 4], [-22, -30], [18, -14], [-38, 28], [-29, 19], [18, -6], [-19, 7], [-16, -15], [-20, -8], [-30, 18], [58, -2], [-57, 1], [-8, 49], [-4, 19], [35, -9], [8, 8], [41, -41], [12, -48], [-38, -22], [-68, 4], [16, -49], [6, -12], [-10, 12], [-2, 16], [-8, -23], [-22, 40], [0, -40], [16, -33], [-40, 7], [-35, 39], [20, 13], [10, -8], [24, 15], [-18, -12], [24, -2], [-4, -28], [36, -33], [-36, 51], [28, -32], [-12, -6], [-8, 7], [-3, -27], [28, -39], [-36, -16], [-30, 20], [2, 16], [16, -24], [44, 11], [54, -10], [-30, 18], [-17, 13], [-23, -31], [-10, -34], [-6, 0], [-36, 16], [-30, 18], [-26, -4], [16, -48], [4, 8], [-36, -18], [26, -42], [17, 37], [60, -9], [-43, -17], [19, -43], [-22, 22], [24, 28], [8, -20], [-6, 8], [-20, -12], [-16, -15], [-47, 1], [-48, 21], [10, 4], [-19, -15], [0, -31], [-22, 56], [-56, 40], [8, 38], [-60, 30], [-39, 51], [-16, -1], [63, -33], [13, -17], [48, -4], [41, -39], [36, -22], [-12, -22], [32, -48], [-36, -17], [18, -30], [-45, -15], [12, -24], [20, 12], [19, -61], [-12, -3], [-44, 52], [36, -24], [18, 30], [-18, 30], [24, 9], [6, -44], [-24, 18], [-88, 4], [22, 0], [48, 2], [28, 29], [-59, 3], [67, 5], [24, -42], [-4, 2], [-12, -6], [6, 42], [-45, 47], [-28, -7], [-48, 20], [22, 20], [0, -16], [9, -27], [-66, 0], [14, 18], [12, -18], [27, -15], [57, 3], [-40, 6], [-28, 7], [-48, -11], [-44, 22], [45, 3], [44, -12], [66, -34], [12, -51], [-32, 44], [6, -36], [-50, 44], [43, 17], [24, -31], [-7, -41], [18, 18], [60, 3], [-19, -21], [59, -19], [-68, 14], [-60, -20], [-60, 18], [0, -36], [-11, 23], [13, 11], [39, -49], [21, 9], [17, 21], [-12, -39], [-42, 8], [-48, 54], [7, -7], [3, -33], [47, 7], [-11, -35], [-2, -32], [33, -23], [18, -24], [-16, 34], [-45, 47], [-56, -6], [44, 23], [0, -60], [-1, 17], [-12, 70], [32, 6], [16, -20], [44, -5], [20, -2], [-42, 76], [-56, 48], [50, 18], [-58, 8], [-57, -11], [57, -45], [-7, 41], [-27, -23], [36, -21], [-73, 13], [-28, 75], [-24, -9], [46, -66], [-36, 0], [-28, 0], [32, 10], [-12, 24], [-8, 53], [-55, 7], [-44, 83], [54, -20], [34, -4], [40, -3], [-16, 8], [20, -42], [-76, 32], [-35, 37], [20, -8], [43, 5], [17, -25], [24, 15], [-32, -15], [50, -20], [-16, 8], [-12, 2], [-26, 0], [60, -21], [6, 48], [-20, 56], [27, -51], [-21, 27], [28, 4], [52, 5], [-6, 52]]; 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_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_2541_2_a_w(: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_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_2541_2_a_w( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2541_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-5, 0, 1]>,<5,R![-1, -1, 1]>,<13,R![-4, -2, 1]>],Snew); return Vf; end function;