// Make newform 2541.2.a.bm 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_bm();". // 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_bm();". // 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, -3, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-1, -1, 1, 0], [1, -2, -1, 1]]; Rf_basisdens := [1, 1, 1, 1]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; 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, 1, 1, 0], [-1, 0, 0, 0], [0, 0, -2, 0], [1, 0, 0, 0], [0, 0, 0, 0], [0, -2, -2, 0], [-2, 4, 0, -2], [2, -2, 0, 2], [-3, -4, 0, 3], [-2, -2, 4, 1], [-4, -2, 2, -2], [-3, 4, 4, -1], [2, 2, -4, -6], [0, -6, 0, 5], [2, 4, 2, -6], [-2, -2, 0, 5], [-6, 0, 2, 2], [-2, -2, -4, 2], [-8, -2, 0, 3], [-6, 4, 4, -3], [2, 0, -2, 0], [4, 2, 0, 1], [-2, -6, 2, 2], [-2, -4, -6, 4], [8, 4, 4, -6], [-6, -2, 0, 4], [6, -4, 0, 6], [5, -8, 0, -1], [3, -4, -8, 1], [-2, 2, 0, -3], [13, 6, 2, -11], [2, 8, 2, -6], [-9, 2, -2, -5], [2, 8, 0, -8], [-14, 8, 4, 4], [-1, 10, -2, 3], [-4, -4, 2, 0], [3, 6, -2, -1], [-2, 4, 4, -4], [-12, -2, 4, 2], [-11, 0, 0, 3], [4, -2, -12, 4], [-6, -12, 4, 7], [0, 4, 0, 5], [-1, -6, 2, 11], [-10, 2, 14, 2], [-1, -10, 2, 11], [10, 4, -6, -4], [-18, 6, 6, 2], [8, 6, -4, -10], [2, -8, 0, -8], [-2, 12, 0, -9], [-2, 6, 12, -4], [-4, -4, 4, -14], [-2, 2, -2, 2], [-6, 4, 4, -11], [14, 8, -6, -8], [-2, 4, 12, -12], [1, 0, -8, 15], [-2, 6, 4, -11], [-20, -2, -2, 6], [8, 4, -2, 8], [4, -4, 6, 8], [-14, 8, 8, 8], [4, -10, -6, 2], [14, -2, -8, -15], [-9, -2, 2, -13], [-5, 8, 20, -3], [14, -8, 0, -7], [-12, 8, 6, 6], [-8, 8, 0, 4], [23, -8, -8, -7], [10, 14, 14, -18], [-6, 12, -4, -8], [-3, 10, -2, -15], [-6, 6, 0, 2], [-5, -2, 2, 19], [22, 4, 0, -6], [-13, -10, -2, 7], [-6, -6, 4, 2], [-10, -10, -8, 8], [3, -8, -12, 1], [-9, -8, 8, 5], [14, -8, -6, 0], [2, 16, -2, -20], [6, -8, -12, 5], [-15, -2, 2, 13], [29, -4, -4, 3], [24, 6, -2, -18], [-4, -2, 0, -7], [16, -4, -18, 8], [0, 12, 2, -6], [20, 2, -12, -13], [-16, -4, -4, 8], [-8, 10, -12, -11], [0, -14, 0, 14], [-4, 8, -14, -8], [0, -8, -10, 18], [-8, -12, 6, -2], [9, -4, -4, -1], [-12, 8, -4, 4], [6, 2, 8, -9], [12, -6, -10, -14], [-18, -8, -4, 4], [1, -10, -2, 17], [-12, 18, 2, -10], [-22, -6, 8, 12], [8, 4, -12, -10], [-11, -16, 0, -1], [-2, -26, 2, 14], [-16, 2, 8, 10], [-12, 24, 8, -13], [-2, -14, 8, 15], [-34, 4, 12, 4], [-7, -10, -18, 5], [-2, 26, 0, -17], [6, -12, -4, 10], [-4, 16, -2, 8], [-1, -6, -2, 11], [-28, -4, 4, 24], [16, 12, 14, -24], [16, 4, -4, 3], [-8, 16, -6, -6], [1, -12, 0, 3], [-22, -22, -12, 32], [17, 6, -2, 5], [-4, -16, -8, 21], [-16, -2, -6, -8], [-4, 0, -4, -12], [-18, -8, -10, 20], [-12, 10, -12, -15], [26, 0, 4, 7], [-17, 14, 18, -9], [-4, 24, 16, -15], [-20, -2, 2, 22], [14, -6, -2, -14], [8, -12, 0, -16], [20, -14, -22, 4], [-4, 6, 8, -16], [7, -22, -22, 11], [-4, -2, 0, 8], [-10, -8, 12, -4], [-16, -6, -4, 5], [24, -12, -36, 4], [-20, 10, 24, -6], [-32, -2, -2, 22], [-30, -2, 12, 16], [18, -4, -22, 6], [2, 16, -14, -22], [-4, 0, -12, 24], [8, 8, -16, -5], [-22, 14, 2, -2], [0, -12, -12, 24], [14, -6, 8, -16], [-4, -6, 0, 9], [4, 0, 24, 4], [12, -26, -8, 11], [-32, -4, 10, 16], [-4, 16, 0, 8], [-12, -8, -4, 18], [-24, 4, 16, 8], [15, 14, -14, 7], [23, 14, -10, -25], [-18, -8, 14, -18], [-9, -2, -10, -5], [-12, -16, 10, 2], [12, 12, -4, -8], [8, 2, 14, -16], [25, 12, -4, 3], [-20, -4, 12, 12], [11, -4, 0, -19], [2, 38, -4, -26], [5, 12, 20, -9], [12, 8, 4, -5], [-26, 4, 10, 26], [2, 18, 18, -12], [4, -4, 4, 16], [10, -30, -8, 11], [-28, -6, 12, -2], [-4, 14, -14, -24], [7, 10, -2, -29], [-14, 32, 6, -18], [-4, -16, -8, 33], [0, 6, 2, -20], [-16, -8, -12, 12], [10, 4, 0, -22], [-18, -16, 24, -4], [8, 10, -8, -30], [-14, 0, 0, 8], [30, -12, -32, 4], [14, 20, -22, 6], [-1, -8, 0, 13], [-28, 12, -4, 0], [6, -22, -12, 2], [1, 4, -8, 23], [42, -16, -8, -10], [-14, -8, 4, 24], [-31, 12, 8, 31], [24, 14, -2, -40], [-2, -30, 6, -2], [15, 2, -10, -25], [10, 28, -10, -8], [16, -18, -6, -16], [-22, 14, 22, 4], [8, 34, -4, -32], [-24, 6, 24, 20], [-44, 26, 10, 6], [-6, 28, 16, -15], [-10, 10, 24, -25], [-16, 2, -8, 16], [4, 8, -4, 19], [-6, -22, -8, 14], [-12, -14, -16, 13], [4, -20, -2, 22], [16, 0, 8, 22], [2, 2, 18, -8], [-8, 22, 8, -13], [12, 42, 10, -28], [38, 0, -4, -31], [-1, 22, 2, -5], [3, -20, -4, -7], [34, -30, -16, -2], [-22, -18, -16, 33], [16, 32, 0, -24], [-12, -2, -6, 28], [20, -16, -24, 7], [26, -16, -6, 12], [22, -12, 12, 9], [-14, -8, -18, 30], [-22, -4, 28, -3], [-8, -40, 8, 17], [-10, -16, -8, 36], [28, -14, 4, -29], [11, -14, -2, 11], [-34, 2, 30, -4], [-2, 0, 20, -34], [24, -20, -10, -4], [26, -8, 0, -12], [34, 0, -4, -37], [16, -4, -4, 16], [-4, -16, 12, -8], [-22, 6, 18, 18], [32, -10, -18, 0], [12, -12, -4, -9], [-4, -16, -16, 38], [12, -4, 24, 6], [-10, -26, -14, 38], [30, 12, -12, -18], [16, -2, -40, -11], [15, -4, -12, -11], [16, 28, 0, -11], [-12, -36, -22, 28], [13, 20, 0, 11], [4, 4, 0, -12], [-12, -32, -6, 42], [23, 0, 12, -15], [12, -8, 8, 17], [-20, 0, 6, -30], [-22, -14, 24, 26], [34, 22, -4, -26], [0, -2, 24, 1], [16, 8, 12, -4], [32, -26, -10, 2], [-4, -8, -28, 30], [30, 2, -8, -24], [-24, 0, 0, 10], [-23, -22, 2, 41], [8, 10, 34, -14], [16, -26, 8, 15], [21, -30, -14, -11], [-8, 14, 2, 14], [28, 8, -36, 8], [-30, 20, 0, -2], [15, -30, -6, -13], [8, -6, -8, -16], [0, -12, -36, 30], [-23, -16, -8, 43], [2, 0, 0, 28], [16, -8, 8, 5], [-30, 20, 4, -2], [22, -6, -2, 2], [9, 18, -22, -47], [-8, 14, 2, -24], [-10, -10, -2, -6], [16, 4, 24, 2], [28, -24, -8, 12], [59, -4, -8, -3], [14, -6, -40, 11], [-12, 4, -12, -38], [-6, 4, -36, -16], [21, -34, 2, -7], [38, 18, 10, -28], [-34, -16, 8, 36], [18, 32, 2, -4], [4, 12, -12, 33], [14, -10, 14, 4], [-14, -4, 2, -10], [-4, -14, 2, -16], [-32, 8, -8, 20], [26, 8, -24, -14], [52, 22, 0, -14], [29, -6, 26, 13], [-4, -20, 20, 32], [20, 0, -40, 8], [-40, -10, 0, 33], [-16, 8, -12, 17], [19, -4, 12, -23], [42, 34, 26, -44], [-18, -8, 12, -19], [-16, 16, 0, 13], [-14, 26, -4, -10], [27, 10, -14, -17], [-35, 10, 14, 5], [-31, 18, 14, -15], [50, 0, 4, -11], [-6, -20, 18, 46], [4, 14, -10, -16], [-16, 16, -16, -44], [20, -6, -24, 30], [-39, 10, 18, 25], [-34, -10, -4, 20], [28, -12, 20, 21], [0, -8, 12, 10], [20, -4, 8, -24], [18, 2, -16, 27], [34, 6, -10, -2], [4, -30, -4, 3], [-12, 38, 8, 6], [-18, 28, 0, -11], [10, -18, -20, 13], [-8, -16, 8, -19], [35, -26, -54, 15], [-2, 18, 20, -14], [18, 6, 16, -24], [-5, 38, -14, -5], [-12, 0, 0, 12], [24, 24, 0, -15], [-14, 32, 14, -36], [4, 18, 2, 4], [-8, -10, 2, -22], [15, -20, -16, 5], [3, -2, 14, 19], [12, 2, 14, -26], [61, 6, -22, 17], [-14, 22, 20, -25], [-22, -16, -28, 27], [2, -10, 34, 30], [20, 42, 0, -21], [-4, -14, 4, 42], [6, 10, 10, 18], [16, 18, 8, -20], [-1, 8, 20, -11], [-6, 10, -20, -19], [22, 20, 18, -12], [-26, -12, 24, -20], [10, -12, -14, -8], [-18, 26, 4, 10], [16, 4, 14, -2], [-18, 46, 16, -25], [1, -8, 16, 3], [32, -28, -28, 17], [28, -36, -4, 16], [64, -12, -4, -8], [-22, 20, -4, 37], [-36, -2, 4, -12], [-11, 0, 24, 35], [-44, 20, 16, 22], [-12, -8, -20, 36], [13, 4, -40, 7], [-30, 0, -4, 7], [22, 8, 22, -26], [-34, 4, -12, -3], [-1, -6, -2, -13], [28, -24, -8, -22], [43, 24, 0, -19], [34, -30, -28, 0], [-35, -12, 16, 11], [-40, 26, 14, -22], [50, 4, 4, -26], [-36, 22, 0, 12], [32, -32, -20, 0], [-22, 26, 0, 17], [67, -10, -26, -13], [7, 18, 6, 11], [8, 30, 10, -42], [0, -40, 12, -10], [-19, -14, -22, 29], [-4, 28, 42, -28], [-23, -6, 10, 5], [-20, -36, -4, 58], [0, 4, -12, 25], [16, 38, 2, -18], [14, 8, 4, -19], [-12, -2, 0, -30], [6, -64, -8, 31], [-53, -8, 16, -15], [18, -38, -20, -2], [-44, -16, 20, 2], [-8, -18, 36, 19], [30, 26, -8, 5], [50, 8, 18, -34], [-43, -14, 6, -3], [-21, 14, 2, 27], [-46, -12, 2, -2], [42, -12, -26, -50], [-44, -24, -30, 38], [44, -12, -36, 7], [-26, -34, 36, 11], [-16, 18, -18, 14], [-20, -24, 6, 66], [-30, 24, 8, 0], [-19, 42, -10, -7], [5, 8, 24, -53], [-64, 10, 16, -9], [-9, 0, -20, 25], [-36, 42, 8, 4], [2, -12, -52, 5], [-42, 24, 10, 50], [-36, -12, 46, -16], [-48, 6, 24, -4], [-30, 18, 20, -5], [-22, -28, -16, 34], [18, 16, -28, 1], [20, -26, -18, -4], [48, -40, -20, 10], [-36, 6, 4, -40], [-4, 16, 8, -52], [-51, 26, 2, -7], [-46, 16, 18, 32], [10, -28, -14, -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_bm();". // 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_bm(:prec:=4) 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_bm();". // 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_bm( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2541_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-1, -7, -5, 1, 1]>,<5,R![16, -24, -8, 4, 1]>,<13,R![16, -16, -8, 6, 1]>],Snew); return Vf; end function;