// Make newform 2541.2.a.bd 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_bd();". // 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_bd();". // 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, 0], [-1, 0], [2, -3], [-1, 0], [0, 0], [-2, 2], [-3, -3], [-1, -1], [5, -5], [2, 0], [4, -7], [-4, -5], [6, 3], [-2, 2], [2, 0], [-4, 10], [10, -2], [4, -8], [8, 0], [-4, 8], [2, 2], [14, 0], [10, -8], [-7, 5], [-10, 4], [-6, 5], [1, -7], [-13, 5], [4, -9], [4, -2], [0, 2], [-12, 6], [10, -2], [13, -11], [20, 0], [2, 0], [0, 6], [14, -12], [8, -10], [-15, -3], [10, -1], [-6, -8], [6, -15], [-20, 1], [-4, 0], [-1, -1], [2, 0], [3, 11], [-2, -12], [18, 2], [-4, 6], [-13, 9], [8, -8], [8, 10], [1, 1], [-17, 9], [2, 16], [-16, -5], [25, -9], [-14, 8], [-28, 9], [-2, -11], [0, 15], [-4, 2], [-26, 10], [8, -22], [-2, 20], [32, -3], [14, -1], [-26, 2], [6, 4], [14, -13], [-8, 15], [12, 7], [-4, 4], [-26, 8], [6, 10], [-8, 14], [12, 4], [-32, 2], [18, -2], [7, -7], [-6, 5], [-14, -2], [4, 11], [13, -5], [16, 16], [-6, -4], [-26, 20], [-8, 12], [16, 2], [24, -8], [-36, -2], [-11, 7], [10, 10], [20, -12], [-14, 5], [-10, -1], [-11, 5], [-12, -3], [30, 4], [2, 12], [-8, -20], [2, 10], [20, -30], [8, -14], [-22, 16], [-18, 9], [13, -9], [-6, 0], [-1, 7], [1, 3], [22, 14], [16, -17], [4, 16], [-14, 14], [1, 29], [2, -20], [0, -16], [14, 13], [18, -6], [-10, -16], [2, -16], [-6, 29], [-7, 29], [18, -26], [24, -27], [34, -18], [-5, 19], [-2, -18], [6, -34], [-6, -7], [-4, -4], [-11, 7], [6, -32], [-34, 12], [-26, -8], [12, -15], [-14, 15], [6, 14], [-16, 20], [2, -20], [18, 4], [-7, 7], [16, 8], [14, 12], [6, 6], [6, 8], [20, -16], [-6, 27], [14, 20], [7, -25], [-2, 6], [-2, 30], [-44, -2], [-24, 8], [12, -12], [-11, -11], [-2, -6], [7, 3], [-38, 9], [34, -24], [-4, -28], [0, 26], [-46, 12], [42, 4], [-4, 14], [4, 30], [-14, 28], [-13, -9], [-18, 37], [-44, 20], [-10, -25], [-35, 3], [28, 4], [2, -9], [12, -20], [14, -44], [-12, -12], [-16, 36], [2, -6], [0, 10], [25, -29], [-6, 13], [3, -23], [-21, 15], [37, -5], [4, 35], [36, -35], [12, 20], [-16, -10], [26, 7], [16, 22], [26, -5], [17, -45], [46, -32], [12, 3], [-20, 17], [38, -4], [-24, 48], [12, -12], [-23, 13], [10, 6], [6, -26], [30, -4], [-18, -1], [-4, 20], [32, 16], [10, -26], [-7, 45], [51, 1], [-35, 25], [-6, -2], [38, -52], [-6, 20], [-10, 10], [34, -34], [-34, 48], [-9, 29], [38, 6], [43, -15], [28, -53], [48, -8], [22, 22], [6, 26], [-27, 51], [2, -15], [22, -1], [-31, 25], [-12, 0], [17, -41], [-12, 16], [6, -22], [-40, -8], [-20, 23], [2, 16], [-38, 28], [-10, 44], [46, -23], [38, 0], [12, -32], [-52, 37], [12, -9], [-39, 11], [-14, 19], [14, -24], [-28, -5], [-34, 52], [-4, 6], [-24, 20], [36, -7], [-26, 37], [-24, -16], [24, -30], [-14, -15], [39, 5], [28, 13], [52, -8], [2, 0], [38, 8], [-16, -4], [-46, 9], [-2, 10], [46, -14], [-17, 43], [-11, 25], [-30, 18], [42, -36], [-4, -20], [14, 6], [-30, -4], [-12, -32], [-16, 16], [-8, -10], [30, -30], [-35, 37], [14, 31], [47, -19], [9, 15], [-50, 50], [40, -4], [20, -36], [26, -56], [-8, -36], [4, 27], [5, 15], [32, 5], [-38, 56], [1, 49], [22, 15], [-46, 12], [-12, 40], [4, 26], [-8, 22], [15, 13], [-70, 9], [-33, -5], [61, -7], [36, 0], [-45, 31], [-4, -2], [-20, 6], [44, -28], [-18, 47], [42, 6], [-77, -7], [-16, 32], [-16, 32], [-29, 33], [-12, 39], [-36, 16], [24, 24], [6, -46], [-66, 14], [-6, -49], [-26, 16], [48, -6], [-12, -36], [-22, 8], [12, -54], [40, -5], [-31, 15], [17, -7], [52, 10], [44, -26], [0, 6], [-50, 50], [59, -21], [-42, 3], [24, -44], [26, -36], [-24, -16], [-53, 19], [2, -4], [30, 22], [28, -46], [35, -79], [35, -17], [-36, -26], [-40, 61], [56, -19], [-84, 2], [23, -57], [52, -8], [30, -58], [54, -25], [24, -40], [-2, 48], [-26, -28], [27, -33], [-28, 71], [-67, 11], [-34, 10], [-56, -11], [-69, 45], [53, 5], [-16, -6], [68, 0], [-52, -10], [-36, 12], [35, -47], [-37, -29], [-1, -27], [23, 9], [37, -59], [12, -19], [18, -4], [48, -2], [-51, 43], [-9, -7], [47, 27], [-65, 37], [-6, -52], [11, 7], [10, -16], [-52, 54], [57, -33], [16, -26], [-34, 33], [56, -44], [-1, 31], [-4, 10], [32, -6], [-40, 16], [-8, 41], [-76, -6], [46, -60], [-40, -24], [-26, -26], [-62, -8], [-85, 17], [57, -75], [29, -33], [-27, 13], [-2, -13], [-23, 19], [56, -19], [14, 3], [-22, 30], [-44, 24], [20, 20], [-40, 14], [-92, 0], [-38, 65], [31, 15], [-24, 45], [-10, 4], [-62, 16], [-52, 9], [72, -20], [-24, -34], [-84, 4], [-3, -9], [-32, -44], [9, 55], [-19, 5], [-18, -11], [-12, -19], [-46, 8], [40, 28], [24, -14], [-10, -16], [46, -21], [-54, 20], [-48, 84], [39, 7], [23, 13], [-4, -44], [16, -63], [-34, 72]]; 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_bd();". // 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_bd(: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_bd();". // 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_bd( : 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]>,<5,R![-11, -1, 1]>,<13,R![-4, 2, 1]>],Snew); return Vf; end function;