// Make newform 1849.2.a.k in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1849_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_1849_2_a_k();". // 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_1849_2_a_k();". // 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, -2, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 0, 1]]; Rf_basisdens := [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_1849_a();" function MakeCharacter_1849_a() N := 1849; order := 1; char_gens := [3]; v := [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_1849_a_Hecke(Kf) return MakeCharacter_1849_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, 0], [0, 0, -1], [1, 0, 1], [1, -1, 2], [-1, 2, -3], [-1, -1, -3], [0, -2, 0], [-5, 1, -1], [1, -4, 2], [-6, 0, 0], [7, -3, 7], [-3, 3, 1], [8, -2, 0], [0, 0, 0], [-8, 2, 1], [2, 3, 3], [-4, 3, -1], [4, -4, 2], [4, -2, -2], [-4, -4, 6], [-3, -1, 5], [-2, -1, 3], [-3, -1, -6], [-3, 2, -9], [-8, 5, 2], [4, 7, -3], [-1, 4, -2], [2, 1, 1], [8, -6, 4], [-13, 9, -12], [-9, 0, -9], [-10, 6, 1], [-2, 7, 4], [-5, 3, -6], [-4, -1, 0], [-6, 13, -8], [11, -5, 9], [-11, 1, -4], [11, -5, 6], [-3, 5, -1], [-10, 12, -8], [16, -9, 2], [-2, -10, 4], [5, 2, 1], [5, 4, -1], [-8, -7, -3], [2, -3, 11], [-12, 11, -4], [-6, 3, -10], [-1, -17, 11], [-16, 12, -12], [1, 9, -8], [1, -2, 2], [-6, -8, -6], [-1, -3, 3], [-10, -3, -7], [7, 17, -6], [-13, 1, -2], [0, -11, 1], [-12, 1, 6], [22, 0, -4], [2, -10, 4], [14, -8, 13], [-1, -7, 1], [-10, 12, -14], [13, 8, -1], [-1, 10, 1], [-1, 7, -15], [4, -13, 13], [25, -7, 12], [-17, -2, -2], [-18, 14, -6], [10, -11, -4], [24, -5, 6], [5, -20, 4], [19, -12, 12], [11, -22, 7], [4, -2, 7], [-8, 10, -8], [-5, -8, 7], [-15, -2, 9], [17, -4, 0], [24, 6, -6], [9, -1, -8], [10, -3, 4], [-2, 6, 12], [-16, 7, -12], [29, 6, 0], [15, -10, -13], [-5, 9, 3], [-4, 2, 21], [3, -23, 7], [0, -4, 7], [-10, 9, 18], [4, -20, 9], [25, -16, 6], [-15, 0, 7], [-4, -11, 19], [14, -17, -4], [2, -3, 14], [-16, 4, -20], [27, -23, 32], [-11, 6, -5], [15, -18, 13], [-2, -9, -15], [4, -2, -21], [-14, 13, 0], [10, -18, 11], [2, 19, -14], [-4, -6, 1], [-4, 0, -21], [-7, -4, 15], [-19, 3, -12], [0, 6, -11], [5, 7, -2], [-7, 21, -22], [-14, 18, -14], [-27, 19, -19], [6, -4, 28], [-20, 2, -15], [10, 6, -15], [29, -5, -2], [-6, -13, 0], [-5, -26, 0], [-9, 14, -12], [15, 14, -5], [22, -1, 12], [8, 14, -7], [-20, -5, -8], [-6, -2, 3], [5, -16, 1], [-32, -3, -6], [12, -9, 3], [-18, 13, -13], [-14, 12, 1], [23, -3, 17], [-1, 37, -17], [-8, 11, -18], [-24, 6, 11], [-13, 27, -20], [-36, 13, -32], [13, -7, 11], [26, 2, 13], [-40, 14, -19], [29, -22, 24], [11, -8, -2], [-13, 8, 21], [-4, 11, 1], [27, -31, 27], [-12, 32, 7], [-11, 0, -2], [9, -26, 2], [16, 5, 23], [-24, 10, -3], [-19, -11, 8], [-3, 11, -2], [-11, 14, 23], [2, 9, -30], [10, -2, 23], [22, -31, 20], [-7, 8, 16], [9, -27, -10], [-3, -1, 12], [18, -10, 6], [14, -18, 13], [-5, -23, 20], [-21, 26, 20], [22, -1, -17], [14, 14, -9], [-33, -20, 10], [-6, -13, 10], [16, -20, 0], [-30, 2, -2], [-7, -9, -6], [4, 2, 10], [34, 8, 1], [1, 21, 1], [-9, 23, -10], [16, -3, 16], [8, -12, 0], [18, 14, 1], [-3, -23, -5], [-12, 15, -23], [-8, -23, 8], [-16, 19, -16], [-26, -2, -17], [-5, -17, -2], [-8, 19, -17], [8, 10, 21], [-28, -10, 10], [17, 12, 0], [-15, -17, 17], [3, 20, -18], [20, -28, 12], [-10, 1, -11], [17, 4, -6], [-23, -3, -26], [-13, 46, -12], [12, 7, -7], [-16, 23, 0], [-20, 4, -2], [4, -30, 18], [-22, 4, 5], [26, -10, -27], [16, -32, 10], [5, -18, -13], [-43, 12, -16], [14, -24, 38], [-43, 1, -15], [30, -21, 26], [20, 14, -22], [-28, -11, -16], [5, -31, 28], [-4, 14, 6], [-21, 1, -15], [2, -9, -38], [2, 37, -22], [-10, -12, -21], [28, -21, 1], [21, -31, 6], [7, 23, -14], [13, 8, 7], [-4, -32, 12], [-7, 53, -20], [4, -17, -32], [34, -14, 50], [48, -8, 10], [-60, -6, 3], [-43, 3, -17], [-22, 44, -19], [35, -2, 0], [2, -14, 19], [-40, 7, 2], [-19, -27, 19], [-26, 8, 13], [8, -8, 14], [6, -15, 30], [-25, 11, 6], [34, 3, 20], [30, -31, 48], [23, 28, -20], [15, -25, -29], [1, 12, 15], [13, 18, -20], [7, -24, -18], [4, -35, 18], [46, 2, -5], [-13, -5, 4], [-17, 25, 17], [8, -33, 34], [-24, 4, 14], [-56, 18, -28], [27, -7, 33], [4, 38, -6], [8, -7, 26], [6, 21, 0], [15, 29, -6], [24, -7, -27], [33, -18, 39], [-11, 45, -19], [-14, -18, 18], [-7, -7, 5], [-27, 22, -3], [36, -22, 3], [24, 4, -10], [1, -21, -7], [15, -11, 37], [-6, -35, -9], [11, 20, 12], [-19, -11, 0], [-7, 39, -22], [-21, -14, -5], [40, -3, 0], [6, -40, 6], [19, 7, -9], [-40, 38, -51], [-1, -6, -24], [2, 42, -11], [-36, -5, -3], [-48, -9, 2], [-32, -18, 19], [-15, -23, 26], [-9, 31, 15], [17, -25, 39], [-28, 44, -5], [-51, 17, -44], [-19, 21, 31], [31, 10, 6], [37, 20, -3], [-50, 1, 7], [29, -21, 15], [-31, 32, 14], [45, -21, 3], [-46, -15, 11], [4, 20, -19], [18, -23, -12], [-4, -6, 26], [24, -14, 0], [-28, 43, -12], [13, -37, 44], [30, -38, 43], [27, -16, -1], [-12, -16, -10], [47, 8, 7], [-19, -28, 15], [42, -20, 33], [-26, -7, -21], [-22, 2, 15], [44, -8, 23], [20, -13, 3], [4, -19, 39], [42, 9, 2], [-12, 14, -14], [31, -33, 45], [7, -15, 31], [-10, -8, -3], [19, -45, 29], [-9, -28, 9], [51, 2, -1], [-17, 18, -22], [-3, 53, -14], [-43, -26, 4], [38, 1, -1], [-26, 17, -22], [-4, 9, -25], [8, -6, 0], [-10, 5, -9], [-39, 21, -5], [-16, -15, -6], [-20, 10, -25], [43, -41, 28], [5, -27, 1], [-42, -1, -22], [30, -42, 17], [-2, 31, -4], [39, -8, 5], [-22, -3, 8], [80, -13, 17], [-31, 31, 22], [-2, -32, 41], [-13, 30, -17], [8, -39, 2], [16, 33, 7], [-23, 18, -36], [0, 6, 17], [-21, 61, -32], [-10, -25, 28], [-40, 9, -23], [-62, 26, -53], [-14, 37, -42], [-79, 2, -2], [8, -30, 10], [-6, 23, -13], [29, -34, 25], [50, -3, -9], [-37, 56, -36], [-58, 10, 15], [-30, -14, -12], [-15, -8, -1], [14, -22, 13], [26, 23, -23], [-63, 30, -17], [-15, 27, 0], [46, -18, 25], [-40, 14, -23], [8, 1, -13], [-21, 30, -5], [-41, 3, -11], [-22, 44, 31], [-34, 42, -15], [-42, -10, 10], [-6, 6, -33], [-67, 5, -11], [0, 27, -3], [25, 18, 14], [-17, -18, -36], [57, -41, 76], [-14, 24, -41], [-22, 5, 32], [-47, 11, -41], [50, 22, -5], [25, 38, -22], [32, -64, 25], [10, -1, -38], [-7, -28, -16], [-31, 41, -52], [-17, 26, -27], [6, 27, -30], [41, -24, 62], [-50, 41, -10], [41, -13, -10], [-7, -13, 4], [9, 41, -23], [73, -9, -1], [-16, 25, 0], [-5, 49, 7], [3, -1, -12], [1, 65, -17], [-7, -12, 24], [-47, 42, -53], [34, -41, 19], [-24, 42, -6], [10, -10, -17], [-28, -41, 14], [-36, -5, 4], [42, -4, 45], [9, -23, -13], [-15, 21, -21], [15, -52, 45], [-65, 18, -4], [12, -1, -5], [-5, 7, -31], [37, -9, 38], [-40, 32, 29], [-48, 22, -41], [10, -43, 38], [21, -9, -42], [74, -14, 21], [-49, -20, 15], [36, -22, -3], [51, -3, 9], [-52, 63, -62], [24, -8, -31], [-11, -51, 27], [-11, -41, 30], [-1, 23, -22], [-21, 11, 26], [-33, 13, -24], [18, -43, 44], [-8, -16, -16]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1849_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1849_2_a_k();". // 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_1849_2_a_k(:prec:=3) chi := MakeCharacter_1849_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_1849_2_a_k();". // 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_1849_2_a_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1849_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![1, -2, -1, 1]>],Snew); return Vf; end function;