// Make newform 5054.2.a.v in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5054_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_5054_2_a_v();". // 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_5054_2_a_v();". // 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 := [5, 0, -5, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, 0, 1, 0], [0, -3, 0, 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_5054_a();" function MakeCharacter_5054_a() N := 5054; order := 1; char_gens := [1445, 1807]; v := [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_5054_a_Hecke(Kf) return MakeCharacter_5054_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, 0, 0], [-1, 1, 0, 0], [0, 0, -1, 1], [1, 0, 0, 0], [1, 0, 0, -2], [1, -1, 1, -1], [1, 1, 1, -1], [0, 0, 0, 0], [1, 0, 0, 2], [-3, 2, 0, 0], [-4, 0, -1, 3], [-7, 2, -2, -2], [1, -1, 2, 4], [2, 0, -3, 2], [2, -2, 5, -3], [1, -4, 2, 0], [-2, -2, -3, -3], [4, 0, 7, -1], [-3, 4, 5, 2], [-4, 0, -6, 4], [1, 5, -3, -1], [2, 0, 5, -6], [0, -6, 0, 2], [-1, 1, 3, -5], [4, -6, -3, -3], [-3, 1, -8, -4], [-5, 1, -2, -4], [3, -8, 2, 0], [-7, 0, -3, 4], [-9, -2, -3, 2], [-4, 2, -3, 2], [-9, 3, 7, 1], [-6, 6, 6, 0], [-14, 0, -5, -3], [3, 6, -7, 2], [-7, -2, -6, -2], [-5, 3, 2, 6], [8, 2, -2, 4], [2, -6, 6, 2], [6, 4, -1, 5], [6, -8, 3, 0], [6, -10, 3, 5], [3, 2, -8, -4], [-16, 0, 2, 2], [1, -6, 1, -6], [-13, 1, -2, 6], [1, 0, 0, 10], [-2, -12, 5, -1], [-4, 2, -7, -5], [-5, 3, 3, -5], [4, 4, 9, 0], [6, 6, 7, -2], [1, 3, -3, -3], [8, 0, 14, -4], [-8, 4, -5, -5], [13, 0, 0, -6], [8, 2, 3, 11], [-4, 4, 1, -3], [-2, -4, 6, -6], [3, 10, 14, 0], [11, -1, 4, 10], [12, 0, -1, -5], [-12, -4, 2, -2], [-10, 2, -3, -3], [-11, -7, -12, -2], [-9, 4, -4, 0], [0, -4, 12, -2], [1, -6, -3, -8], [10, -2, 4, 12], [-4, -2, -1, 1], [-7, 7, -2, -6], [9, 2, 6, 6], [-7, -11, 0, 2], [-16, 0, -7, -12], [-3, -2, 13, 4], [16, 0, -7, 1], [-7, -2, 1, -4], [-15, 1, 9, -3], [-2, -2, 6, 2], [3, 1, -9, 5], [-15, -9, -8, 8], [-19, 12, -4, -6], [-10, 2, -12, -6], [-15, 3, -13, 1], [-10, 14, -5, -5], [-6, 10, -1, 2], [10, -2, 2, 4], [14, 2, 10, 8], [2, -2, 12, -6], [7, -10, 9, 6], [14, 6, 13, 5], [-16, 2, -14, 4], [-9, -20, 4, 2], [-12, 4, -14, 0], [5, -6, 0, 2], [-8, -4, -18, -8], [24, -2, 28, 4], [10, 8, 11, -1], [3, -5, 8, -10], [-3, -2, 14, 4], [15, 2, 4, -10], [-9, 6, -2, 16], [-4, 4, -6, -18], [-22, 4, -15, 6], [4, 6, 10, -2], [-4, -4, -21, -7], [-8, -8, -9, -5], [7, 11, -13, 15], [13, 4, 6, 0], [-18, 16, 15, 7], [12, 4, 26, 8], [-12, 4, 18, 4], [-7, -4, -7, 16], [-1, 1, -2, 10], [-10, 14, -10, -12], [-5, 12, -1, 4], [-18, 12, -8, -4], [-3, 3, -7, 5], [-12, -2, -27, -2], [5, 0, 4, -12], [-8, -14, -9, 9], [-15, 4, -3, 0], [16, 8, 6, -12], [18, 2, -11, -6], [4, 4, 19, 7], [2, -20, 3, 8], [13, -2, 18, 8], [-22, -10, -17, 3], [-14, 6, 5, 3], [32, -8, -5, -5], [14, -10, 0, -12], [1, 16, -3, -2], [-10, -4, -4, -6], [-24, -4, 6, -12], [-22, -10, 1, 7], [4, -8, -10, -16], [-3, -3, 4, -4], [-3, 13, 11, 21], [1, 15, -8, -6], [-33, -6, -15, 6], [-4, 18, -18, -2], [-19, 8, -16, -18], [-13, -16, -2, -2], [19, -6, 1, 8], [-12, 2, -17, 13], [0, 6, -19, -1], [-12, -10, -13, 13], [-19, -9, 4, 0], [14, 18, 5, 5], [17, 4, 24, -6], [-6, 4, -5, 8], [11, 5, 2, 24], [21, 0, -3, 2], [2, 8, -10, -2], [0, 16, -1, 10], [10, 12, 29, -2], [9, 18, -1, 0], [-28, -4, 15, -1], [-23, 1, -16, -6], [-25, 5, -23, -17], [6, -10, -4, 12], [37, 8, 6, -2], [-3, 0, -5, -14], [2, -8, -17, -5], [-1, 16, 6, 6], [21, -9, 28, -4], [-36, 4, -12, 2], [3, 5, -11, 5], [13, 4, -2, -12], [33, 9, 7, 1], [-11, -12, -17, -6], [10, 14, -14, 8], [4, 16, 17, -2], [-1, 2, -17, -12], [8, 6, 16, 4], [2, -8, 32, 2], [7, 10, 9, -20], [5, 6, 16, 8], [33, 7, 19, 1], [8, -12, -16, -12], [-4, 2, -17, -8], [4, 6, 13, 5], [-33, 0, -7, 12], [20, 2, -16, -2], [-26, 12, -4, 6], [20, -6, 0, -2], [-12, 2, -30, -6], [8, -4, 14, -4], [-25, 10, 14, 10], [-5, -9, -15, 13], [13, 13, 0, -6], [16, 8, 12, 6], [-34, 6, -27, -10], [51, 5, 1, 5], [19, 2, 10, 8], [-39, -7, -10, 0], [-27, 2, 9, -6], [13, -2, -5, -8], [-22, -2, -17, 9], [9, 13, 13, -3], [-1, 2, 12, 20], [2, 2, -13, -15], [46, 4, 13, -13], [17, 13, 12, -18], [15, -3, -15, -5], [-27, 11, -36, 0], [20, 0, -2, 4], [11, -14, 33, 4], [2, -10, 39, -8], [-7, 15, -28, 0], [-3, 0, 18, 12], [6, -2, -3, 1], [24, 20, -7, 10], [12, -2, 4, 18], [21, 9, 42, -2], [-37, -9, -33, 7], [-4, -4, 20, -18], [22, -6, 0, 2], [2, -2, -34, -6], [-33, -6, -33, 4], [36, -6, 40, 0], [7, -15, 21, 1], [-1, 0, -3, 30], [45, 2, 7, 14], [-22, 12, -26, 2], [9, -16, 32, -2], [30, -8, -3, -11], [9, -12, 25, -12], [-1, 19, -11, 3], [0, 18, 13, 4], [1, 6, -6, 26], [-32, -2, -34, -8], [-35, 0, 0, -10], [-11, 20, -13, -18], [0, -2, 16, 12], [16, 0, 15, 31], [5, 7, 10, 14], [-29, -2, -14, 16], [12, 8, 18, -14], [5, 19, -17, 11], [9, -4, 10, -8], [47, 17, 14, -14], [50, 2, -1, 1], [-22, 20, -7, -8], [14, -12, 4, 4], [-33, 9, -34, -16], [33, -5, 6, 14], [-32, -8, -35, 5], [-27, -16, -14, 22], [-1, 0, -2, -2], [10, 30, 17, -2], [-22, -6, -24, 18], [-7, 2, -16, -26], [12, -24, 21, 3], [16, 12, 23, -3], [7, -8, 3, -38], [-26, 4, -17, -6], [3, -15, 22, -16], [19, -3, 7, 15], [29, 9, 17, 1], [-11, 16, 16, -10], [6, 14, 5, -2], [-2, 0, -14, -30], [40, 2, 0, 8], [21, 3, -17, 11], [-27, -3, -36, 0], [11, -24, 5, 6], [0, -6, -2, -20], [18, 16, 7, -8], [1, 4, -16, -12], [10, -6, 7, 7], [29, -20, 11, 14], [12, 20, -13, 7], [-5, -4, -11, -2], [-38, 2, -27, 9], [-8, 8, -28, 24], [27, -20, -26, -12], [-23, -18, 2, 0], [-32, -6, 7, -14], [6, 22, 1, -17], [26, -8, -12, -8], [27, 0, 8, 32], [-10, 6, -1, 17], [-37, -1, 17, 1], [3, 5, -11, -23], [38, 0, 29, -6], [-8, -16, -5, -6], [-4, 12, -19, 30], [36, -12, 49, 7], [16, -22, -33, -7], [42, 14, -2, 8], [7, -17, 39, -5], [-16, -2, -22, 14], [2, -4, -8, 24], [3, 0, 4, -16], [12, -24, 7, 11], [0, -2, -14, 18], [27, 9, 22, -4], [-10, 8, 8, -20], [28, -14, 3, 11], [20, -20, 21, 13], [-23, 24, 19, 10], [-9, -24, 11, -32], [-10, 18, -48, 8], [-10, 20, -26, 6], [-12, -10, 15, 20], [-52, -6, -27, 12], [39, -5, -6, 18], [-5, -2, 3, -28], [8, -28, 4, 0], [39, -15, 31, 13], [42, 8, -11, 4], [-11, -18, -34, 4], [-24, 2, 16, -2], [-27, 8, 4, 0], [-11, 5, -23, -19], [-13, -23, 20, -6], [13, -28, 26, 8], [-1, 39, -2, -16], [-37, -8, -8, -6], [-11, -9, 1, -21], [-30, 2, -40, -2], [-16, -36, -17, 11], [8, 10, -31, -10], [-25, -10, 2, -14], [-4, -4, -3, 37], [-6, -26, 25, -6], [26, 4, 38, 8], [14, -28, -11, 4], [-38, -4, 14, 2], [15, 12, 21, 6], [14, -28, 4, -4], [-38, 0, -47, 17], [-18, 0, -38, -12], [-5, 8, -8, 30], [3, -7, -15, 9], [-36, -14, -50, 8], [36, 4, 13, 1], [27, 21, 10, 4], [-31, -15, -16, 12], [-30, 32, 16, 2], [-31, -14, 12, -2], [-6, -8, -2, 10], [2, 18, 14, 20], [3, -12, -2, -18], [10, -24, 4, 2], [10, 8, -8, 6], [12, 2, 11, -2], [3, 15, 0, -10], [-25, -15, -49, 9], [13, 17, -2, -20], [8, 8, -5, -30], [15, -12, 0, -6], [44, 4, 4, -12], [-2, -8, -9, 22], [11, -1, -1, 5], [0, 2, -38, 4], [-6, 24, -14, 2], [-2, -12, -13, 20], [-2, 36, 33, 8], [-31, -4, -11, 14], [-13, -7, -26, 10], [34, 16, 14, 0], [1, 20, -3, 38], [4, -6, -12, -6], [-61, -8, -16, 16], [-18, -4, -27, 31], [-20, -4, 8, 0], [-36, -10, -16, -18], [8, -42, -10, 4], [33, 19, 22, 12], [52, -4, 26, 22], [8, 0, -2, -18], [1, 35, 21, -11], [32, 0, 10, 22], [-12, -4, 4, 10], [-3, -8, 13, 34], [-2, -18, -20, -26], [-35, -11, -20, -4], [7, 33, 13, -3], [-77, -5, -13, 7], [-51, -6, -4, 12], [-7, -6, -38, -14], [55, 2, 29, -22], [-23, -3, 26, -8], [-14, 20, 9, 29], [8, 16, -3, 26], [39, -7, 42, 8], [23, 2, -2, 22], [-42, 8, -22, -30], [-24, -2, -26, 4], [34, 0, 4, -36], [4, -24, -20, -2], [-26, 6, -41, 3], [-23, -12, -40, -4], [9, -18, 7, -40], [-2, 26, 17, 7], [-9, -7, -19, 7], [-12, 16, -5, 10], [-7, -20, 29, -24], [-34, 16, -15, -13], [1, 12, -1, 0], [-16, 16, 0, 8], [30, 20, 6, 0], [4, 12, 19, 11], [-10, -24, 15, 1], [-36, 22, -11, -6], [19, -10, -2, -16], [-20, 12, 6, 18], [-4, 20, 15, 1], [4, 0, 24, 10], [-14, -24, -23, 14], [-4, -26, 29, -4], [-5, 4, 15, -20], [-15, -12, 16, 22], [-6, -6, -33, -31], [15, 4, 20, -16], [20, -18, 11, -37], [14, 0, 34, -16], [-30, 12, -17, -5], [39, -6, 33, 14], [-15, -31, 6, -14], [-1, -8, -30, -16], [-27, 21, -3, 17], [30, -22, -17, -1], [7, 9, -36, 6], [24, 4, -28, 8], [-36, 8, -32, 18], [-6, 12, -47, 25], [-4, 0, 2, -28]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5054_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5054_2_a_v();". // 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_5054_2_a_v(:prec:=4) chi := MakeCharacter_5054_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_5054_2_a_v();". // 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_5054_2_a_v( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5054_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, -6, 1, 4, 1]>,<5,R![1, 2, -6, -2, 1]>,<13,R![1, -8, -11, -2, 1]>],Snew); return Vf; end function;