// Make newform 9801.2.a.bl in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9801_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_9801_2_a_bl();". // 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_9801_2_a_bl();". // 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 := [4, 5, -6, -1, 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, -5, 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_9801_a();" function MakeCharacter_9801_a() N := 9801; order := 1; char_gens := [9560, 244]; 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_9801_a_Hecke(Kf) return MakeCharacter_9801_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, 0, 1, 0], [0, 0, 0, 0], [-1, 1, 0, 1], [0, -1, 0, 0], [0, 0, 0, 0], [-1, -1, -1, 1], [-1, -2, 1, 0], [-2, -2, 0, -1], [-4, 1, 0, -1], [-1, 0, -1, 1], [0, 0, -1, 1], [1, -1, -2, -2], [-1, 3, -1, 0], [5, 0, 1, 0], [2, -2, 1, 0], [1, 3, 0, 1], [-1, 1, 0, -1], [-4, 1, 0, 0], [4, -3, -1, -2], [1, -3, 2, 0], [-6, 1, 2, 1], [-2, -2, -2, -1], [5, 2, -1, 3], [0, 4, 2, 0], [7, 2, -2, 2], [-6, 4, -2, -3], [3, 0, -4, -3], [3, 2, -1, 4], [-3, 2, 1, -2], [2, 1, 1, 1], [6, 3, -4, -2], [-8, 0, 2, 0], [-8, 9, 3, 0], [0, -2, -2, -3], [1, 4, -1, 2], [-5, 0, 1, -2], [7, -3, -2, 0], [-10, 4, -1, -5], [-3, 4, -1, 3], [0, -3, 4, 4], [0, 3, 1, -2], [10, -1, 1, 3], [2, 4, -5, 3], [-2, -4, -2, 2], [-12, 1, 4, 1], [2, -1, 7, -1], [2, 0, 4, -3], [-18, -2, -2, 1], [9, 1, 5, 0], [6, -3, -4, -1], [2, 7, 2, -4], [2, -7, 0, -2], [-6, 10, 0, -1], [-10, 3, 6, 3], [-11, 0, -3, -1], [-1, 2, 1, 3], [-2, 3, -5, 0], [-10, 3, 4, -4], [-11, -2, 1, -8], [-3, 0, -3, -6], [9, -3, -1, -3], [-4, -5, -4, 2], [-5, 1, -3, -5], [-3, -3, -4, -4], [-5, 7, -1, -1], [0, 1, -6, -1], [-5, 0, 10, 1], [9, -4, -3, 1], [-19, 2, 3, 0], [-12, 2, -2, 6], [15, -4, -1, 4], [4, 7, 4, -1], [11, -7, -10, -4], [-11, -6, 9, -1], [10, -7, 2, -8], [-11, 5, -2, -9], [13, -10, -2, 1], [-4, -8, 1, -3], [11, 2, 4, 10], [-9, 5, -7, -2], [0, -5, -1, 5], [5, 0, 4, 5], [12, -11, -6, -2], [2, -11, -4, -2], [6, -6, 8, 6], [-16, -1, 7, 0], [-11, 2, -8, -3], [0, 5, -10, 7], [-4, 3, -8, 9], [-11, -6, -1, 2], [-10, -9, -5, 3], [-10, -10, 6, 2], [-15, 6, 6, 1], [0, 3, 8, 1], [10, -12, -3, -7], [-5, 7, -1, 0], [4, -10, -6, -6], [-6, 0, 5, 2], [3, -4, -7, -3], [5, -18, -3, 0], [2, -5, 4, -9], [12, 13, -4, 11], [-1, -9, -1, -3], [20, 0, 6, 2], [16, -1, 8, -5], [-5, 4, 6, 5], [11, 3, 6, -6], [-5, -3, 7, 6], [-8, -12, 0, 5], [-27, 0, 1, -3], [12, 11, -12, 1], [-15, 4, 7, 1], [15, -2, -8, -4], [5, 2, 14, 1], [-20, -6, -3, -6], [3, -8, 11, 3], [-4, -15, 0, -6], [-5, 1, -3, -8], [18, -6, 7, -1], [16, -2, 2, -6], [2, 4, 7, -3], [8, -3, -2, 5], [-18, 5, -2, 6], [35, -2, 0, -4], [-7, -15, 6, 0], [-4, 4, -6, 7], [-1, -6, 12, 0], [-7, -1, -6, 4], [-7, -2, -2, -2], [-21, -1, -9, -1], [2, -3, 2, 6], [18, -10, -12, -8], [5, 3, 0, 7], [-1, 4, 2, 9], [5, 3, -1, 4], [4, 0, 4, -8], [4, -3, 10, -1], [16, -9, -6, -4], [-21, -4, 0, -8], [-4, -7, -2, -7], [-1, 3, 11, -1], [14, 13, -2, 3], [-14, 5, 6, 3], [14, -15, 2, -7], [18, -1, -9, -2], [-11, 4, -5, -5], [-10, -4, -12, -3], [-37, 0, -5, -5], [-20, 6, -5, -1], [-3, -5, 3, 4], [-10, 5, 4, -1], [-18, -12, 9, -7], [-11, 10, -6, 7], [5, 13, -9, 7], [0, 8, 4, -6], [-7, 1, 0, 0], [29, 1, -9, -2], [22, -1, -7, -1], [12, 8, 10, -7], [-13, 9, -7, 1], [7, -3, 14, -6], [-7, 3, -15, -5], [-10, 9, 4, 1], [-9, 11, 5, 0], [-11, 9, 9, 10], [-13, 13, 6, 0], [15, 12, 1, -6], [-28, 15, 8, 9], [0, 3, 0, -1], [8, 13, -6, 12], [-27, -3, 3, -8], [28, 17, -3, 2], [8, -3, -12, -3], [37, 6, -1, 2], [-4, 5, 11, -5], [14, 2, -7, -4], [1, -11, 1, -7], [-12, -8, 6, -3], [21, -11, -1, 11], [-10, 11, 10, -6], [-3, 19, -5, 6], [2, -2, -6, 1], [17, 3, 9, 11], [39, 2, 3, -4], [-11, -12, 8, 0], [-28, 10, 9, 3], [-7, 0, -7, -2], [-3, -9, 4, 1], [-18, 22, 6, 12], [0, -22, 6, -6], [8, -11, 6, -14], [34, 4, 4, -1], [-21, -10, 10, -6], [-4, 4, -10, 8], [10, 6, -8, -7], [-21, -4, 6, -13], [3, -16, 11, -6], [-1, -10, 0, -3], [-1, -19, 7, -13], [-38, 4, 0, 6], [24, -14, 0, -5], [-44, 2, 4, 1], [19, -1, -4, 11], [34, 0, -2, -3], [-17, -9, 10, -5], [-7, -1, 2, 1], [18, -4, 10, 3], [-8, 19, -4, -5], [-7, 7, -1, 8], [-8, 1, -4, -9], [23, -9, 7, 9], [16, -7, -2, 10], [19, 1, -10, 1], [-34, 3, -2, -5], [-14, -2, 2, 7], [1, 7, -2, -2], [-4, -4, 4, -8], [-5, 11, -1, 14], [4, -22, -5, -5], [16, 10, 1, 0], [-15, 19, -1, 13], [10, -15, -4, 6], [13, 14, -6, 3], [17, -9, -2, 16], [-16, 24, 0, -7], [-29, -3, -3, 3], [13, -23, -11, 3], [-9, 0, 8, 5], [6, -10, 12, 6], [-17, 23, 3, -4], [-9, 24, 2, 1], [17, -1, 3, -4], [-1, 1, 3, -8], [22, -7, 14, -9], [-1, 1, 1, -15], [-2, -11, -2, -16], [12, 19, -7, 0], [16, -10, 18, 9], [13, -18, -1, -17], [4, -11, -9, 6], [8, -2, -1, -15], [-38, -8, 8, -3], [-15, -21, 8, 2], [16, -4, -11, -5], [8, 9, -14, 8], [-1, 11, 15, 4], [17, -20, -1, -3], [-31, 4, -9, 11], [0, -16, -6, -16], [15, 14, 3, 7], [-50, -1, -10, 0], [-25, 2, -5, 3], [-1, -2, -6, -15], [8, 13, -5, -4], [-9, 26, 7, 0], [-34, 8, 6, 3], [15, 21, -14, 9], [-22, 12, 10, -3], [7, -2, -1, -11], [-6, 8, 2, 5], [-11, 20, -5, 17], [1, -11, 11, -8], [-10, 5, -10, 2], [2, 7, -8, -14], [23, 1, 14, -1], [18, -20, 5, 5], [-5, 4, -21, 2], [2, -7, 21, -2], [12, 10, 12, -3], [29, 7, 3, -3], [13, 17, -4, -6], [16, -7, -7, -4], [-12, 18, -3, 5], [-4, -29, 6, 5], [-6, -1, 0, 10], [4, 18, -12, 20], [0, 4, 20, -3], [-9, 10, 11, -2], [-19, 9, -3, 9], [3, 14, -5, 0], [27, -4, 7, -13], [8, -5, 7, -7], [9, 2, 5, -10], [18, 0, 2, 13], [-26, -5, -18, 12], [2, 18, -5, -3], [-28, 4, -3, -12], [39, -5, 9, -8], [7, 5, -18, -1], [-36, 24, 4, -9], [4, 5, -2, 9], [38, 12, 3, -2], [-38, 5, -2, -2], [-39, 2, -7, -17], [-7, 22, 1, -5], [5, 7, 21, -2], [15, -2, 14, -4], [0, 11, 14, -1], [16, -3, -8, -1], [10, 7, -12, 4], [35, 18, -7, 10], [-35, 10, -1, -2], [-4, -1, -24, -10], [-5, 8, 18, 10], [27, -25, -3, -3], [-12, -22, 22, -1], [-1, -8, -12, -21], [-4, -1, -2, 8], [34, -10, -7, 5], [11, 10, 9, 3], [1, 2, -7, -4], [12, 11, 10, 4], [9, -18, 11, 5], [12, 0, -11, 5], [-10, -5, 6, -9], [-6, -6, 16, -6], [-5, 7, -20, -7], [7, -24, -7, 5], [5, 19, -16, -5], [8, 18, 10, 4], [-22, 15, 4, -5], [-12, 3, 14, 10], [24, 6, -8, 8], [5, 1, 0, 7], [34, -19, -22, -8], [-31, -8, -3, 1], [17, 2, 16, -4], [26, -8, -2, 17], [-13, -5, -5, -2], [0, -19, -16, 12], [-5, -9, -1, -14], [27, 2, 23, 4], [-25, 5, 7, -5], [31, -17, -9, -5], [-59, -11, -1, 1], [-29, 8, -16, 11], [-22, -18, -9, 10], [-15, 10, -13, 20], [8, 9, -9, -8], [-1, 2, -9, 24], [22, -13, -16, -9], [16, -5, 21, -4], [-34, 24, 8, 1], [-4, -6, 32, 8], [59, 12, -13, 0], [-10, -16, 16, -14], [25, -7, 14, 1], [22, 9, -10, -5], [1, 2, -9, -6], [15, -5, -2, 23], [6, -30, 4, -3], [-40, -18, 11, 1], [6, 8, 11, 16], [12, 16, -12, 16], [-4, 11, -6, -16], [8, -14, 12, 0], [-21, 14, -8, 24], [-12, 27, 10, 5], [16, 23, 0, 13], [18, 10, 7, 19], [14, 13, -10, -8], [-45, -2, -1, 3], [-11, -8, 23, -2], [-3, 6, 9, -16], [6, 12, -10, 14], [10, -14, 11, 3], [21, 24, -15, 0], [11, -5, -7, 1], [-11, 1, 21, 8], [40, 15, -9, 10], [1, -9, 6, -14], [-10, 5, 16, 10], [28, 2, -18, -7], [16, 12, -26, -6], [6, 18, -8, 0], [-15, 25, 2, 17], [-13, -3, -19, -7], [15, 13, -5, 13], [8, -14, 12, 18], [41, -21, 1, -3], [21, -1, -26, -11], [21, 1, -3, -8], [-29, -1, -5, 3], [-4, -12, -10, 17], [5, 0, -7, 5], [41, -8, 19, 22], [-5, 6, -4, -6], [-33, -22, 6, -15], [21, -2, 5, -1], [-47, 11, 22, 6], [-26, 1, -9, -9], [-1, 8, -2, 9], [-16, 2, 17, 8], [19, 18, 5, -4], [-62, 3, 10, 2], [25, -6, -1, 22], [15, -7, -5, 17], [-11, -19, 9, -21], [4, 9, -13, 11], [2, -18, -20, 10], [14, -23, -16, -11], [23, -6, 24, -6], [-29, -3, 1, -1], [13, -21, -4, 6], [-17, 4, 7, -16], [-17, -2, -17, -18], [-25, 22, 2, 2], [41, -4, 15, 21], [-11, -3, 6, -28], [-31, 9, -5, 20], [-3, 23, 17, -5], [-27, 21, 1, 12], [24, -9, 11, 0], [3, 5, -7, -9], [31, -12, -9, 9], [-22, -8, -3, -21], [-8, 32, 2, -9], [4, -21, 1, -19], [29, 5, 17, -13], [-57, -12, 13, -5], [8, -1, 9, 12], [0, 6, -20, -13], [-28, -16, -8, -7], [44, -12, -20, -13], [-4, 6, -1, 27], [-49, 12, 2, -2], [20, -7, 7, 20], [-37, -10, 21, -6], [39, -1, -4, 6], [29, -22, -3, 18]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9801_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9801_2_a_bl();". // 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_9801_2_a_bl(:prec:=4) chi := MakeCharacter_9801_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_9801_2_a_bl();". // 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_9801_2_a_bl( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9801_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![13, 8, -9, -1, 1]>,<5,R![31, -29, -9, 4, 1]>,<7,R![4, -5, -6, 1, 1]>,<17,R![-236, -169, -24, 5, 1]>],Snew); return Vf; end function;