// Make newform 9800.2.a.j in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9800_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_9800_2_a_j();". // 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_9800_2_a_j();". // 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 Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_9800_a();" function MakeCharacter_9800_a() N := 9800; order := 1; char_gens := [7351, 4901, 1177, 5001]; v := [1, 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_9800_a_Hecke(Kf) return MakeCharacter_9800_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], [-2], [0], [0], [4], [2], [3], [0], [-3], [-6], [9], [0], [5], [6], [-9], [6], [8], [8], [-14], [11], [2], [9], [6], [11], [11], [-8], [-15], [-8], [-14], [-15], [-8], [12], [-17], [-6], [-20], [20], [-10], [24], [0], [-16], [4], [-8], [-13], [-5], [18], [21], [22], [-11], [22], [-8], [22], [11], [18], [-16], [-18], [3], [-2], [7], [-8], [-13], [-2], [-14], [24], [1], [31], [12], [-2], [29], [-2], [14], [-15], [0], [-16], [-6], [4], [-21], [34], [38], [18], [-31], [30], [20], [-31], [-21], [35], [-12], [-27], [-2], [24], [-11], [30], [9], [-23], [30], [38], [16], [12], [45], [-22], [-30], [40], [36], [26], [-11], [-12], [-14], [-8], [13], [9], [26], [13], [-26], [17], [-10], [-15], [-35], [16], [48], [-30], [14], [44], [43], [6], [14], [-22], [12], [-8], [29], [-29], [-14], [-16], [27], [-32], [34], [-43], [14], [20], [-8], [-8], [10], [-10], [20], [4], [-14], [-32], [-29], [38], [-22], [20], [7], [4], [-37], [-2], [20], [12], [47], [7], [-6], [2], [-26], [4], [10], [-13], [22], [3], [-48], [-31], [-44], [37], [-36], [-54], [16], [0], [9], [43], [-21], [-48], [12], [0], [-10], [32], [30], [12], [-6], [7], [-28], [34], [46], [49], [0], [-6], [66], [14], [22], [-20], [-6], [-34], [32], [34], [28], [36], [-37], [-22], [55], [30], [-24], [11], [26], [-41], [60], [-1], [-40], [-27], [-18], [-45], [-27], [-1], [-14], [4], [-12], [64], [55], [-3], [45], [20], [-6], [-31], [-63], [-37], [-10], [-24], [36], [9], [39], [28], [13], [-34], [22], [-56], [-63], [-36], [34], [4], [54], [-75], [-11], [31], [-66], [44], [-57], [8], [-50], [47], [62], [66], [-48], [58], [2], [18], [-33], [36], [6], [18], [26], [-33], [-4], [-20], [22], [-6], [-26], [30], [-74], [65], [-41], [14], [-73], [48], [-6], [11], [-32], [31], [19], [-11], [74], [-78], [-15], [67], [54], [48], [-45], [0], [8], [-3], [-70], [16], [-30], [-56], [44], [-54], [-66], [-25], [22], [-25], [26], [12], [-25], [10], [12], [24], [-16], [-75], [-60], [-70], [-38], [-63], [70], [-38], [11], [73], [15], [2], [-38], [-82], [-20], [-89], [-70], [78], [80], [-23], [-46], [40], [4], [-55], [-14], [-12], [48], [-74], [41], [-42], [-49], [-64], [27], [-44], [-84], [-24], [66], [20], [34], [28], [18], [64], [83], [-20], [79], [28], [-83], [-21], [54], [43], [4], [8], [-66], [-39], [64], [0], [-26], [88], [25], [-6], [-32], [-10], [-51], [-60], [-17], [-56], [-60], [-80], [21], [-27], [3], [-84], [78], [8], [34], [40], [-19], [-97], [58], [-4], [-47], [-85], [88], [0], [8], [95], [19], [56], [34], [-34], [-94], [60], [14], [-23], [-58], [-30], [67], [98], [-3], [70], [-20], [62], [98], [30], [78], [-55], [-30], [-36], [-87], [-65], [72], [50], [30], [8], [-90], [63], [26], [-38], [90], [94], [-27], [61], [24], [-96], [-109], [-32], [9], [31], [16], [18], [93], [96], [-73], [-38], [33], [-25], [13], [-40], [-73], [106], [-18], [-62], [-21], [64], [-63], [77], [-22], [4], [6], [46], [109], [-10], [9], [88], [78], [4], [65], [41], [44], [93], [10], [-68], [-54], [-63], [-17]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9800_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9800_2_a_j();". // 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_9800_2_a_j(:prec:=1) chi := MakeCharacter_9800_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(3361) - 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_9800_2_a_j();". // 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_9800_2_a_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9800_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![2, 1]>,<11,R![-4, 1]>,<13,R![-2, 1]>,<19,R![0, 1]>,<23,R![3, 1]>],Snew); return Vf; end function;