// Make newform 9800.2.a.k in Magma, downloaded from the LMFDB on 28 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_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_9800_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 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_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_9800_2_a_k(: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_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_9800_2_a_k( : 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;