// Make newform 9800.2.a.s 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_s();". // 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_s();". // 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], [-1], [0], [0], [3], [-6], [-5], [-1], [7], [2], [5], [-3], [2], [4], [5], [1], [-15], [5], [9], [0], [7], [1], [12], [-7], [-2], [-3], [-15], [-9], [-5], [18], [-8], [-5], [-11], [12], [-17], [-5], [-9], [-13], [-12], [-13], [-13], [10], [-11], [-3], [6], [13], [-4], [-24], [11], [-23], [-11], [20], [17], [-16], [-21], [9], [9], [7], [17], [6], [-13], [6], [4], [-15], [-1], [-3], [29], [10], [-15], [-2], [3], [21], [23], [-11], [24], [-3], [-29], [-17], [27], [-27], [-20], [34], [3], [-2], [-21], [31], [-26], [17], [30], [16], [-3], [-31], [-7], [-24], [-35], [-32], [9], [33], [-7], [-41], [8], [5], [-41], [-33], [-13], [-33], [-20], [-45], [-17], [-22], [13], [-43], [-30], [9], [-16], [-17], [-44], [15], [-35], [44], [-23], [26], [39], [-27], [-5], [-50], [-29], [39], [32], [19], [-5], [16], [53], [-10], [-3], [-22], [19], [-17], [18], [39], [-12], [-9], [-47], [36], [-11], [24], [34], [39], [-1], [-17], [-23], [14], [20], [-3], [1], [16], [-11], [-7], [-6], [33], [-37], [-22], [-48], [35], [-3], [-21], [15], [-41], [30], [19], [31], [-42], [9], [17], [21], [34], [-16], [15], [16], [41], [47], [-44], [26], [-41], [9], [9], [-59], [-13], [-21], [45], [-1], [-36], [17], [-63], [-67], [3], [19], [1], [22], [-47], [-61], [-24], [-37], [53], [-36], [39], [31], [39], [26], [-31], [-3], [54], [-40], [21], [37], [-3], [-51], [-11], [3], [2], [63], [44], [23], [-21], [28], [-46], [51], [23], [61], [-3], [41], [-47], [16], [-5], [28], [15], [13], [-39], [-68], [36], [25], [-41], [-73], [-5], [66], [23], [-56], [-59], [-21], [4], [-26], [-47], [29], [42], [79], [9], [-25], [-13], [66], [3], [25], [60], [21], [34], [27], [55], [10], [50], [-60], [-51], [-7], [5], [11], [-51], [-18], [-31], [-29], [55], [-1], [-33], [3], [-37], [-24], [6], [-55], [-3], [81], [-10], [57], [66], [-45], [-77], [73], [-40], [-74], [-43], [-21], [-46], [-45], [8], [-49], [9], [51], [36], [-15], [-86], [9], [62], [61], [1], [65], [23], [-5], [65], [-8], [77], [-76], [67], [-26], [-18], [-59], [-59], [-66], [-4], [49], [25], [-35], [13], [-57], [30], [51], [-51], [-88], [55], [83], [-4], [-38], [43], [-26], [73], [21], [-6], [30], [-64], [37], [-36], [-55], [-1], [20], [-85], [-77], [53], [18], [63], [75], [-6], [63], [37], [-27], [24], [70], [69], [69], [-75], [-57], [13], [42], [-51], [-38], [-1], [-65], [-21], [22], [-43], [93], [29], [0], [-61], [-23], [-74], [-19], [34], [8], [9], [28], [51], [-73], [23], [43], [-32], [-46], [7], [-33], [-99], [-83], [-71], [65], [50], [-20], [-21], [-99], [-59], [51], [-9], [85], [55], [1], [22], [-69], [27], [39], [-27], [-92], [-23], [66], [-27], [13], [67], [102], [13], [-9], [-61], [-48], [12], [30], [63], [-49], [-42], [25], [53], [-23], [20], [-31], [-60], [94], [9], [35], [47], [76], [40], [-9], [35], [-46], [3], [-62], [18], [84], [87], [-107], [-59], [71], [0], [-59], [49], [-7], [-78], [-89], [-7], [31], [17], [-45], [-103], [-19], [-5], [-85], [-7], [4], [-63], [-93], [-64], [89], [-92], [-100], [-90]]; 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_s();". // 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_s(: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_s();". // 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_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9800_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![1, 1]>,<11,R![-3, 1]>,<13,R![6, 1]>,<19,R![1, 1]>,<23,R![-7, 1]>],Snew); return Vf; end function;