// Make newform 9702.2.a.bg in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9702_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_9702_2_a_bg();". // 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_9702_2_a_bg();". // 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_9702_a();" function MakeCharacter_9702_a() N := 9702; order := 1; char_gens := [4313, 199, 5293]; v := [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_9702_a_Hecke(Kf) return MakeCharacter_9702_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], [-2], [0], [-1], [2], [-3], [7], [-7], [-5], [2], [3], [6], [11], [-7], [-4], [11], [10], [-4], [-5], [-8], [-8], [14], [2], [15], [3], [-10], [18], [4], [2], [-19], [-8], [6], [17], [17], [-5], [1], [6], [-10], [14], [-9], [-10], [-8], [24], [15], [10], [20], [-14], [10], [-10], [-9], [-10], [-4], [5], [-14], [30], [-20], [8], [24], [3], [24], [-3], [28], [27], [15], [30], [-14], [8], [24], [8], [10], [-18], [0], [-22], [-16], [31], [-6], [-3], [32], [32], [3], [-17], [40], [5], [5], [41], [-36], [40], [13], [-24], [13], [18], [18], [-26], [-40], [-30], [4], [-18], [-20], [16], [-11], [9], [-42], [13], [-43], [-18], [-12], [-21], [0], [0], [36], [2], [-20], [-26], [-10], [-12], [40], [0], [-10], [-20], [-11], [-10], [-21], [39], [18], [-27], [-31], [9], [-14], [22], [20], [-4], [-40], [15], [-30], [28], [4], [-37], [-50], [-14], [-28], [50], [-4], [-10], [-17], [24], [46], [21], [-22], [-28], [26], [36], [8], [-20], [-24], [13], [43], [4], [26], [15], [-27], [-10], [-3], [-36], [-44], [49], [8], [30], [-16], [-60], [8], [-46], [-42], [-6], [4], [10], [3], [26], [40], [14], [34], [-2], [-53], [-63], [9], [-44], [-20], [38], [-20], [2], [-13], [48], [-50], [-6], [48], [6], [40], [-15], [58], [-56], [22], [-33], [62], [-10], [55], [-56], [-70], [40], [6], [48], [12], [-48], [26], [60], [-36], [-65], [-68], [-66], [-8], [-12], [-32], [-5], [-6], [56], [-30], [28], [-24], [-27], [-37], [-18], [31], [-41], [35], [-67], [46], [18], [43], [-50], [-28], [56], [11], [12], [58], [-43], [-49], [54], [20], [25], [15], [-50], [58], [59], [29], [-62], [62], [20], [10], [-37], [4], [-20], [-72], [18], [64], [-30], [-28], [-42], [58], [-6], [19], [34], [-15], [2], [-26], [-31], [-20], [-18], [-3], [-40], [68], [22], [-48], [-32], [72], [52], [27], [-45], [-74], [43], [16], [-57], [64], [25], [30], [-42], [52], [-26], [-68], [18], [82], [-13], [-44], [57], [-31], [-1], [88], [-38], [-60], [11], [53], [26], [0], [-18], [-57], [34], [72], [-12], [65], [48], [51], [33], [17], [41], [-67], [-20], [59], [-70], [2], [-54], [-6], [-6], [46], [-6], [11], [-56], [4], [-12], [-39], [19], [-23], [43], [46], [-90], [33], [74], [-8], [-6], [53], [22], [42], [-6], [5], [65], [-28], [9], [-52], [-39], [-20], [8], [-39], [-48], [-56], [43], [24], [72], [-48], [-79], [-63], [-56], [-62], [-65], [-36], [82], [90], [0], [47], [45], [24], [-90], [62], [10], [76], [-76], [0], [45], [-47], [-76], [32], [59], [-21], [33], [79], [52], [-59], [-8], [-7], [-8], [31], [-84], [-40], [-101], [30], [0], [-40], [-72], [-55], [-32], [-53], [-57], [20], [51], [-78], [25], [37], [-9], [-70], [64], [-64], [52], [90], [22], [-8], [78], [-89], [52], [-5], [-68], [-28], [39], [-34], [36], [29], [-46], [-60], [80], [-39], [-30], [-54], [46], [6], [33], [101], [-48], [-70], [-30], [64], [7], [-61], [-78], [-70], [70], [-41], [-95], [108], [45], [50], [40], [-70], [36], [-8], [-78], [88], [58], [50], [-93], [36], [54], [43], [83], [-46], [-50], [-1], [-63], [87], [38], [17], [-8], [60], [88], [-78], [48], [74], [17], [14], [49], [98], [78], [-75], [111], [-50], [68], [-108], [68], [53], [59], [48], [74], [-56], [-60], [-75], [8], [-87], [-16], [97], [-48], [-79], [-110], [83], [72], [-2], [41], [62], [-50], [16], [-38], [33], [-97], [4], [40], [-72], [-68], [38], [40], [107], [32], [38], [-60], [79], [-85], [-112], [-8], [-72], [-54], [43], [48], [-55], [60], [67], [86], [114], [15], [104], [-32], [38], [92], [-58], [72], [16], [35], [49], [-106], [55], [77], [-48], [-36], [-7], [18], [4]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9702_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9702_2_a_bg();". // 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_9702_2_a_bg(:prec:=1) chi := MakeCharacter_9702_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(4027) - 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_9702_2_a_bg();". // 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_9702_2_a_bg( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9702_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![2, 1]>,<13,R![-2, 1]>,<17,R![3, 1]>,<19,R![-7, 1]>,<23,R![7, 1]>,<29,R![5, 1]>],Snew); return Vf; end function;