// Make newform 9408.2.a.bb in Magma, downloaded from the LMFDB on 30 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9408_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_9408_2_a_bb();". // 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_9408_2_a_bb();". // 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_9408_a();" function MakeCharacter_9408_a() N := 9408; order := 1; char_gens := [1471, 6469, 3137, 4609]; 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_9408_a_Hecke(Kf) return MakeCharacter_9408_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], [1], [0], [-1], [0], [8], [-4], [4], [5], [-7], [-8], [-4], [-10], [-6], [1], [9], [-2], [-2], [6], [-2], [-9], [-3], [6], [1], [2], [16], [-15], [-10], [4], [13], [-19], [18], [10], [-6], [7], [20], [-20], [18], [-18], [-20], [20], [0], [-11], [-18], [-20], [-22], [19], [7], [24], [-8], [12], [-15], [1], [-18], [-30], [-17], [-11], [24], [6], [-6], [3], [-4], [-28], [-1], [5], [-20], [1], [-28], [6], [24], [6], [3], [4], [28], [34], [34], [4], [-32], [-23], [24], [-2], [0], [-30], [-19], [-29], [-24], [15], [-14], [-40], [-12], [18], [-31], [27], [-34], [12], [-9], [-14], [8], [-22], [-8], [-9], [29], [-12], [-14], [-23], [15], [36], [30], [-19], [-25], [-44], [6], [-10], [33], [38], [2], [-10], [31], [-16], [18], [-19], [-27], [37], [-12], [-27], [-14], [-10], [-19], [-14], [-34], [-38], [-7], [26], [0], [19], [18], [-38], [-51], [-32], [-2], [1], [16], [43], [-20], [8], [42], [6], [-18], [-46], [-32], [-18], [28], [-4], [12], [-6], [-32], [-34], [29], [-51], [-40], [-6], [7], [59], [50], [24], [-5], [10], [-1], [-3], [57], [42], [-4], [-35], [23], [-52], [10], [51], [-40], [42], [-41], [15], [-50], [-22], [14], [9], [46], [46], [51], [4], [-41], [57], [12], [-18], [19], [10], [46], [26], [26], [34], [15], [0], [-38], [-10], [-24], [-69], [16], [9], [-48], [-18], [-38], [61], [-1], [39], [20], [22], [13], [-28], [58], [-2], [0], [56], [-42], [19], [-61], [8], [-18], [-36], [13], [36], [-2], [44], [-33], [42], [14], [24], [29], [-13], [-12], [-44], [-4], [60], [-69], [50], [-24], [0], [-63], [20], [32], [-12], [26], [10], [14], [38], [39], [39], [26], [-64], [-57], [-13], [-8], [-69], [-38], [-64], [-30], [46], [-53], [-72], [40], [-5], [64], [18], [83], [49], [-53], [8], [33], [44], [-66], [40], [18], [59], [56], [-16], [-2], [32], [-65], [27], [-17], [20], [-6], [87], [52], [-12], [-16], [37], [48], [-10], [-76], [-42], [10], [-45], [25], [51], [22], [34], [-69], [30], [58], [-10], [42], [78], [-24], [56], [32], [42], [-15], [40], [3], [16], [24], [-22], [-77], [0], [-12], [-43], [-50], [-14], [18], [79], [10], [19], [40], [48], [-22], [4], [28], [-42], [15], [-5], [-30], [-72], [75], [37], [45], [-75], [-74], [2], [38], [-69], [36], [19], [62], [-81], [14], [62], [-66], [1], [-60], [86], [-62], [32], [74], [84], [84], [-21], [-14], [-88], [-31], [33], [8], [66], [5], [-16], [-20], [32], [24], [-43], [-14], [90], [-50], [-46], [72], [-48], [-2], [-90], [80], [-76], [-32], [-36], [9], [38], [-21], [-58], [30], [23], [-33], [-54], [-20], [-45], [-76], [84], [-4], [-42], [79], [27], [-2], [-86], [-14], [-19], [-55], [-65], [-11], [70], [-14], [-33], [48], [80], [40], [-18], [-49], [-18], [-54], [93], [-34], [10], [-47], [24], [30], [-28], [-30], [-45], [-12], [14], [-74], [24], [93], [24], [-54], [-79], [-47], [34], [60], [-58], [-5], [30], [-16], [-16], [78], [-10], [2], [50], [31], [-68], [-67], [3], [28], [39], [104], [56], [-46], [24], [-29], [76], [98], [77], [44], [-43], [-28], [6], [-101], [20], [54], [77], [40], [-30], [-81], [-8], [-76], [-57], [37], [26], [-47], [3], [25], [-31], [-24], [89], [2], [47], [46], [68], [63], [110], [-81], [-86], [8], [-35], [-47], [26], [45]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9408_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9408_2_a_bb();". // 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_9408_2_a_bb(:prec:=1) chi := MakeCharacter_9408_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(3581) - 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_9408_2_a_bb();". // 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_9408_2_a_bb( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9408_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-1, 1]>,<11,R![1, 1]>,<13,R![0, 1]>,<17,R![-8, 1]>,<19,R![4, 1]>,<31,R![7, 1]>],Snew); return Vf; end function;