// Make newform 9408.2.a.bu in Magma, downloaded from the LMFDB on 29 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_bu();". // 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_bu();". // 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], [-3], [0], [3], [4], [0], [-4], [0], [-9], [1], [-8], [0], [-10], [6], [3], [3], [10], [-10], [6], [2], [1], [-9], [6], [-1], [18], [-8], [-3], [-14], [0], [-5], [-9], [18], [2], [-18], [1], [4], [-16], [-6], [-18], [12], [-8], [0], [-19], [-6], [-20], [14], [19], [-27], [4], [-24], [24], [-1], [27], [6], [6], [-21], [-11], [-8], [6], [14], [-33], [8], [-24], [-31], [-9], [20], [-7], [12], [-26], [24], [-30], [19], [-8], [8], [-18], [6], [4], [24], [-25], [0], [22], [-12], [-34], [-35], [-33], [12], [-1], [-30], [-8], [36], [18], [-41], [-33], [2], [12], [3], [-18], [-4], [-26], [8], [-3], [39], [-36], [-34], [23], [21], [24], [18], [11], [7], [16], [-6], [-34], [7], [-30], [-34], [18], [-3], [-24], [-14], [29], [33], [33], [8], [15], [10], [18], [13], [10], [50], [-42], [7], [-38], [12], [-19], [-6], [50], [33], [0], [2], [3], [40], [15], [4], [24], [10], [-42], [50], [-6], [-32], [-6], [32], [24], [8], [-6], [-8], [-6], [35], [9], [0], [-6], [1], [-39], [42], [-36], [13], [-14], [-49], [33], [-27], [58], [60], [5], [-41], [-36], [2], [-15], [16], [-26], [49], [45], [34], [-18], [54], [-3], [34], [-22], [-13], [-12], [-31], [-51], [20], [6], [-33], [-54], [-10], [-62], [30], [-42], [21], [-8], [-14], [26], [48], [-33], [16], [-27], [24], [-58], [-46], [9], [1], [69], [36], [2], [-5], [0], [-18], [-30], [40], [-56], [-30], [-11], [-15], [64], [42], [36], [13], [-36], [-74], [-16], [49], [6], [-58], [-60], [35], [-15], [60], [72], [12], [56], [-5], [70], [12], [60], [25], [60], [32], [-24], [-2], [-54], [-30], [2], [-69], [51], [22], [-52], [3], [29], [16], [-9], [-62], [-44], [-6], [-70], [81], [-12], [-4], [-39], [16], [-34], [-43], [-17], [5], [16], [-3], [16], [38], [48], [54], [37], [-12], [-20], [50], [24], [-25], [9], [7], [-12], [6], [45], [-36], [-60], [4], [9], [-80], [-42], [36], [-46], [-58], [-39], [7], [63], [2], [50], [39], [58], [-6], [-14], [-6], [-30], [60], [32], [-48], [38], [-21], [-48], [53], [-24], [-64], [-46], [-33], [16], [48], [-37], [62], [-22], [-18], [-75], [-2], [81], [-32], [-24], [-10], [-12], [-20], [54], [41], [19], [82], [-36], [87], [-77], [-9], [9], [10], [-46], [6], [15], [-4], [11], [42], [55], [10], [42], [30], [-45], [12], [-6], [46], [24], [30], [12], [56], [35], [42], [-56], [-7], [-51], [-64], [66], [-9], [-80], [16], [24], [-12], [11], [66], [62], [6], [30], [16], [-84], [50], [-42], [-32], [4], [-4], [12], [-55], [6], [-9], [-58], [-78], [23], [79], [42], [80], [33], [76], [48], [4], [-6], [-21], [-29], [-62], [90], [26], [-81], [-1], [21], [9], [26], [-70], [75], [84], [-56], [72], [6], [-51], [106], [78], [-33], [-70], [-78], [-27], [48], [38], [12], [86], [87], [104], [66], [10], [-48], [-19], [-20], [26], [25], [51], [-102], [-68], [18], [53], [-6], [104], [12], [74], [34], [2], [-42], [-45], [-36], [77], [33], [28], [-27], [-104], [72], [-70], [-32], [87], [16], [-22], [-43], [-44], [-9], [-24], [-70], [61], [-12], [-42], [101], [0], [-26], [93], [64], [96], [45], [11], [30], [-49], [-105], [-71], [-27], [76], [-93], [98], [25], [-62], [-72], [-25], [-18], [75], [-86], [-28], [-33], [25], [-58], [-105]]; 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_bu();". // 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_bu(: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_bu();". // 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_bu( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9408_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![3, 1]>,<11,R![-3, 1]>,<13,R![-4, 1]>,<17,R![0, 1]>,<19,R![4, 1]>,<31,R![-1, 1]>],Snew); return Vf; end function;