// Make newform 8016.2.a.j in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8016_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_8016_2_a_j();". // 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_8016_2_a_j();". // 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_8016_a();" function MakeCharacter_8016_a() N := 8016; order := 1; char_gens := [3007, 2005, 5345, 673]; 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_8016_a_Hecke(Kf) return MakeCharacter_8016_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], [-3], [-6], [2], [0], [0], [-4], [-6], [9], [-1], [2], [8], [1], [-3], [-3], [-12], [-1], [-8], [10], [12], [-17], [-15], [-13], [5], [-14], [-6], [-6], [4], [5], [-12], [1], [-1], [-11], [-12], [-18], [7], [-1], [-12], [10], [-16], [13], [-12], [6], [-24], [18], [-17], [-25], [20], [-1], [16], [-16], [0], [-10], [-27], [15], [-14], [3], [7], [8], [12], [-21], [31], [14], [-6], [-4], [5], [29], [9], [-21], [24], [16], [25], [-29], [-24], [18], [26], [24], [26], [32], [38], [-13], [37], [32], [-9], [-35], [22], [22], [22], [14], [24], [4], [-32], [-24], [36], [-8], [-10], [30], [1], [25], [22], [-36], [0], [39], [-17], [-27], [-6], [-23], [27], [-18], [-8], [22], [16], [-9], [-18], [-24], [4], [-12], [-9], [-30], [20], [22], [0], [-52], [-40], [-17], [20], [32], [-38], [45], [-33], [-8], [-10], [-5], [-10], [39], [-28], [-18], [-15], [-31], [18], [50], [-32], [-35], [37], [36], [-45], [30], [12], [6], [-52], [-2], [-30], [-2], [-27], [-1], [31], [-18], [-38], [30], [-50], [32], [7], [-32], [0], [32], [12], [9], [38], [47], [-26], [-29], [-5], [46], [22], [-24], [25], [0], [5], [-37], [-54], [-51], [8], [-42], [14], [-18], [0], [57], [16], [-16], [-21], [-30], [52], [-50], [-9], [9], [34], [45], [-13], [30], [-33], [15], [28], [-48], [-26], [20], [20], [36], [-32], [-7], [-68], [46], [-28], [-20], [29], [58], [30], [9], [35], [-38], [35], [-4], [-12], [-68], [-26], [-57], [0], [-2], [-74], [-61], [-20], [58], [50], [-18], [72], [0], [20], [-51], [16], [60], [-56], [24], [17], [50], [64], [19], [39], [34], [-6], [58], [2], [-58], [-2], [50], [60], [-53], [-74], [3], [61], [56], [21], [53], [-73], [30], [32], [50], [-12], [7], [54], [-50], [2], [-66], [-11], [50], [-64], [75], [-69], [12], [-8], [52], [40], [-62], [-34], [-37], [12], [-49], [-54], [-51], [0], [42], [-47], [-51], [36], [18], [8], [-49], [27], [-73], [-64], [14], [9], [61], [20], [-40], [18], [77], [56], [-62], [56], [-14], [-48], [66], [-76], [-24], [-10], [-9], [5], [-6], [-60], [-8], [-59], [30], [56], [-32], [-43], [70], [-80], [79], [-4], [22], [-48], [-4], [58], [44], [92], [-20], [-64], [70], [7], [-44], [-82], [-51], [-92], [-54], [-66], [34], [64], [-70], [18], [56], [-8], [-81], [-82], [-11], [-6], [-15], [-27], [90], [-39], [-22], [92], [66], [-86], [53], [82], [39], [68], [-29], [-37], [-20], [56], [68], [-98], [29], [36], [30], [-49], [-34], [58], [34], [-35], [24], [15], [-90], [90], [36], [85], [-32], [-16], [-6], [-54], [36], [-13], [-102], [60], [-61], [-27], [25], [-11], [-88], [6], [26], [-18], [7], [8], [75], [28], [43], [-20], [0], [22], [-78], [-52], [-6], [-96], [18], [37], [-77], [-81], [-6], [61], [-27], [-50], [-35], [22], [108], [29], [-56], [-12]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8016_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8016_2_a_j();". // 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_8016_2_a_j(:prec:=1) chi := MakeCharacter_8016_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(2999) - 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_8016_2_a_j();". // 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_8016_2_a_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8016_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-3, 1]>,<7,R![3, 1]>,<11,R![6, 1]>,<13,R![-2, 1]>],Snew); return Vf; end function;