// Make newform 1216.2.a.k in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1216_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_1216_2_a_k();". // 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_1216_2_a_k();". // 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_1216_a();" function MakeCharacter_1216_a() N := 1216; order := 1; char_gens := [191, 837, 705]; 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_1216_a_Hecke(Kf) return MakeCharacter_1216_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], [0], [1], [1], [3], [4], [-3], [1], [8], [0], [-2], [8], [0], [11], [7], [-2], [6], [1], [-10], [-2], [5], [2], [0], [6], [-12], [-14], [14], [10], [2], [12], [-16], [13], [5], [-5], [15], [18], [-2], [-4], [-12], [-18], [-16], [-6], [-19], [-14], [6], [-7], [-22], [-28], [-16], [11], [-29], [5], [4], [-11], [-20], [11], [18], [-24], [15], [-2], [15], [16], [-30], [15], [-26], [10], [-26], [-8], [25], [7], [26], [-27], [-4], [24], [-10], [18], [-19], [23], [-30], [-22], [36], [-30], [28], [-28], [28], [9], [-12], [-1], [-33], [-17], [33], [32], [-38], [4], [29], [24], [-28], [42], [-26], [-19], [-4], [7], [4], [-24], [40], [15], [-3], [6], [30], [26], [-8], [-9], [-27], [4], [25], [28], [-29], [-27], [-25], [30], [-48], [-10], [4], [16], [45], [50], [26], [41], [7], [-42], [23], [-6], [20], [41], [45], [-29], [46], [-44], [22], [-29], [16], [-9], [9], [-30], [-20], [-22], [-2], [-26], [-5], [4], [46], [-7], [7], [-18], [-30], [54], [16], [-10], [13], [-18], [-4], [18], [-16], [-52], [-2], [-36], [8], [-61], [-18], [-21], [12], [-14], [-28], [-50], [-28], [-53], [-20], [23], [8], [-35], [-17], [12], [-14], [-46], [41], [-5], [-6], [50], [60], [-12], [32], [-21], [10], [-44], [28], [-24], [45], [14], [54], [-29], [-12], [48], [-58], [40], [20], [-45], [47], [-12], [57], [-62], [-26], [38], [29], [-24], [-16], [8], [0], [4], [-42], [-39], [62], [-40], [-28], [19], [-18], [17], [46], [54], [-30], [15], [-11], [-74], [-42], [-18], [33], [57], [73], [45], [-36], [-38], [-2], [25], [-16], [-38], [0], [27], [-57], [18], [40], [-45], [34], [-29], [12], [-6], [-3], [36], [43], [-8], [-12], [-38], [76], [6], [-35], [62], [-33], [46], [6], [3], [34], [21], [50], [52], [33], [45], [-82], [80], [-41], [-14], [-70], [19], [-74], [-48], [-37], [-2], [-77], [45], [19], [64], [41], [-80], [2], [-13], [-16], [-70], [58], [30], [48], [81], [54], [-35], [-39], [34], [-79], [70], [25], [14], [-28], [-76], [49], [-34], [-59], [-66], [44], [16], [40], [-2], [-63], [34], [55], [-39], [40], [5], [-54], [22], [58], [22], [32], [-4], [-24], [-45], [41], [-90], [-8], [-77], [63], [13], [-34], [64], [-1], [-44], [30], [49], [36], [-10], [-54], [-44], [78], [60], [-16], [-25], [-88], [68], [21], [-54], [66], [-44], [24], [-16], [69], [-48], [55], [22], [12], [-30], [21], [44], [-54], [94], [-70], [92], [40], [-81], [16], [29], [-25], [-44], [0], [-69], [-29], [-20], [-74], [-5], [-33], [11], [28], [80], [44], [-6], [69], [-100], [46], [-70], [-68], [65], [58], [-18], [-34], [44], [-28], [-18], [50], [-73], [-48], [74], [74], [31], [-5], [-32], [-48], [-15], [-86], [-35], [-46], [-47], [-6], [42], [-100], [-10], [42], [-34], [46], [8], [63], [12], [-4], [84], [-48], [-25], [67], [-101]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1216_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1216_2_a_k();". // 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_1216_2_a_k(:prec:=1) chi := MakeCharacter_1216_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_1216_2_a_k();". // 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_1216_2_a_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1216_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![0, 1]>,<5,R![-1, 1]>,<7,R![-1, 1]>],Snew); return Vf; end function;