// Make newform 3822.2.a.be in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3822_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_3822_2_a_be();". // 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_3822_2_a_be();". // 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_3822_a();" function MakeCharacter_3822_a() N := 3822; order := 1; char_gens := [2549, 3433, 1471]; 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_3822_a_Hecke(Kf) return MakeCharacter_3822_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], [1], [-1], [0], [-1], [-1], [-6], [-4], [-6], [3], [-11], [4], [12], [-8], [-8], [-5], [-5], [12], [16], [6], [-10], [7], [-17], [-12], [13], [10], [0], [7], [12], [-4], [-7], [1], [-8], [0], [-10], [13], [-18], [-8], [-24], [6], [12], [-16], [-6], [15], [-18], [8], [12], [7], [-15], [6], [6], [-6], [-15], [17], [-6], [-18], [-9], [-7], [-10], [4], [10], [-7], [6], [2], [13], [-3], [-8], [25], [20], [14], [30], [-22], [-7], [26], [16], [-14], [22], [-18], [-18], [25], [16], [-36], [-38], [-14], [-17], [-39], [20], [-11], [34], [-24], [12], [14], [23], [-35], [-28], [-6], [13], [28], [16], [32], [36], [-17], [21], [-4], [28], [1], [5], [-36], [-10], [11], [29], [0], [-36], [-8], [-29], [-12], [20], [-4], [-11], [-12], [22], [-39], [-35], [-27], [-40], [-13], [6], [46], [19], [2], [-26], [-32], [25], [-38], [-28], [31], [-42], [-12], [-3], [12], [-40], [-11], [44], [-25], [-2], [-16], [26], [54], [-8], [-4], [18], [30], [30], [36], [-36], [-10], [32], [-20], [-13], [15], [36], [-44], [-11], [-29], [42], [-26], [-5], [-28], [-27], [-15], [-63], [0], [-50], [47], [-15], [14], [32], [-5], [-24], [-46], [13], [-21], [-30], [42], [40], [17], [16], [10], [25], [52], [29], [-11], [4], [-30], [53], [32], [10], [-26], [54], [42], [-45], [-40], [-24], [66], [-44], [7], [40], [-5], [48], [30], [-46], [-55], [-9], [-15], [0], [-34], [37], [-16], [-38], [-6], [50], [4], [-18], [-27], [19], [40], [-42], [2], [-17], [-48], [-22], [-28], [-51], [4], [-26], [-76], [-19], [31], [-8], [26], [-36], [24], [-59], [-10], [10], [34], [13], [-48], [-44], [26], [64], [38], [-48], [2], [39], [-33], [16], [-38], [27], [-53], [-20], [21], [-40], [-74], [32], [54], [31], [-56], [64], [-43], [12], [64], [-53], [-35], [57], [-80], [-3], [-76], [26], [-84], [-60], [-19], [-58], [-36], [34], [20], [35], [-9], [11], [20], [18], [3], [24], [12], [-26], [-63], [48], [74], [-4], [22], [54], [-9], [-71], [53], [14], [-34], [61], [-6], [-14], [-58], [-84], [46], [-54], [26], [48], [74], [-25], [16], [-29], [18], [-40], [38], [51], [-20], [14], [3], [14], [32], [-80], [9], [-32], [-3], [-36], [12], [-28], [-60], [24], [-6], [-57], [-3], [-48], [72], [-43], [-71], [45], [-49], [18], [-74], [24], [11], [68], [13], [-14], [-31], [56], [-84], [0], [-33], [-90], [16], [-26], [14], [66], [12], [-94], [-1], [18], [-88], [-55], [-85], [42], [46], [-13], [80], [-74], [-20], [-48], [-19], [36], [38], [-54], [82], [20], [-58], [-30], [0], [-20], [4], [28], [-64], [15], [-54], [39], [62], [12], [-81], [-33], [80], [26], [51], [22], [-8], [-56], [42], [5], [-35], [66], [-2], [54], [71], [19], [101], [11], [-34], [-90], [-25], [-22], [-44], [10], [-44], [-75], [-58], [-8], [33], [-98], [46], [15], [38], [-38], [26]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3822_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3822_2_a_be();". // 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_3822_2_a_be(:prec:=1) chi := MakeCharacter_3822_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_3822_2_a_be();". // 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_3822_2_a_be( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3822_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![1, 1]>,<11,R![1, 1]>,<17,R![6, 1]>,<29,R![-3, 1]>],Snew); return Vf; end function;