// Make newform 3822.2.a.bb in Magma, downloaded from the LMFDB on 28 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_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_3822_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_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], [-3], [0], [-5], [1], [3], [3], [-3], [-9], [10], [1], [-4], [-13], [-12], [-10], [6], [-3], [0], [6], [-9], [-16], [-14], [-4], [-2], [2], [-19], [-8], [15], [0], [0], [7], [21], [4], [-8], [1], [-7], [-2], [-3], [12], [22], [-10], [17], [10], [-22], [23], [-7], [14], [0], [-4], [-4], [-14], [2], [-5], [26], [-4], [-12], [20], [-32], [6], [4], [2], [4], [2], [-22], [18], [-4], [7], [-6], [28], [10], [-16], [0], [-14], [-26], [9], [-26], [6], [34], [13], [21], [30], [-18], [16], [-23], [30], [5], [-18], [-41], [29], [23], [-5], [-24], [-2], [-30], [-40], [-17], [-7], [-2], [-13], [-36], [12], [-45], [18], [20], [10], [-26], [36], [11], [28], [-7], [-9], [19], [23], [41], [-12], [49], [40], [29], [38], [-38], [-5], [-38], [-15], [20], [-26], [-10], [-40], [-35], [2], [-2], [-42], [-14], [-8], [-44], [-27], [31], [-33], [42], [-36], [-9], [32], [-18], [15], [-11], [0], [-16], [54], [-36], [-10], [46], [43], [-49], [-44], [28], [19], [-6], [36], [8], [30], [-13], [36], [23], [12], [-9], [-33], [22], [26], [-8], [-58], [-9], [48], [-24], [-10], [51], [-30], [-44], [-34], [52], [35], [47], [-60], [-26], [30], [-42], [46], [-8], [55], [38], [9], [20], [-25], [-12], [-59], [-30], [35], [-38], [-16], [48], [-30], [20], [49], [8], [48], [16], [-26], [14], [-40], [59], [34], [-27], [-28], [-46], [-50], [-47], [41], [66], [27], [-42], [6], [-28], [50], [5], [-64], [49], [44], [-13], [40], [-3], [13], [64], [-36], [35], [0], [26], [1], [25], [52], [74], [-30], [-12], [32], [-55], [-74], [-60], [42], [-40], [-38], [42], [-57], [-67], [-78], [6], [0], [36], [1], [-44], [-52], [36], [-29], [30], [34], [54], [-3], [-14], [-60], [32], [-4], [-62], [14], [-42], [-10], [-17], [4], [-44], [52], [57], [-43], [20], [-20], [-60], [-25], [-10], [-56], [23], [-48], [-41], [-24], [-64], [-27], [9], [33], [-43], [-20], [26], [30], [-72], [-42], [-31], [-23], [50], [76], [-2], [-26], [-39], [58], [12], [-65], [-36], [-6], [-46], [0], [34], [-29], [-78], [0], [32], [-42], [33], [6], [-32], [-70], [58], [-11], [-27], [-70], [-80], [-21], [-39], [1], [-11], [-75], [-24], [0], [68], [-30], [33], [16], [-74], [-55], [-62], [-42], [69], [34], [-24], [32], [-73], [20], [-52], [38], [4], [-38], [64], [-8], [-16], [51], [-5], [0], [72], [-66], [-57], [63], [21], [-12], [-2], [-10], [35], [-53], [57], [24], [-34], [-42], [-57], [63], [-6], [80], [-21], [5], [63], [44], [-30], [7], [-13], [-66], [-64], [17], [-45], [-47], [-88], [-20], [46], [66], [36], [46], [38], [-54], [-40], [-57], [11], [24], [24], [96], [45], [28], [-46], [-75], [2], [57], [-1], [16], [-48], [3], [0], [-7], [41], [-24], [-16], [-53], [-66], [40], [-95], [62], [20], [-12], [-25], [42], [-48], [66], [35], [32]]; 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_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_3822_2_a_bb(: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_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_3822_2_a_bb( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3822_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![3, 1]>,<11,R![5, 1]>,<17,R![-3, 1]>,<29,R![9, 1]>],Snew); return Vf; end function;