// Make newform 3822.2.a.v 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_v();". // 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_v();". // 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_v();". // 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_v(: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_v();". // 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_v( : 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;