// Make newform 3381.2.a.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3381_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_3381_2_a_l();". // 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_3381_2_a_l();". // 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_3381_a();" function MakeCharacter_3381_a() N := 3381; order := 1; char_gens := [2255, 346, 442]; 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_3381_a_Hecke(Kf) return MakeCharacter_3381_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 := [[2], [-1], [-4], [0], [-5], [2], [0], [5], [-1], [-2], [-6], [6], [-5], [8], [9], [9], [-9], [5], [4], [12], [0], [-10], [18], [-10], [18], [-5], [19], [4], [-12], [-2], [9], [5], [9], [-12], [3], [-19], [-7], [13], [-19], [-2], [8], [14], [-23], [-19], [-2], [-3], [13], [-10], [18], [13], [-12], [12], [-17], [-10], [-17], [7], [18], [8], [7], [2], [20], [4], [8], [15], [25], [30], [29], [22], [2], [2], [14], [32], [-27], [-34], [-30], [-6], [6], [-30], [3], [-14], [18], [8], [27], [11], [36], [-36], [6], [10], [-2], [-35], [6], [-32], [-20], [14], [-20], [36], [19], [30], [-19], [11], [-8], [46], [-40], [-17], [8], [2], [17], [18], [42], [0], [-8], [34], [18], [20], [-10], [-9], [31], [28], [24], [24], [5], [49], [-36], [36], [4], [-15], [-6], [24], [-41], [6], [12], [15], [-32], [8], [21], [-14], [-20], [23], [-46], [-6], [-2], [-4], [27], [-21], [-26], [-46], [-8], [-1], [40], [-30], [-17], [-6], [32], [16], [44], [24], [2], [-26], [27], [-20], [32], [-55], [28], [18], [-3], [-42], [5], [56], [-38], [-17], [-45], [-30], [-9], [32], [34], [-14], [19], [-50], [45], [26], [-49], [30], [41], [-41], [51], [22], [49], [-4], [1], [8], [2], [36], [32], [-21], [19], [12], [-34], [-18], [24], [-51], [-19], [-8], [-18], [17], [-62], [61], [0], [18], [0], [2], [-27], [-15], [-2], [-10], [40], [50], [-1], [38], [40], [-66], [-61], [-55], [20], [-52], [13], [17], [14], [-52], [-32], [24], [37], [20], [-40], [-44], [51], [-36], [71], [-34], [36], [-8], [-39], [-24], [-24], [-29], [63], [35], [-10], [-2], [-26], [57], [44], [12], [73], [65], [26], [-34], [20], [39], [-63], [-26], [68], [11], [-50], [-47], [-35], [-35], [-75], [-10], [64], [4], [-16], [-10], [-10], [2], [44], [-26], [-32], [27], [-56], [4], [1], [-48], [-26], [26], [46], [30], [-50], [-15], [40], [30], [-17], [86], [-78], [12], [34], [-76], [-4], [-11], [55], [-14], [6], [-78], [34], [-44], [-6], [-21], [12], [-21], [-29], [10], [-9], [74], [-27], [-24], [-37], [43], [75], [84], [-73], [-36], [19], [82], [3], [-1], [-39], [-25], [-20], [-20], [47], [15], [-86], [-24], [-20], [-27], [76], [11], [38], [-66], [-50], [32], [-62], [35], [33], [10], [-31], [34], [-28], [-17], [18], [-26], [-22], [27], [28], [-57], [34], [-89], [42], [-18], [38], [-6], [72], [-43], [72], [89], [-68], [-17], [-23], [10], [-30], [84], [-60], [18], [-57], [-17], [22], [20], [-23], [7], [-12], [-12], [-68], [90], [-77], [8], [-60], [19], [-13], [68], [92], [-12], [40], [-21], [66], [64], [-67], [-62], [-55], [-18], [30], [-54], [-8], [-52], [-2], [-78], [57], [-70], [24], [-10], [-32], [-12], [-74], [-10], [47], [88], [42], [-18], [-42], [16], [12], [48], [-78], [47], [62], [4], [76], [75], [1], [30], [66], [51]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3381_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3381_2_a_l();". // 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_3381_2_a_l(:prec:=1) chi := MakeCharacter_3381_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_3381_2_a_l();". // 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_3381_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3381_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-2, 1]>,<5,R![4, 1]>,<11,R![5, 1]>,<13,R![-2, 1]>],Snew); return Vf; end function;