// Make newform 2646.2.a.w in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2646_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_2646_2_a_w();". // 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_2646_2_a_w();". // 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_2646_a();" function MakeCharacter_2646_a() N := 2646; order := 1; char_gens := [785, 1081]; v := [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_2646_a_Hecke(Kf) return MakeCharacter_2646_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], [0], [0], [0], [-6], [-5], [6], [4], [-6], [-6], [1], [-1], [-6], [-1], [-6], [6], [-6], [1], [-1], [-12], [-2], [-1], [6], [0], [-17], [12], [19], [-18], [5], [-12], [-13], [-12], [6], [7], [12], [17], [22], [11], [-18], [0], [0], [10], [0], [-1], [12], [-11], [-13], [-8], [-6], [-5], [-24], [-24], [-17], [18], [-6], [-6], [0], [7], [17], [-12], [13], [18], [-17], [-12], [22], [-18], [20], [2], [24], [1], [-36], [-6], [28], [26], [-1], [6], [12], [13], [6], [-29], [-24], [14], [18], [25], [-8], [24], [24], [-19], [18], [8], [12], [12], [-40], [30], [-25], [6], [30], [42], [-23], [-10], [-37], [0], [-42], [36], [-16], [-23], [0], [36], [-36], [-47], [16], [11], [30], [25], [-7], [-18], [25], [-12], [18], [0], [-50], [-34], [12], [-12], [-17], [-48], [17], [12], [-5], [-17], [-49], [-18], [-16], [29], [0], [10], [-24], [31], [-30], [30], [16], [-6], [-31], [12], [-14], [18], [46], [-12], [-23], [-36], [-13], [30], [-40], [-18], [-37], [6], [-55], [48], [37], [18], [-18], [6], [35], [-36], [-6], [-24], [5], [-23], [-31], [-54], [-12], [-50], [0], [-22], [31], [-6], [-7], [-12], [7], [-35], [-4], [-48], [29], [-42], [-36], [60], [-7], [4], [-31], [-36], [-14], [-36], [47], [-6], [18], [36], [-10], [53], [36], [18], [12], [-35], [-41], [19], [-60], [-42], [-29], [-24], [-12], [31], [-37], [-6], [-19], [-48], [18], [61], [-4], [12], [60], [-42], [41], [61], [-48], [-16], [6], [-73], [6], [-42], [-32], [-54], [-34], [-20], [-31], [30], [-5], [54], [46], [36], [-54], [-54], [-24], [-20], [40], [-25], [-30], [-6], [-11], [24], [-4], [12], [29], [30], [78], [-29], [18], [-18], [5], [-65], [78], [34], [56], [-42], [-17], [-35], [66], [-11], [-18], [-60], [-4], [24], [70], [-25], [7], [-1], [-14], [25], [-12], [-25], [11], [-42], [-18], [17], [-54], [25], [13], [48], [-7], [42], [7], [-48], [-42], [60], [-24], [6], [59], [18], [55], [12], [30], [37], [49], [84], [-52], [42], [29], [5], [30], [-50], [6], [-22], [60], [-72], [-6], [23], [6], [-29], [30], [24], [-53], [-6], [28], [-82], [42], [29], [84], [-17], [-19], [49], [18], [6], [-7], [-36], [40], [42], [-55], [-30], [47], [60], [-77], [1], [35], [84], [-30], [32], [-54], [-72], [55], [44], [72], [42], [67], [65], [12], [-8], [35], [-48], [-60], [-84], [36], [-48], [-73], [-12], [-12], [24], [-65], [35], [-54], [29], [11], [30], [55], [42], [-6], [64], [-79], [-66], [42], [61], [-30], [-26], [-66], [66], [-43], [90], [-5], [48], [5], [-77], [-40], [-18], [-55], [-42], [-60], [-59], [-24], [-67], [79], [12], [89], [6], [-50], [48], [32], [-18], [18], [-83], [53], [48], [-17], [18], [10], [102], [90], [-1], [-79], [36], [-36], [88], [-36], [-72], [-54], [-11], [-72], [-84], [-38], [-102], [-18], [-30], [-92], [0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2646_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2646_2_a_w();". // 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_2646_2_a_w(:prec:=1) chi := MakeCharacter_2646_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_2646_2_a_w();". // 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_2646_2_a_w( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2646_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![0, 1]>,<11,R![6, 1]>,<13,R![5, 1]>,<17,R![-6, 1]>],Snew); return Vf; end function;