// Make newform 2240.2.a.v in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2240_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_2240_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_2240_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_2240_a();" function MakeCharacter_2240_a() N := 2240; order := 1; char_gens := [1471, 1541, 897, 1921]; v := [1, 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_2240_a_Hecke(Kf) return MakeCharacter_2240_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 := [[0], [1], [1], [-1], [5], [-1], [3], [6], [-6], [9], [0], [-6], [8], [-6], [3], [12], [-8], [4], [4], [8], [10], [-3], [12], [-16], [7], [14], [-9], [-2], [11], [-6], [8], [6], [-16], [-18], [14], [-19], [2], [-6], [-9], [-19], [-4], [20], [11], [8], [12], [0], [-13], [-25], [13], [-16], [8], [-7], [18], [-14], [6], [18], [-10], [4], [-14], [-13], [-29], [1], [27], [-14], [29], [-30], [20], [-26], [10], [-14], [-33], [0], [-29], [0], [20], [12], [-25], [29], [9], [14], [-36], [19], [23], [-30], [-34], [-30], [-33], [-28], [24], [20], [-33], [-6], [26], [-33], [25], [-31], [-18], [-6], [-4], [9], [-20], [-24], [-4], [34], [36], [17], [-8], [-7], [-33], [-34], [17], [22], [10], [-2], [-9], [34], [-47], [0], [-30], [25], [-8], [32], [-33], [20], [-20], [21], [41], [50], [-16], [-5], [37], [-36], [-35], [-8], [-46], [-10], [-1], [-11], [-35], [-23], [-38], [7], [32], [-2], [-16], [-26], [26], [-14], [28], [20], [10], [-16], [16], [-8], [30], [-40], [25], [36], [7], [-4], [52], [20], [22], [-28], [18], [9], [-16], [1], [37], [-18], [24], [-10], [27], [-30], [10], [36], [9], [15], [60], [28], [-8], [0], [6], [6], [-40], [-14], [16], [25], [25], [-12], [66], [-6], [-17], [44], [28], [-15], [-1], [32], [-61], [23], [18], [-10], [-10], [-2], [-6], [33], [-10], [32], [22], [12], [-8], [-26], [0], [-3], [6], [48], [-14], [40], [-46], [-54], [62], [-74], [45], [-18], [53], [66], [55], [-9], [-17], [28], [-24], [-42], [72], [-63], [37], [-73], [20], [40], [23], [-6], [-24], [32], [59], [-54], [9], [-2], [32], [18], [19], [-66], [50], [-38], [52], [-16], [18], [9], [-46], [-4], [-33], [-63], [-18], [-78], [-60], [-61], [42], [-50], [-1], [-14], [36], [-38], [-60], [-8], [55], [28], [-2], [-53], [12], [30], [-31], [-46], [-25], [13], [32], [-38], [-3], [-33], [50], [18], [-60], [-12], [-33], [47], [-36], [16], [10], [84], [-26], [-49], [36], [57], [-53], [-62], [-85], [32], [15], [-50], [34], [48], [-48], [32], [0], [21], [-34], [-26], [78], [-56], [26], [23], [-37], [45], [-2], [44], [-26], [8], [62], [20], [-81], [-4], [-52], [32], [-70], [16], [-46], [-29], [12], [57], [-13], [-14], [66], [-77], [38], [24], [50], [51], [-66], [-12], [-38], [-56], [-14], [-3], [-88], [4], [-25], [11], [-25], [-10], [18], [-78], [-82], [86], [-28], [-72], [84], [-7], [36], [38], [13], [-70], [69], [65], [76], [24], [-27], [10], [-26], [-48], [-5], [9], [54], [21], [18], [-66], [-68], [-34], [44], [31], [-25], [-17], [-4], [-19], [-55], [-23], [51], [68], [23], [-64], [-16], [76], [13], [-50], [0], [-44], [-94], [7], [36], [-26], [86], [15], [76], [-52], [-65], [50], [-54], [36], [58], [24], [63], [84], [2], [15], [67], [90], [-74], [12], [69], [-19], [-56], [66], [-28], [60]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2240_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2240_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_2240_2_a_v(:prec:=1) chi := MakeCharacter_2240_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_2240_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_2240_2_a_v( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2240_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-1, 1]>,<11,R![-5, 1]>,<13,R![1, 1]>,<19,R![-6, 1]>],Snew); return Vf; end function;