// Make newform 6240.2.a.u in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6240_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_6240_2_a_u();". // 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_6240_2_a_u();". // 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_6240_a();" function MakeCharacter_6240_a() N := 6240; order := 1; char_gens := [1951, 2341, 2081, 2497, 5761]; v := [1, 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_6240_a_Hecke(Kf) return MakeCharacter_6240_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], [0], [0], [-1], [2], [4], [-4], [-6], [-8], [-6], [-2], [-4], [0], [6], [0], [-2], [-8], [0], [6], [4], [12], [6], [-10], [10], [-8], [12], [18], [-14], [8], [-12], [-14], [-4], [-6], [0], [-22], [-8], [0], [-18], [-12], [-10], [0], [-2], [-14], [-28], [20], [-24], [20], [-14], [-6], [-8], [10], [20], [-14], [4], [-30], [8], [18], [-2], [4], [-6], [8], [24], [-6], [-6], [-20], [-14], [28], [-14], [-6], [0], [-32], [2], [12], [16], [-14], [-14], [6], [10], [-4], [10], [0], [-30], [-12], [36], [6], [-34], [18], [16], [20], [16], [24], [20], [-20], [-12], [18], [26], [-12], [26], [36], [-6], [-36], [-22], [20], [6], [20], [2], [16], [26], [0], [-6], [18], [20], [16], [-6], [-16], [-12], [38], [12], [34], [-22], [22], [44], [12], [10], [34], [40], [8], [18], [-28], [16], [-36], [18], [-10], [-30], [-6], [-16], [-42], [-6], [20], [-22], [-24], [-20], [22], [0], [42], [-38], [-28], [24], [2], [10], [-36], [-28], [20], [-32], [4], [14], [10], [34], [-4], [-22], [-56], [20], [-46], [-16], [36], [-14], [-22], [-26], [0], [26], [-8], [-34], [-12], [42], [44], [42], [-56], [-34], [-16], [-52], [34], [42], [0], [-54], [-30], [-32], [-6], [48], [-38], [-4], [-36], [2], [12], [42], [42], [18], [18], [4], [18], [28], [42], [18], [-8], [-42], [0], [60], [6], [-44], [-46], [58], [-16], [28], [32], [-38], [0], [-6], [-64], [10], [62], [40], [62], [24], [12], [-10], [10], [16], [-32], [48], [34], [52], [-16], [42], [-20], [24], [-30], [-30], [-44], [-56], [-12], [28], [-32], [-70], [26], [0], [-16], [0], [-28], [68], [34], [14], [-24], [58], [-42], [-48], [78], [64], [-26], [-2], [32], [-12], [66], [18], [10], [-76], [-46], [54], [-64], [-2], [-18], [-40], [46], [44], [-38], [0], [76], [42], [50], [-60], [44], [-8], [-20], [-30], [-8], [48], [58], [-62], [24], [-6], [-62], [-20], [-78], [24], [-62], [-54], [20], [14], [36], [8], [-22], [18], [44], [-12], [4], [-82], [-28], [54], [56], [-46], [52], [-70], [66], [12], [-48], [42], [16], [-48], [-2], [2], [68], [62], [90], [32], [-22], [-46], [-36], [-16], [-12], [-90], [34], [-10], [4], [-60], [12], [12], [50], [-22], [26], [-64], [-86], [-22], [58], [-4], [90], [52], [38], [8], [64], [-66], [-4], [-10], [-14], [80], [14], [26], [8], [-72], [18], [-24], [-38], [-6], [-12], [-56], [4], [-30], [10], [-16], [-54], [-60], [-20], [48], [-38], [4], [-22], [8], [-48], [-74], [26], [-62], [-14], [-14], [72], [10], [12], [-64], [88], [-14], [-96], [-28], [10], [-14], [-72], [52], [56], [-14], [32], [-62], [76], [42], [50], [18], [88], [66], [-6], [-28], [-94], [-58], [-32], [-104], [-38], [-66], [-44], [-20], [-62], [-30], [80], [16], [74], [84], [-22], [106], [-48], [44], [62], [-14], [100], [-18], [-20], [56]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6240_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6240_2_a_u();". // 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_6240_2_a_u(:prec:=1) chi := MakeCharacter_6240_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_6240_2_a_u();". // 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_6240_2_a_u( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6240_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![0, 1]>,<11,R![0, 1]>,<17,R![-2, 1]>,<19,R![-4, 1]>],Snew); return Vf; end function;