// Make newform 4410.2.a.bl in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4410_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_4410_2_a_bl();". // 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_4410_2_a_bl();". // 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_4410_a();" function MakeCharacter_4410_a() N := 4410; order := 1; char_gens := [3431, 2647, 1081]; 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_4410_a_Hecke(Kf) return MakeCharacter_4410_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], [1], [0], [4], [-6], [4], [-6], [0], [6], [4], [8], [10], [-2], [10], [-14], [-4], [8], [6], [2], [10], [16], [-8], [2], [-2], [-6], [8], [-12], [-18], [14], [-8], [4], [-6], [-10], [-6], [-16], [14], [-10], [6], [-2], [24], [8], [18], [-10], [-22], [16], [16], [-16], [-20], [-4], [6], [22], [-10], [-12], [16], [-4], [18], [12], [-12], [8], [4], [2], [-4], [-12], [10], [-2], [16], [-26], [-20], [-28], [0], [-34], [-16], [32], [0], [-30], [-22], [26], [16], [-18], [-36], [14], [-30], [22], [-28], [4], [12], [-14], [2], [-4], [-32], [36], [8], [-24], [-24], [30], [38], [-38], [-16], [18], [30], [-2], [-44], [24], [-36], [2], [4], [12], [-6], [30], [32], [48], [-10], [10], [-40], [-4], [-16], [2], [38], [-20], [8], [-26], [6], [-52], [26], [-2], [-14], [36], [0], [-22], [40], [0], [-40], [40], [6], [26], [-54], [-52], [-42], [24], [22], [30], [4], [-4], [4], [-32], [-38], [-48], [-30], [-44], [-28], [-26], [6], [-2], [-10], [-30], [8], [-34], [22], [58], [-4], [30], [-8], [-36], [6], [-38], [-8], [-14], [22], [14], [44], [-32], [6], [6], [4], [14], [4], [18], [-56], [20], [20], [-12], [-40], [-44], [36], [54], [40], [-32], [-50], [-60], [6], [-44], [16], [10], [-12], [-36], [22], [-44], [0], [-14], [58], [64], [30], [-54], [36], [58], [4], [36], [36], [34], [6], [-6], [16], [-32], [-24], [10], [-32], [-30], [-20], [6], [50], [40], [24], [36], [52], [-70], [-40], [2], [-40], [8], [4], [42], [40], [-60], [-28], [6], [38], [74], [-16], [8], [28], [10], [48], [34], [-12], [-60], [-8], [-60], [56], [-52], [52], [-30], [-12], [-54], [18], [64], [-10], [-52], [42], [-18], [12], [-68], [32], [-2], [24], [58], [-6], [-14], [-14], [-6], [0], [66], [-38], [48], [-14], [-8], [-28], [-46], [54], [-28], [2], [80], [-42], [-12], [-32], [-66], [34], [-54], [28], [-42], [42], [28], [-2], [76], [-12], [78], [-28], [26], [-60], [28], [54], [38], [16], [36], [68], [42], [-4], [0], [-6], [32], [-6], [6], [88], [38], [-12], [-38], [84], [30], [-42], [-4], [74], [22], [-50], [-24], [-54], [70], [20], [24], [-44], [-6], [62], [54], [-64], [-24], [-52], [-48], [-86], [84], [2], [-32], [-32], [-74], [6], [-64], [-30], [-80], [84], [10], [0], [-54], [46], [-54], [30], [64], [-30], [-24], [-12], [44], [-18], [36], [32], [22], [68], [4], [-4], [-38], [2], [-44], [-62], [-12], [26], [-20], [-42], [28], [-68], [4], [54], [46], [18], [-42], [14], [46], [-24], [18], [30], [-6], [-48], [58], [38], [-58], [98], [86], [-92], [36], [-62], [-86], [8], [38], [48], [-30], [-12], [-42], [28], [12], [-22], [-40], [92], [-36], [-84], [-84], [94], [54], [-36], [52], [62], [62], [6], [-72], [96], [22], [-62], [-62], [-48], [-44], [-6], [-70], [20], [-12], [-10], [-80]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4410_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4410_2_a_bl();". // 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_4410_2_a_bl(:prec:=1) chi := MakeCharacter_4410_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_4410_2_a_bl();". // 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_4410_2_a_bl( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4410_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-4, 1]>,<13,R![6, 1]>,<17,R![-4, 1]>,<19,R![6, 1]>,<29,R![-6, 1]>,<31,R![-4, 1]>],Snew); return Vf; end function;