// Make newform 5070.2.a.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5070_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_5070_2_a_e();". // 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_5070_2_a_e();". // 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_5070_a();" function MakeCharacter_5070_a() N := 5070; order := 1; char_gens := [1691, 4057, 1861]; 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_5070_a_Hecke(Kf) return MakeCharacter_5070_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], [-1], [1], [-2], [-4], [0], [8], [6], [6], [-4], [0], [2], [2], [-4], [0], [-10], [-4], [-10], [-12], [8], [8], [8], [-12], [14], [16], [-16], [-12], [12], [-12], [20], [4], [10], [6], [-8], [10], [0], [22], [16], [4], [22], [-10], [10], [0], [4], [18], [0], [-20], [2], [4], [-4], [24], [-16], [-2], [6], [12], [2], [-24], [-16], [-10], [6], [-4], [26], [-12], [24], [6], [30], [-10], [14], [-24], [28], [14], [24], [-36], [-2], [18], [12], [8], [-14], [-30], [26], [10], [8], [0], [-2], [-32], [16], [6], [-8], [6], [26], [28], [-32], [-26], [42], [-38], [10], [-18], [-26], [36], [8], [-20], [-2], [24], [6], [40], [-12], [-36], [-22], [24], [22], [-28], [10], [6], [10], [-12], [18], [-44], [-30], [42], [2], [-48], [26], [-30], [44], [14], [32], [4], [36], [16], [-38], [-30], [-12], [-16], [-6], [-22], [-10], [-38], [32], [-10], [2], [38], [10], [-52], [12], [-10], [40], [-46], [4], [36], [12], [14], [-18], [-28], [-2], [52], [-20], [40], [30], [18], [34], [-4], [24], [14], [38], [-30], [52], [0], [-26], [14], [54], [40], [-20], [-20], [-52], [32], [-50], [62], [18], [32], [26], [-10], [10], [-54], [-38], [-4], [-12], [2], [52], [50], [-8], [58], [-20], [28], [2], [-8], [-4], [18], [54], [-38], [-14], [14], [0], [54], [-34], [-12], [-30], [56], [0], [-38], [20], [-58], [-12], [-48], [-44], [-64], [46], [24], [6], [-12], [14], [-62], [60], [-30], [14], [-48], [18], [-16], [-4], [-8], [-28], [70], [-44], [16], [38], [36], [-28], [-14], [30], [-26], [20], [4], [68], [-32], [64], [-46], [0], [-22], [-72], [-26], [-54], [26], [-66], [-48], [-6], [54], [24], [30], [-72], [38], [48], [64], [-28], [-44], [-22], [2], [-8], [54], [-42], [36], [74], [-22], [32], [12], [40], [-30], [6], [-12], [-16], [34], [-74], [66], [-4], [18], [-28], [-44], [-60], [-50], [-6], [52], [-6], [-36], [-12], [18], [40], [-30], [-24], [72], [66], [-38], [-8], [-42], [-22], [-64], [32], [-12], [-44], [44], [-22], [56], [-70], [-10], [54], [14], [76], [-16], [-6], [-28], [-40], [-16], [82], [-24], [-56], [48], [-54], [34], [-34], [2], [68], [42], [70], [-80], [-62], [-56], [-12], [-34], [84], [8], [-18], [26], [-28], [6], [-84], [-54], [-8], [66], [-46], [-26], [76], [40], [18], [10], [92], [42], [72], [34], [16], [-56], [84], [-32], [-80], [-42], [58], [-22], [-24], [52], [22], [18], [-14], [-22], [-30], [-64], [4], [-44], [32], [2], [-40], [-80], [36], [-42], [46], [-70], [30], [-62], [-62], [42], [24], [-52], [50], [60], [-6], [-26], [6], [44], [52], [96], [-74], [-64], [18], [44], [-26], [-80], [-92], [26], [-70], [-42], [-16], [34], [86], [-56], [-40], [14], [-26], [-72], [80], [2], [44], [-72], [100], [74], [-22], [52], [22], [-32], [14], [-56], [62], [-44], [-90], [34], [64]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5070_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5070_2_a_e();". // 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_5070_2_a_e(:prec:=1) chi := MakeCharacter_5070_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_5070_2_a_e();". // 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_5070_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5070_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![2, 1]>,<11,R![4, 1]>,<17,R![-8, 1]>,<31,R![0, 1]>],Snew); return Vf; end function;