// Make newform 6080.2.a.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6080_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_6080_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_6080_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_6080_a();" function MakeCharacter_6080_a() N := 6080; order := 1; char_gens := [191, 5701, 1217, 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_6080_a_Hecke(Kf) return MakeCharacter_6080_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], [-2], [-1], [4], [4], [0], [6], [1], [8], [6], [-8], [8], [-2], [0], [12], [-4], [-8], [14], [2], [-8], [-2], [4], [-12], [6], [0], [6], [-6], [-18], [2], [-16], [-18], [20], [-10], [-4], [6], [12], [6], [-24], [-6], [20], [0], [10], [-24], [16], [-6], [8], [-16], [2], [6], [6], [10], [24], [-2], [-12], [12], [-24], [-6], [-16], [-26], [30], [4], [-4], [22], [-8], [-14], [-32], [16], [8], [4], [10], [6], [8], [36], [12], [-4], [22], [-6], [14], [10], [-14], [-20], [-2], [4], [16], [12], [-36], [-6], [-22], [18], [16], [4], [24], [26], [-12], [12], [24], [-26], [-30], [-22], [-10], [-26], [-2], [18], [-22], [-4], [18], [-28], [-6], [-44], [30], [18], [-34], [6], [28], [-16], [-2], [-28], [24], [-26], [16], [46], [44], [-20], [34], [-12], [-2], [50], [-40], [-20], [-6], [-28], [30], [20], [34], [-42], [14], [-48], [-34], [-44], [-6], [-44], [2], [-52], [-6], [42], [-36], [-22], [-28], [4], [-34], [-52], [14], [8], [-18], [46], [-44], [16], [-30], [22], [10], [-52], [36], [-40], [28], [-28], [-34], [-20], [22], [6], [6], [-12], [-6], [-32], [-22], [16], [38], [-4], [6], [6], [22], [-40], [-12], [56], [28], [32], [50], [-28], [42], [50], [24], [28], [-20], [-20], [22], [-48], [8], [58], [66], [-6], [-36], [-18], [44], [16], [-10], [-20], [-42], [16], [54], [50], [-12], [-54], [-26], [-40], [10], [32], [-66], [28], [-46], [-10], [-54], [38], [40], [38], [-40], [38], [10], [-52], [-8], [-50], [12], [2], [4], [28], [30], [-4], [28], [-66], [-46], [-12], [-20], [42], [28], [-20], [-34], [-72], [-72], [12], [-56], [76], [4], [58], [2], [-16], [-46], [-38], [36], [58], [-22], [-28], [38], [34], [42], [-42], [-44], [66], [64], [30], [30], [-34], [58], [58], [34], [6], [-80], [24], [-4], [64], [-26], [78], [44], [70], [-24], [48], [50], [20], [-32], [26], [-24], [56], [26], [42], [-56], [-72], [-16], [-44], [-58], [48], [-30], [-84], [64], [46], [52], [24], [54], [-60], [-16], [34], [-10], [-40], [66], [-36], [-26], [18], [26], [-40], [38], [44], [84], [82], [-30], [-52], [-2], [78], [-10], [-18], [6], [-48], [-78], [6], [6], [-18], [36], [-56], [32], [-20], [40], [82], [84], [-54], [32], [44], [-66], [-78], [-48], [-12], [-8], [-22], [-22], [-64], [-66], [28], [-60], [46], [22], [-2], [-28], [40], [-28], [46], [-26], [-10], [6], [74], [68], [-28], [-48], [-2], [-66], [78], [-12], [-64], [-68], [2], [8], [-34], [12], [48], [34], [50], [-64], [42], [18], [92], [18], [72], [-62], [-72], [-2], [16], [18], [34], [12], [-4], [-36], [44], [-12], [-84], [30], [-4], [10], [50], [26], [-38], [36], [-54], [56], [38], [-30], [-94], [4], [-76], [66], [-66], [60], [78], [-14], [-100], [30], [-66], [-34], [-54], [36], [60], [-12], [-56], [-28], [18], [14], [-20], [32]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6080_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6080_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_6080_2_a_e(:prec:=1) chi := MakeCharacter_6080_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_6080_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_6080_2_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6080_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![2, 1]>,<7,R![-4, 1]>,<11,R![-4, 1]>],Snew); return Vf; end function;