// Make newform 5520.2.a.k in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5520_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_5520_2_a_k();". // 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_5520_2_a_k();". // 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_5520_a();" function MakeCharacter_5520_a() N := 5520; order := 1; char_gens := [4831, 1381, 1841, 4417, 1201]; 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_5520_a_Hecke(Kf) return MakeCharacter_5520_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], [-2], [0], [6], [-8], [1], [-8], [8], [-2], [6], [-6], [4], [-6], [0], [2], [2], [-14], [6], [-16], [0], [16], [-4], [0], [0], [-20], [6], [-6], [2], [20], [-22], [-4], [14], [20], [6], [8], [-16], [14], [-24], [-22], [-24], [10], [-10], [0], [12], [-2], [4], [-10], [-10], [-6], [10], [2], [-26], [-24], [-12], [0], [-32], [8], [22], [-14], [4], [34], [-32], [2], [4], [8], [-12], [2], [-22], [0], [-16], [-26], [-4], [24], [18], [8], [-20], [-10], [-18], [2], [-8], [-36], [20], [-20], [6], [-28], [8], [-26], [16], [-16], [38], [-36], [-28], [-32], [40], [-24], [22], [-30], [12], [6], [-16], [12], [36], [14], [4], [-42], [22], [-10], [-10], [-38], [14], [-32], [40], [-24], [14], [-12], [-46], [-10], [30], [46], [-10], [20], [20], [-6], [38], [-42], [44], [22], [28], [16], [16], [-14], [-22], [6], [-6], [14], [-2], [-30], [4], [44], [42], [-12], [34], [-28], [32], [-46], [-28], [28], [40], [32], [8], [-12], [50], [60], [40], [-42], [-36], [22], [60], [34], [-42], [18], [-26], [40], [-28], [24], [-14], [-42], [2], [-26], [-48], [-28], [36], [44], [-4], [52], [52], [46], [22], [-38], [16], [6], [-16], [18], [40], [-38], [-10], [-2], [34], [-60], [-60], [40], [24], [10], [-10], [-10], [-46], [36], [10], [28], [52], [42], [42], [38], [32], [12], [-30], [-52], [46], [0], [16], [8], [42], [54], [-14], [-46], [-48], [18], [-58], [56], [58], [-28], [28], [30], [42], [-42], [-40], [0], [-4], [-68], [40], [-2], [22], [-48], [38], [-22], [16], [14], [-32], [20], [-14], [26], [-34], [6], [26], [30], [-64], [-8], [-6], [0], [24], [-26], [-22], [-36], [14], [-2], [10], [-62], [-16], [36], [-50], [66], [-30], [20], [18], [-72], [-38], [-30], [30], [-58], [28], [8], [-2], [-14], [60], [-22], [70], [-30], [0], [-56], [-48], [-70], [8], [2], [-24], [-54], [60], [38], [14], [-4], [-78], [-6], [-4], [54], [80], [6], [0], [-4], [-32], [6], [-24], [36], [-16], [-74], [52], [14], [60], [20], [-12], [-74], [64], [-52], [64], [-78], [-12], [46], [4], [34], [-24], [48], [4], [10], [-66], [10], [40], [-76], [-56], [6], [62], [18], [-20], [-12], [-16], [12], [-70], [30], [78], [-60], [44], [22], [0], [0], [54], [16], [-34], [-72], [36], [22], [36], [2], [40], [28], [10], [2], [36], [-42], [-58], [-64], [34], [-66], [24], [50], [-68], [-66], [-38], [-40], [78], [20], [-92], [-56], [50], [32], [-52], [-60], [-12], [56], [-24], [86], [-38], [-38], [22], [82], [4], [44], [-44], [32], [-14], [-16], [14], [34], [-92], [-28], [-104], [-56], [-40], [36], [84], [-72], [46], [86], [-28], [90], [48], [4], [-86], [18], [-26], [64], [-14], [-26], [-80], [56], [0], [68], [-54], [2], [-62], [-40], [58], [6], [-16], [96], [34], [78], [-4], [10], [28], [46]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5520_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5520_2_a_k();". // 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_5520_2_a_k(:prec:=1) chi := MakeCharacter_5520_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_5520_2_a_k();". // 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_5520_2_a_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5520_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![0, 1]>,<11,R![2, 1]>,<13,R![0, 1]>,<17,R![-6, 1]>,<19,R![8, 1]>],Snew); return Vf; end function;