// Make newform 5610.2.a.bf in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_5610_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_5610_2_a_bf();". // 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_5610_2_a_bf();". // 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_5610_a();" function MakeCharacter_5610_a() N := 5610; order := 1; char_gens := [1871, 3367, 1531, 3301]; 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_5610_a_Hecke(Kf) return MakeCharacter_5610_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], [-4], [1], [-2], [1], [0], [0], [2], [4], [-6], [-2], [4], [-8], [-14], [4], [2], [-16], [-12], [-6], [-8], [-12], [-6], [6], [-14], [-8], [4], [-14], [6], [16], [-20], [-6], [20], [18], [-16], [-14], [4], [-12], [18], [20], [18], [8], [-22], [2], [20], [4], [-24], [28], [14], [26], [0], [14], [-12], [-30], [-8], [-30], [16], [-22], [18], [4], [22], [4], [-12], [14], [-30], [-12], [18], [20], [-22], [2], [24], [8], [-2], [-4], [24], [38], [2], [10], [-14], [-12], [-2], [8], [-22], [-8], [24], [-14], [-22], [-6], [-24], [24], [32], [-8], [-40], [28], [20], [14], [-30], [-20], [-46], [-12], [30], [-28], [26], [28], [-22], [-8], [2], [24], [30], [28], [-10], [46], [-4], [-24], [-6], [-20], [24], [-14], [-24], [46], [-6], [2], [-36], [-12], [-22], [-22], [-12], [32], [6], [8], [-36], [-4], [2], [-54], [-30], [-30], [28], [-14], [6], [20], [10], [-40], [-36], [-10], [-36], [-6], [58], [20], [-56], [-14], [-38], [-32], [36], [44], [-20], [-16], [42], [-38], [18], [-4], [10], [8], [-4], [-30], [48], [4], [18], [-10], [-6], [24], [-34], [0], [-38], [-24], [50], [-20], [-30], [8], [18], [48], [36], [-22], [-6], [-24], [30], [-38], [64], [38], [24], [-34], [28], [-20], [6], [28], [-18], [38], [50], [34], [24], [-46], [-64], [34], [2], [-20], [-6], [32], [12], [54], [-4], [-46], [54], [8], [48], [24], [26], [16], [26], [40], [66], [2], [64], [-46], [8], [-12], [-22], [-18], [20], [0], [36], [-30], [-52], [56], [42], [-16], [-56], [18], [-46], [36], [-24], [-36], [8], [32], [-34], [18], [0], [-40], [12], [72], [-48], [-10], [14], [-8], [26], [-10], [-8], [2], [4], [-38], [-6], [-60], [44], [-38], [50], [54], [-12], [6], [58], [4], [54], [-14], [-16], [-14], [-32], [-46], [-16], [80], [-62], [34], [0], [0], [-60], [12], [-70], [-28], [-24], [38], [-78], [56], [50], [66], [12], [-6], [44], [10], [-30], [56], [74], [-20], [28], [-6], [-58], [72], [44], [52], [-10], [-24], [-86], [-16], [86], [4], [-54], [-42], [24], [72], [50], [84], [-24], [-50], [-30], [20], [-50], [10], [16], [-30], [-14], [-20], [-76], [20], [-78], [-78], [18], [88], [-36], [-60], [-68], [30], [50], [-30], [-64], [-6], [42], [26], [48], [26], [-20], [-30], [-80], [-16], [42], [-32], [22], [-50], [-44], [-6], [-38], [72], [-4], [42], [-48], [50], [6], [-32], [92], [-72], [-46], [-30], [-32], [-74], [36], [-4], [-44], [-62], [-8], [34], [84], [-8], [-62], [66], [26], [-14], [-54], [60], [-54], [-4], [-96], [-40], [50], [52], [-16], [42], [26], [68], [-48], [72], [-38], [-64], [-102], [60], [66], [-46], [66], [88], [6], [-22], [96], [-6], [18], [72], [28], [-70], [-26], [32], [-36], [-86], [-22], [64], [8], [-82], [-8], [14], [-54], [72], [0], [-82], [-78], [44], [-74], [52], [-80]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_5610_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_5610_2_a_bf();". // 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_5610_2_a_bf(:prec:=1) chi := MakeCharacter_5610_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_5610_2_a_bf();". // 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_5610_2_a_bf( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5610_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![4, 1]>,<13,R![2, 1]>,<19,R![0, 1]>,<23,R![0, 1]>],Snew); return Vf; end function;