// Make newform 5610.2.a.x in Magma, downloaded from the LMFDB on 28 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_x();". // 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_x();". // 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], [-2], [1], [4], [1], [-2], [-6], [6], [-8], [2], [6], [6], [-4], [4], [-2], [14], [12], [-10], [0], [-10], [12], [-2], [-14], [10], [-8], [12], [10], [4], [20], [-12], [2], [-16], [22], [0], [2], [12], [16], [18], [2], [10], [0], [4], [-10], [16], [16], [8], [20], [-14], [-18], [16], [28], [-6], [10], [32], [26], [12], [18], [-26], [0], [6], [-22], [-18], [-14], [-2], [12], [-32], [-4], [-12], [34], [24], [-8], [-12], [-4], [32], [0], [-34], [-20], [6], [-20], [-18], [-12], [14], [-26], [30], [-16], [-2], [30], [16], [-18], [0], [16], [-40], [20], [-12], [-32], [-28], [-30], [-18], [-28], [-42], [44], [-22], [24], [-2], [-18], [30], [24], [-40], [-10], [-20], [8], [4], [16], [12], [36], [-4], [34], [-16], [10], [36], [6], [-12], [-28], [38], [-34], [-30], [-24], [-4], [14], [28], [-20], [-18], [30], [-38], [0], [12], [12], [34], [-20], [42], [-20], [28], [-42], [-18], [-14], [18], [-44], [-8], [2], [32], [56], [-24], [36], [-42], [0], [40], [-42], [-14], [-8], [-50], [-4], [54], [-14], [-30], [20], [22], [16], [14], [56], [-6], [-48], [-58], [24], [-44], [4], [54], [20], [16], [-32], [-28], [6], [6], [12], [24], [54], [-44], [24], [-16], [-38], [12], [-16], [-16], [-44], [52], [-16], [-2], [-22], [0], [42], [-2], [-38], [-46], [50], [-60], [-64], [0], [18], [-16], [20], [-60], [16], [2], [12], [-30], [-8], [-54], [10], [72], [-20], [2], [-30], [36], [56], [20], [-36], [-10], [24], [68], [46], [-40], [-40], [30], [-8], [48], [-22], [50], [-28], [16], [-60], [-26], [-24], [74], [30], [-28], [20], [-52], [-2], [-24], [-36], [70], [60], [-42], [-34], [32], [-74], [-76], [-18], [-26], [26], [-28], [-26], [34], [-24], [80], [-48], [18], [64], [66], [18], [-4], [74], [-32], [-26], [56], [30], [32], [30], [68], [0], [-8], [28], [52], [-52], [-60], [-54], [-6], [64], [2], [42], [12], [42], [60], [-10], [78], [-72], [-20], [-4], [82], [50], [6], [-14], [24], [12], [22], [18], [26], [72], [-48], [44], [-54], [-18], [-44], [-88], [70], [-82], [16], [50], [-58], [64], [-2], [-2], [-16], [6], [62], [20], [-20], [52], [70], [58], [78], [-66], [52], [4], [-72], [-46], [-66], [-18], [-52], [-14], [10], [-14], [40], [12], [-60], [-42], [-4], [0], [22], [78], [-2], [24], [-26], [52], [-54], [-36], [-36], [-78], [-72], [-2], [14], [8], [-44], [-88], [-42], [78], [88], [-40], [-74], [-28], [24], [34], [36], [-2], [72], [-48], [-70], [66], [-26], [-38], [-14], [18], [42], [-80], [-6], [-72], [-26], [-60], [-24], [66], [70], [-70], [20], [48], [-68], [-96], [-30], [84], [42], [-26], [-22], [12], [-60], [-18], [-90], [6], [-62], [-12], [-48], [96], [-18], [30], [80], [26], [-74], [56], [-48], [12], [-96], [40], [90], [46], [-72], [26], [-32], [-4], [6], [32], [84]]; 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_x();". // 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_x(: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_x();". // 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_x( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_5610_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![2, 1]>,<13,R![-4, 1]>,<19,R![2, 1]>,<23,R![6, 1]>],Snew); return Vf; end function;