// Make newform 9360.2.a.be in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9360_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_9360_2_a_be();". // 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_9360_2_a_be();". // 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_9360_a();" function MakeCharacter_9360_a() N := 9360; order := 1; char_gens := [8191, 2341, 2081, 5617, 5761]; 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_9360_a_Hecke(Kf) return MakeCharacter_9360_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], [0], [1], [-2], [-4], [-1], [4], [-6], [0], [4], [10], [-2], [6], [8], [-8], [4], [12], [2], [10], [0], [-6], [-12], [-4], [-14], [-14], [-12], [4], [4], [-2], [-16], [8], [-16], [-10], [-12], [18], [-10], [18], [-6], [-24], [-24], [0], [-10], [-12], [-26], [-18], [-8], [0], [14], [-20], [-10], [8], [0], [22], [12], [-8], [-4], [-12], [-14], [10], [30], [4], [2], [22], [-4], [2], [-18], [-18], [14], [32], [-6], [6], [-16], [8], [22], [2], [-24], [-8], [-18], [-22], [30], [-16], [22], [-40], [-2], [36], [20], [-30], [-22], [14], [-18], [8], [-24], [22], [36], [38], [12], [-42], [-12], [12], [14], [36], [26], [-4], [-28], [12], [38], [-20], [-14], [8], [10], [8], [6], [-18], [-22], [14], [40], [-14], [-28], [0], [36], [26], [-18], [36], [4], [-34], [-40], [-50], [-20], [24], [34], [-30], [24], [-20], [-34], [-18], [2], [50], [22], [12], [-28], [-30], [10], [4], [-20], [-2], [-40], [30], [48], [-16], [-48], [-42], [28], [4], [-56], [-44], [12], [-44], [-10], [54], [-6], [12], [48], [-58], [-52], [-54], [-16], [48], [-58], [-10], [-40], [-44], [-46], [-28], [34], [-28], [-16], [26], [10], [-12], [-10], [-10], [40], [-34], [2], [-24], [24], [-42], [46], [46], [-64], [34], [36], [-20], [18], [-36], [56], [-22], [2], [-6], [24], [-2], [-40], [34], [-46], [-60], [-44], [34], [52], [18], [52], [30], [8], [-4], [12], [0], [-14], [-20], [64], [56], [50], [10], [-46], [-38], [46], [32], [62], [-4], [-48], [-72], [4], [-58], [-28], [-2], [24], [-4], [0], [70], [2], [48], [-60], [-20], [28], [36], [34], [-6], [-20], [38], [4], [-30], [-48], [-38], [46], [0], [-74], [0], [-12], [-30], [6], [44], [-66], [-24], [0], [-42], [-34], [66], [-8], [-26], [54], [-18], [48], [-46], [-70], [-62], [4], [22], [42], [4], [-14], [18], [-36], [8], [66], [24], [-54], [14], [56], [66], [-38], [-18], [52], [-24], [-56], [-70], [-12], [50], [-40], [48], [28], [60], [46], [34], [62], [-28], [24], [-20], [-78], [60], [-14], [-8], [70], [-36], [-42], [-60], [-64], [48], [22], [28], [0], [66], [0], [-40], [78], [-48], [58], [82], [-6], [-42], [34], [-12], [-56], [22], [72], [12], [-36], [-30], [-36], [-14], [14], [-2], [-52], [46], [24], [-30], [-20], [-66], [64], [-78], [58], [0], [20], [22], [2], [-62], [-8], [-86], [-72], [80], [12], [24], [-24], [78], [-48], [4], [60], [72], [86], [14], [-30], [46], [80], [-16], [-24], [-16], [44], [-30], [-84], [40], [46], [24], [-70], [18], [38], [-26], [42], [-26], [-16], [-10], [6], [-10], [-12], [-58], [74], [-36], [88], [24], [74], [-10], [0], [40], [78], [-14], [-92], [6], [18], [-18], [92], [34], [18], [6], [-36], [-6], [-36], [0], [-8], [-34], [72], [-48], [-28], [-46], [8], [-12], [38], [-104], [8], [46], [-102], [-32], [-62], [70], [-52], [38], [4], [88], [32], [22], [-60], [6], [-14], [-48], [-58], [-84], [34], [34], [-84], [-14], [-108], [8], [32], [34], [46], [-38], [-88], [-20], [62], [-38], [48], [-90], [48], [-26], [58], [-20], [50], [60], [-70], [74], [-70], [24], [36], [72], [0], [70], [-76], [-72], [-50], [72], [-38], [-40], [66], [108], [30], [-26], [28], [38], [108], [-50], [-24], [-10], [-12], [6], [16], [-10], [8], [26], [-24], [112], [42], [-22], [30], [-60], [0], [70], [10], [-46], [62], [74], [-20], [48], [64], [106], [-12], [-76], [-24], [-106], [42], [52], [118], [-8], [110], [-32], [-52], [74], [38], [-48], [-84], [22], [40], [-34], [48], [100], [40], [4], [-78], [-28], [-56], [-94], [72], [10], [22], [58], [18], [-72], [-40], [-54], [-84], [0], [-38], [28], [-4], [-106], [-46], [-24], [68], [-4], [16], [-60], [2], [16]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9360_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9360_2_a_be();". // 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_9360_2_a_be(:prec:=1) chi := MakeCharacter_9360_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(4027) - 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_9360_2_a_be();". // 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_9360_2_a_be( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9360_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![2, 1]>,<11,R![4, 1]>,<17,R![-4, 1]>,<19,R![6, 1]>,<31,R![-10, 1]>],Snew); return Vf; end function;