// Make newform 9576.2.a.y in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9576_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_9576_2_a_y();". // 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_9576_2_a_y();". // 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_9576_a();" function MakeCharacter_9576_a() N := 9576; order := 1; char_gens := [7183, 4789, 5321, 4105, 1009]; 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_9576_a_Hecke(Kf) return MakeCharacter_9576_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], [4], [-1], [-4], [2], [4], [-1], [0], [-4], [-8], [-2], [6], [12], [10], [0], [12], [-2], [-8], [-6], [14], [-8], [-6], [-2], [2], [12], [-8], [14], [-14], [16], [-4], [22], [6], [8], [22], [4], [-2], [4], [12], [-6], [10], [-2], [-8], [-14], [10], [20], [-12], [16], [24], [-14], [-6], [-28], [26], [-26], [-6], [12], [-18], [-8], [10], [-16], [8], [34], [20], [-6], [26], [12], [24], [-2], [-32], [-26], [36], [-24], [36], [-34], [-32], [-4], [-18], [-2], [-28], [6], [-30], [2], [10], [2], [16], [24], [-20], [-26], [-20], [-24], [14], [6], [-8], [-16], [28], [-6], [-18], [10], [4], [-18], [-4], [46], [24], [-24], [-4], [18], [38], [24], [6], [-38], [40], [-14], [-2], [8], [8], [12], [-16], [18], [-6], [-6], [2], [2], [-2], [-22], [-28], [10], [26], [14], [-8], [10], [12], [-22], [12], [-34], [32], [-6], [-6], [-20], [-30], [22], [-12], [-2], [-56], [6], [46], [24], [-2], [6], [44], [-2], [-2], [-36], [-12], [-20], [-36], [46], [-40], [4], [-38], [-18], [-12], [44], [8], [-28], [44], [56], [56], [62], [-2], [36], [30], [10], [-44], [54], [-16], [24], [-28], [2], [-8], [-26], [32], [-12], [26], [-58], [-24], [-36], [-14], [36], [14], [-50], [30], [-52], [16], [14], [44], [66], [-26], [54], [-28], [22], [-40], [8], [-34], [-50], [-10], [20], [4], [-34], [18], [-4], [46], [4], [48], [48], [-40], [-50], [0], [-70], [6], [-18], [34], [-40], [-64], [-40], [12], [-10], [2], [-42], [-64], [24], [-10], [-4], [32], [0], [-44], [-66], [38], [-14], [24], [-52], [-54], [44], [-4], [-66], [-54], [-34], [12], [-8], [20], [-48], [-18], [48], [-36], [54], [-24], [-12], [-58], [28], [-26], [-46], [-56], [70], [58], [-46], [8], [-36], [72], [56], [20], [-18], [-82], [-4], [46], [-24], [-62], [36], [-56], [-26], [10], [-58], [-84], [80], [-22], [-74], [-76], [-44], [-62], [36], [4], [-22], [38], [10], [-24], [20], [14], [16], [48], [-60], [24], [-16], [-70], [-12], [-8], [54], [20], [30], [-34], [18], [-12], [50], [26], [-58], [-12], [-32], [-64], [58], [6], [-62], [-66], [-18], [36], [-2], [78], [-36], [-14], [-22], [-52], [-28], [18], [-14], [62], [-20], [-4], [26], [28], [-26], [78], [14], [30], [-88], [-66], [62], [82], [32], [-12], [-54], [10], [-84], [40], [-84], [-28], [-42], [50], [48], [-90], [-54], [-30], [66], [-78], [-14], [86], [48], [26], [-74], [72], [-62], [72], [44], [-46], [-20], [-76], [36], [8], [0], [-74], [-60], [-88], [74], [-24], [-62], [70], [6], [8], [-82], [76], [-60], [-40], [34], [68], [-52], [6], [-6], [-36], [28], [90], [-2], [88], [62], [72], [78], [34], [14], [-72], [-30], [94], [12], [70], [-84], [-60], [42], [-14], [6], [6], [20], [-86], [48], [10], [8], [-44], [24], [4], [38], [36], [-20], [70], [18], [-6], [-62], [-40], [-22], [6], [-100], [44], [96], [22], [-76], [-26], [-10], [-16], [40], [94], [6], [-90], [66], [-42], [-20], [80], [-88], [-26], [-110], [36], [-96], [96], [-64], [-46], [-36], [26], [-68], [-54], [36], [36], [20], [-26], [-66], [-52], [50], [88], [50], [54], [72], [40], [110], [-44], [-98], [18], [14], [50], [-76], [-114], [60], [106], [-50], [-98], [-22], [8], [32], [50], [72], [-76], [-108], [-34], [-96], [114], [86], [-36], [-42], [100], [-18], [4], [68], [70], [40], [96], [48], [-90], [-8], [114], [80], [-14], [-116], [-102], [46], [-46], [-28], [68], [-42], [-66], [-74], [-6], [-32], [74], [-96], [-28], [24], [-46], [86], [-98], [-32], [78], [-26], [-32], [-84]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9576_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9576_2_a_y();". // 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_9576_2_a_y(:prec:=1) chi := MakeCharacter_9576_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(3833) - 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_9576_2_a_y();". // 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_9576_2_a_y( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9576_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-4, 1]>,<11,R![4, 1]>,<13,R![-2, 1]>,<17,R![-4, 1]>],Snew); return Vf; end function;