// Make newform 9576.2.a.u 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_u();". // 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_u();". // 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], [2], [1], [-6], [0], [4], [-1], [-4], [2], [-2], [0], [10], [-4], [-4], [14], [0], [-2], [12], [0], [-2], [10], [6], [10], [-2], [6], [6], [0], [-8], [-6], [2], [2], [8], [12], [22], [10], [-6], [4], [0], [14], [-4], [20], [16], [6], [18], [-4], [8], [-6], [8], [-10], [-16], [20], [-26], [-2], [6], [-12], [-18], [-28], [22], [-18], [-4], [6], [28], [-12], [-22], [30], [8], [14], [18], [2], [0], [24], [-28], [-16], [0], [24], [-2], [-14], [-6], [-18], [14], [-20], [-24], [-14], [-6], [-22], [-26], [10], [18], [-4], [30], [4], [-26], [-18], [-20], [40], [-6], [18], [-16], [10], [-8], [22], [-20], [6], [-28], [30], [-38], [12], [16], [-26], [34], [-38], [-12], [12], [20], [34], [4], [0], [38], [8], [-28], [46], [-18], [-36], [20], [-14], [22], [36], [-40], [34], [-20], [32], [-22], [34], [36], [34], [-14], [52], [18], [12], [36], [18], [44], [12], [-4], [24], [-26], [-54], [52], [32], [32], [40], [28], [-48], [44], [16], [4], [16], [54], [-30], [-2], [-2], [-24], [-8], [-6], [0], [-30], [-14], [42], [34], [-20], [0], [24], [-62], [-30], [44], [20], [6], [-26], [22], [4], [-52], [16], [-2], [-24], [-6], [32], [-28], [-6], [-24], [-46], [-18], [-64], [-30], [34], [-6], [-2], [2], [4], [-20], [14], [18], [8], [-38], [-66], [-18], [-16], [-36], [-4], [-56], [-18], [-30], [4], [48], [-16], [18], [-24], [-46], [-24], [-26], [-48], [-14], [-22], [-60], [52], [-58], [54], [-40], [34], [42], [34], [56], [-50], [34], [-44], [-24], [14], [-42], [-2], [24], [8], [20], [-16], [4], [6], [-56], [52], [44], [36], [-76], [-22], [-28], [-8], [-30], [70], [-6], [38], [20], [-6], [-6], [-10], [-12], [18], [16], [-24], [44], [-50], [76], [36], [-66], [-8], [16], [74], [-48], [-30], [-4], [-22], [-68], [50], [-6], [0], [32], [-72], [24], [-4], [60], [78], [-62], [48], [6], [6], [-26], [62], [-44], [44], [-14], [6], [86], [60], [44], [-10], [18], [-24], [-24], [-12], [-22], [0], [-68], [16], [-34], [-4], [74], [-14], [28], [-40], [-86], [58], [64], [-62], [84], [80], [50], [-30], [46], [-40], [-34], [-32], [4], [48], [6], [90], [-66], [-68], [-74], [-76], [-58], [-56], [-62], [26], [-32], [-24], [-48], [6], [26], [2], [-72], [-62], [28], [-80], [-10], [32], [-38], [-6], [54], [48], [42], [0], [-58], [0], [-24], [-22], [24], [-72], [-36], [84], [-22], [30], [-6], [94], [50], [56], [60], [-50], [52], [78], [40], [-60], [-82], [-16], [70], [22], [44], [-20], [-72], [44], [-72], [-40], [-82], [-28], [-8], [-82], [58], [-18], [-60], [-80], [26], [22], [-50], [-20], [-90], [0], [-4], [-18], [-18], [38], [-16], [-58], [22], [-48], [90], [-90], [78], [-52], [100], [-6], [-90], [16], [-86], [-60], [-56], [-18], [-28], [-48], [96], [-6], [-18], [88], [-72], [92], [-56], [-82], [-22], [68], [48], [-50], [0], [-22], [44], [-16], [80], [42], [20], [92], [96], [2], [-18], [108], [-72], [2], [20], [40], [72], [62], [32], [-2], [-42], [-20], [40], [-2], [-26], [-84], [50], [-16], [-32], [92], [-14], [82], [18], [-24], [-20], [22], [-88], [-32], [30], [-24], [-72], [-10], [-56], [12], [18], [22], [6], [86], [50], [84], [-54], [-30], [-76], [104], [58], [-100], [64], [50], [78], [-2], [26], [-92], [-78], [-24], [-24], [18], [-64], [50], [-20], [-88], [-40], [80], [-22], [-100], [88], [86], [20], [66], [94], [4], [30], [-46], [94], [80], [106], [-82], [20], [-10], [-72], [-42], [-14], [86], [-2], [44], [-14], [-24], [-102]]; 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_u();". // 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_u(: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_u();". // 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_u( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9576_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![-2, 1]>,<11,R![6, 1]>,<13,R![0, 1]>,<17,R![-4, 1]>],Snew); return Vf; end function;