// Make newform 9900.2.a.w in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_9900_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_9900_2_a_w();". // 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_9900_2_a_w();". // 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_9900_a();" function MakeCharacter_9900_a() N := 9900; order := 1; char_gens := [4951, 5501, 2377, 4501]; 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_9900_a_Hecke(Kf) return MakeCharacter_9900_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], [0], [2], [-1], [2], [4], [-6], [0], [8], [-8], [-10], [-8], [2], [-8], [-2], [-12], [10], [-12], [-8], [-6], [-2], [16], [14], [2], [16], [-4], [0], [-10], [6], [10], [-12], [-2], [-6], [-20], [-10], [-6], [-16], [12], [0], [12], [-10], [-16], [22], [24], [-16], [18], [4], [-24], [-14], [24], [-4], [-18], [-4], [-30], [-4], [2], [10], [-14], [8], [26], [20], [-2], [16], [18], [-14], [8], [-14], [0], [-14], [-6], [8], [4], [-34], [16], [0], [14], [-18], [18], [-10], [12], [6], [4], [18], [34], [12], [-42], [-22], [-12], [-4], [-20], [-28], [8], [0], [4], [-20], [2], [-6], [34], [-18], [2], [20], [-36], [-20], [-10], [-34], [44], [-16], [-24], [-30], [-14], [-2], [-10], [-32], [0], [6], [16], [-16], [6], [20], [30], [22], [8], [44], [-8], [36], [34], [-40], [8], [26], [-34], [-40], [-32], [-54], [20], [34], [-2], [-14], [-38], [20], [18], [-16], [12], [-48], [18], [0], [6], [-28], [-8], [-48], [38], [-46], [-44], [-48], [0], [-8], [50], [-6], [38], [24], [52], [36], [-14], [-20], [38], [8], [-56], [-2], [62], [46], [-36], [-62], [24], [42], [-28], [18], [-26], [-30], [-14], [42], [32], [12], [42], [-20], [-16], [-6], [50], [-32], [-58], [-60], [-58], [56], [4], [30], [4], [-14], [34], [34], [12], [-8], [-20], [54], [42], [-26], [28], [2], [20], [52], [-36], [-4], [22], [-30], [-8], [20], [0], [14], [-50], [0], [56], [54], [14], [30], [34], [36], [-28], [34], [14], [0], [6], [24], [-50], [-38], [22], [12], [4], [8], [-22], [-52], [60], [0], [-60], [38], [-56], [22], [8], [16], [8], [-36], [38], [-72], [-14], [-32], [-8], [-10], [36], [72], [10], [70], [-30], [-34], [38], [-28], [-42], [46], [66], [36], [-30], [18], [-58], [56], [18], [-56], [-66], [26], [-6], [-72], [-12], [2], [74], [-20], [68], [-72], [72], [-22], [66], [-48], [18], [-56], [72], [-24], [-18], [-4], [16], [-20], [-18], [36], [-8], [54], [16], [46], [54], [36], [54], [36], [-60], [50], [36], [14], [-48], [6], [-4], [46], [32], [-44], [-12], [-26], [28], [0], [-74], [-36], [-78], [-26], [76], [-48], [-24], [2], [-60], [-4], [-52], [-72], [-18], [18], [70], [-48], [-2], [36], [66], [24], [10], [58], [22], [74], [4], [-4], [30], [-8], [70], [-40], [64], [-2], [-14], [-42], [-30], [-30], [58], [-4], [96], [-8], [12], [-64], [6], [-64], [96], [-24], [60], [6], [-44], [34], [-54], [84], [-32], [72], [-64], [26], [-14], [36], [52], [14], [-36], [2], [-42], [26], [2], [-4], [6], [-48], [-24], [102], [-14], [-88], [-82], [-42], [-12], [-44], [24], [26], [14], [-58], [-88], [0], [82], [54], [34], [-78], [-12], [-98], [30], [-16], [68], [-68], [22], [76], [44], [54], [-54], [14], [-36], [44], [-70], [20], [-86], [14], [24], [-48], [54], [70], [-20], [-24], [-40], [-4], [-10], [40], [52], [-16], [26], [18], [-50], [42], [-56], [6], [12], [18], [-30], [96], [-26], [-72], [-34], [84], [-98], [-50], [14], [32], [24], [-20], [98], [46], [-10], [72], [-38], [62], [28], [-92], [-36], [-10], [-50], [-62], [74], [36], [-96], [112], [86], [108], [-96], [-42], [92], [2], [98], [-84], [-88], [18], [26], [-108], [2], [36], [20], [-60], [-18], [36], [-8], [-74], [-90], [-8], [6], [16], [80], [82], [-52], [50], [-2], [-78], [20], [-6], [60], [-46], [-6], [-6], [24], [28], [-94], [-2], [-96], [-12], [14], [66], [10], [26], [30], [-6], [-96], [36], [62], [-94], [24], [72], [2], [72], [70], [-56], [44], [-82], [-34], [42], [-38], [20], [-14], [-36], [-26], [78], [50], [74], [76], [-18], [-16], [64], [-32], [-28], [100], [-108], [42], [-60], [64], [-74], [96], [-30], [20], [-18], [76], [-86], [-64], [50], [6], [40], [36], [6], [-102], [-42], [-96], [70], [36], [-12], [-106], [44], [64], [-22], [-50], [-108], [-18], [70], [30], [-38], [-60], [14], [84], [108], [102], [-16], [-94], [-44], [44], [-98]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_9900_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_9900_2_a_w();". // 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_9900_2_a_w(:prec:=1) chi := MakeCharacter_9900_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(4297) - 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_9900_2_a_w();". // 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_9900_2_a_w( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_9900_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![-2, 1]>,<13,R![-2, 1]>,<17,R![-4, 1]>],Snew); return Vf; end function;