
// Make newform 9576.2.a.co in Magma, downloaded from the LMFDB on 17 April 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_co();".
// 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_co();".
// 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
        poly := [-2, 4, 4, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 2, 0, 0, 0], [-4, -8, 2, 2, 0], [-2, -6, 8, 2, -2], [6, 0, -10, 0, 2]];
    Rf_basisdens := [1, 1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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<X, Y> := 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<T> := 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, 0, 0], [0, 0, 0, 0, 0], [0, -1, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, 0, 0, 1], [2, 0, 0, -1, 0], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 0, 0, -1, 0], [0, -1, 0, -1, 0], [0, 0, 0, -1, 1], [2, 0, 0, 1, 1], [2, -2, 0, 0, 0], [0, 0, 0, 0, 0], [2, -2, -1, 1, 2], [-4, 1, 0, 1, 0], [0, 3, 1, -1, -3], [6, 0, 0, 0, 0], [4, -1, 1, 1, 2], [2, 0, -1, 2, 1], [2, 1, 1, 1, 1], [0, -1, -1, 0, -1], [2, -1, 0, 1, 2], [-2, -1, -1, 1, -1], [2, -2, 0, 2, 1], [-8, 3, 0, 0, 0], [0, 1, -1, 0, 2], [-2, 2, 1, 2, -1], [6, -2, 0, 1, -1], [4, -1, 0, 1, 4], [-4, -1, 1, 2, 1], [-2, 1, 0, -1, -4], [-10, 0, 0, 2, 0], [4, -1, -1, 3, 3], [-6, -2, 0, 1, 1], [4, -3, -1, -2, -1], [14, -4, 0, 0, 2], [12, -1, 1, 1, 3], [0, 1, 1, -1, -1], [2, 1, 1, -3, 3], [-2, -3, 0, 3, 4], [-2, 1, 1, 0, -3], [0, -2, -2, -3, -2], [10, -2, -2, 0, 0], [-2, 4, 0, -1, -3], [0, 1, 1, 1, 1], [0, 1, 1, -3, -4], [0, 2, 2, 1, -3], [4, -1, -1, -3, 1], [10, 2, 0, 0, 2], [-6, -2, 0, 2, -2], [4, 4, 2, -3, -4], [2, 0, -2, 4, 1], [-2, 3, 2, 1, -2], [2, 1, -1, -1, -3], [-12, 0, 2, -3, -2], [-2, 3, 1, 1, 1], [0, -2, -2, -2, 2], [10, -5, -1, 1, -1], [4, 3, 0, 1, 0], [-4, 7, -1, -1, -1], [2, 1, 1, 3, 1], [4, 3, 1, -1, -3], [2, 2, -1, 1, 4], [6, -7, -1, 1, 3], [-4, 2, 1, 4, 1], [4, 1, -1, -1, 2], [-2, 4, 2, 0, 0], [-4, -4, -2, -4, 1], [-2, -2, -2, -2, 4], [-8, 6, -1, 0, 0], [8, -4, 0, -1, 2], [-4, 2, 0, -2, -4], [18, 0, 0, 1, -1], [-4, 3, 1, -3, 0], [4, -2, 0, 2, 8], [2, 2, 0, 3, 5], [-2, 0, 0, -2, -4], [4, 4, 1, -4, 1], [-10, -1, 1, 5, 6], [2, 1, -2, 5, 0], [-2, 0, 2, -3, 1], [-6, -2, 1, 0, -1], [10, -6, 0, -2, 1], [8, -4, 0, -1, 5], [-12, 4, -2, 0, -1], [20, 0, 1, -4, 3], [2, 6, 0, -2, -4], [8, 1, -2, -4, -4], [-16, -2, 2, -2, -2], [6, 7, -2, 1, 0], [6, 0, -1, -3, -2], [0, 5, 1, -2, -1], [-4, -6, 0, 4, 5], [0, -3, -1, -5, -5], [-6, 6, -1, -3, -4], [6, 5, -1, 3, -1], [10, 5, -1, -1, 3], [-4, -1, -3, -1, -3], [2, 7, 1, 5, -3], [-16, -3, 1, -1, -2], [2, 0, -2, 3, 3], [20, -1, -1, -1, -1], [20, 1, 2, -1, -4], [8, -1, 1, -3, -1], [-6, 4, 0, -4, 4], [6, -1, 2, -1, 2], [4, 0, -1, -2, 6], [6, 4, -3, 2, 3], [2, -8, 2, 2, -3], [-16, -3, -1, 0, 4], [10, -1, -1, 3, -3], [14, 4, 0, 0, 2], [-12, 1, -3, 3, 3], [8, -7, -1, 1, 3], [12, -6, -1, -2, -3], [16, 0, 2, -2, 0], [-6, 8, 1, 5, -4], [14, 2, -2, -3, -3], [14, -5, 2, 1, -2], [18, -4, 0, 3, -2], [-6, 6, 2, -4, -6], [14, -1, 1, 1, -1], [-10, -1, -2, -3, -4], [-12, -6, 2, 2, 2], [14, -6, -2, 3, -1], [6, 4, -2, 6, 0], [6, -6, 1, 1, -2], [0, -10, 2, 4, 4], [2, 8, 2, -4, -4], [-4, 1, -1, -5, -1], [-30, -2, 1, -2, -1], [-12, -9, 1, 4, 3], [-10, 0, -4, 2, 6], [12, -4, -1, 4, 0], [-10, 1, 3, -7, -5], [-22, 5, 1, -1, -3], [-20, 5, -1, 3, -7], [26, -9, -1, -1, 5], [-10, 4, -4, -2, -2], [-4, 4, -4, -2, 4], [-14, -4, -2, 1, 5], [8, -1, -3, 5, 5], [10, -11, 2, 3, 2], [14, 1, 1, -4, 3], [-20, 3, 5, -3, -1], [6, -2, 2, 2, -2], [-18, 3, -1, 5, -1], [8, -3, 3, 3, -7], [6, -2, 3, -10, 1], [10, -6, -2, 3, 3], [24, 4, -3, 6, 2], [8, 5, -5, 3, 3], [4, -8, 6, 2, 0], [-8, 5, 1, 3, 0], [-18, -8, -3, 6, 11], [-24, 0, 0, 4, -4], [8, 12, 1, 0, -4], [-10, -1, 1, 3, -1], [-2, 1, 3, -1, 3], [-16, 2, 2, 0, 3], [24, -1, 2, 1, -6], [-16, -3, 3, 5, 3], [8, -5, 1, 1, 3], [-4, 0, 1, -8, 1], [-20, 5, -1, 5, -1], [0, -1, -1, -6, 1], [22, -6, -2, 2, 4], [-26, 4, 2, 2, 0], [8, 3, 4, -2, -2], [18, -11, -2, 3, 0], [10, -2, -2, 3, 2], [-12, -6, 0, 3, 0], [-6, -2, 4, -2, -6], [-16, -1, 1, -2, -6], [12, 0, -1, -6, 2], [4, -1, 1, 1, 9], [6, -4, -4, -1, -1], [24, -2, 2, -3, 9], [22, -6, -2, 2, -2], [16, 4, 4, -6, -4], [4, 6, -2, -4, -8], [-26, 8, 2, 3, -3], [18, -2, 0, 0, -6], [-8, -2, 6, -5, -2], [0, -1, -4, -6, 2], [-18, 6, -4, 1, -7], [12, -1, -3, 5, 9], [6, -2, 0, 8, 4], [6, 8, 3, -3, -8], [6, 1, 3, -3, -8], [-40, -2, -2, 0, 3], [-20, 5, -1, 7, 0], [22, -1, -3, -3, -1], [8, -6, 6, -2, 1], [-10, 4, 0, 6, 0], [10, 2, -4, 2, 2], [6, 2, 2, -4, 4], [0, 10, -1, 0, -4], [-18, 6, -3, 3, 0], [-12, 5, 4, -3, 4], [8, 1, -5, 0, 0], [-10, 1, 1, -6, -5], [-2, -9, -3, 3, 6], [-2, 3, -2, 3, 2], [-8, -5, 4, -8, 0], [-4, 14, 4, 8, -6], [-6, 3, 0, 1, 2], [-6, 6, 4, 6, 0], [4, 9, 3, 7, 1], [26, 2, -4, 6, 4], [-16, 3, -4, 0, -8], [-8, -1, 5, -3, 1], [24, -4, -6, -2, 0], [-4, 7, 1, -3, 11], [-34, 3, 1, 5, 0], [-8, 5, 3, -5, -1], [-10, 4, 0, 2, -4], [-18, -4, -3, 2, 3], [-30, 6, 0, -1, -3], [22, -12, -2, -1, 1], [-8, -1, -3, 4, 0], [20, -5, 0, 1, 10], [-16, 0, -4, -4, -4], [-12, -6, -2, 4, 4], [6, 0, 0, -4, 4], [10, 7, 1, 5, 7], [14, 6, -5, 6, 5], [-8, -4, -4, 1, 11], [-12, 2, 4, 2, 3], [-14, 1, 3, -1, 7], [-12, 11, 5, -7, -9], [16, -9, -1, 0, 1], [12, -1, 4, 5, 6], [24, -9, -3, 3, 1], [14, -6, 5, -1, 6], [-22, 1, 1, 5, -7], [-10, 2, -2, 7, 5], [-4, 6, 4, -8, -5], [20, -6, 0, -6, 8], [-14, -4, 5, 0, -3], [-4, 2, 2, 6, 2], [16, 1, 5, -1, -1], [2, -2, 2, 1, 3], [-6, 5, 3, 3, -3], [22, -4, -1, -1, 0], [20, 8, -2, -2, 4], [-24, 12, -2, 2, -8], [16, -3, -3, 5, 10], [-32, 10, -2, 5, -8], [-6, 1, -1, 3, -1], [-12, -8, -1, -2, 2], [-20, 16, 2, 1, -4], [6, -9, -3, -1, 2], [-36, 5, 4, 4, 0], [-16, 12, 4, -8, -11], [-6, 5, 3, -3, 5], [-28, -1, 1, -5, -9], [14, -5, 1, -7, -1], [14, -5, -3, -1, 3], [-8, -3, 5, -4, 1], [-2, -2, -3, -10, -7], [-22, -2, 0, -2, -10], [34, -8, -4, 5, 2], [-20, 20, 3, -4, -12], [-12, 11, 5, -3, -7], [12, 9, 4, 5, 0], [4, 2, 5, 8, -4], [0, 11, 3, -5, -14], [-6, -16, 2, 5, -5], [6, 1, 1, 8, 9], [16, -7, 1, 1, -6], [42, 7, -1, -9, -5], [-24, 2, -2, -4, -10], [-22, 2, 4, -8, 3], [24, -1, -5, 5, 5], [-20, 6, 0, 4, -1], [-22, 8, 4, -1, -3], [10, 8, 4, -4, -12], [46, -7, -4, 5, 4], [8, 3, 7, -5, -1], [-24, 9, -1, 7, -5], [-30, 0, 5, 7, 2], [14, -15, -3, 4, 15], [8, -11, -1, 1, 7], [-20, -2, 0, 9, -2], [-10, 11, 5, 5, -3], [-8, 5, 2, -5, 4], [-32, -7, 1, 1, -3], [38, 2, 2, 0, -8], [2, 6, 4, -3, 5], [18, -5, 0, 1, 6], [-4, 3, -4, -7, -12], [28, 8, -4, 4, 4], [18, 0, 4, -1, 5], [-36, -9, 2, -8, -4], [0, 16, 0, 5, -1], [-24, -5, -4, -2, 2], [28, -7, 1, 1, 1], [-8, 6, 4, 8, 10], [-6, -4, 4, 4, -6], [0, -3, -2, -3, -10], [0, 0, -4, 6, 0], [-10, -2, -3, 10, 5], [-8, -8, 0, 8, 0], [-18, 0, 6, 2, -2], [-22, 9, 2, -1, 0], [18, 10, -2, -9, 0], [20, 2, 4, -1, -6], [22, -4, 2, -10, 4], [18, 6, -1, -1, 8], [2, 0, -2, -5, 5], [20, -2, -1, -6, 5], [12, 11, 1, 3, -4], [-24, 2, -2, -5, -4], [-30, -6, 4, 0, -1], [14, 1, 0, -13, -2], [-22, -6, -3, -8, 7], [-2, 4, 2, 4, 0], [-2, 6, 2, 2, 16], [-20, -4, 0, 2, -2], [18, 6, 4, -2, -12], [30, 3, 1, -9, -3], [20, -7, 3, -4, 1], [18, 8, -6, -6, -4], [18, 16, -2, 0, 1], [-8, -15, 1, -1, 4], [-36, 2, 2, -2, -4], [-30, -2, -3, 12, 3], [-2, 2, -2, 1, 9], [18, -5, -3, -3, -3], [20, 2, 1, -10, 7], [-12, 16, -2, -2, -8], [-14, 3, 4, 7, 14], [12, 14, 2, -2, -12], [6, 13, 0, -1, -6], [-14, 4, 4, -3, -11], [-30, -2, 0, 0, -8], [10, 7, 3, 3, -15], [-8, -6, 2, -8, -4], [10, -2, 2, 7, 9], [-10, 4, -8, 2, 4], [18, -9, 3, 5, -7], [-8, 17, 1, -4, 5], [44, 4, -1, -12, 1], [-2, -1, -6, 5, 2], [18, -18, 4, 4, 4], [0, 5, -3, 3, -6], [-4, 5, 3, 3, 5], [-32, 9, 2, 10, -2], [4, -14, -6, -4, 8], [-10, 2, 4, -4, -2], [-10, 6, -2, 9, -5], [0, -13, -3, -4, 8], [34, -16, -4, 5, 3], [-22, 7, -3, 1, -11], [18, -14, -5, 9, 6], [18, -7, 2, 7, -6], [-6, -10, 0, -8, 6], [-38, -8, 3, -6, -7], [-6, -9, 1, 5, -3], [-20, -2, 5, -8, 0], [10, -2, -3, 8, -5], [2, -1, 4, 5, -10], [0, -6, 4, 6, 8], [18, -10, -2, 6, 8], [4, 17, 0, 6, 2], [4, -7, -5, 0, -9], [-14, -4, -4, -10, 10], [-24, 2, 6, 8, 5], [4, -11, -1, -5, -1], [-4, -2, 0, -7, 10], [16, -4, 1, -8, -5], [-16, -4, -8, -4, 4], [-6, 1, 3, 13, 5], [40, -15, -1, 3, 5], [8, 14, 2, -11, -14], [-10, 7, -7, 3, 1], [20, -10, 1, 6, 10], [26, 8, 6, -4, -19], [14, -7, -1, 11, 5], [-26, -4, -4, 6, -2], [-16, 1, -5, 1, 7], [22, -2, 2, 2, 2], [-12, 3, 5, 1, -11], [4, -14, -4, 0, 2], [-16, -7, -1, -7, -9], [-30, -10, 4, 10, 8], [48, -6, -2, 6, -2], [-16, 1, -3, 15, 3], [-42, 6, 0, -6, 2], [-22, -11, 1, 3, 3], [0, -6, 2, 0, 3], [-12, -2, 2, 4, 12], [10, 0, -1, -4, -15], [6, 4, -2, 6, 10], [-40, -3, 3, -2, 4], [10, 0, -6, 2, 0], [20, 9, -1, -5, -10], [-38, -10, 4, 7, 5], [34, 0, -4, 9, 14], [-2, -8, 0, -2, -6], [-8, -5, 3, -10, 5], [-6, -5, -3, 5, 7], [6, 7, 1, 5, 1], [4, -2, 4, 8, 10], [-10, 8, 4, 4, -8], [-8, 17, 4, -1, -18], [-4, -4, 0, 4, 10], [66, 5, 2, -5, -4], [10, 20, -2, -6, 1], [2, 4, -6, 5, 11], [54, 4, 3, -8, -7], [-28, 7, 5, 1, 7], [-18, -9, 1, 1, 1], [-28, 9, 8, -10, -2], [26, 2, 1, -2, -13], [-56, 12, -4, 1, -15], [-16, -6, -5, 10, 18], [0, -14, 2, 8, 12], [20, 0, 7, -16, 1], [-26, 21, 5, 6, -1], [20, -14, -4, -5, 4], [20, -14, -6, -6, 6], [-18, 11, 1, -3, -2], [18, -17, -1, 3, -11], [2, -8, -7, -8, -9], [-22, 2, 0, -2, -6], [-24, -4, -2, 6, 8], [34, 0, 1, 5, 10], [-2, -15, -1, -1, -12], [-24, -14, -2, 14, 17], [-36, 3, 5, -3, -5], [-4, 1, 3, -3, 7], [-18, -2, -6, 8, 4], [-32, 8, 1, -8, 0], [-46, 6, 0, -2, -12], [-6, 0, 8, -17, -5], [-12, 3, 1, 11, -10], [0, -10, -6, -10, -2], [22, -15, -8, -3, 2], [-58, 6, -2, -6, -20], [-14, -6, 6, 1, -3], [26, 10, -3, -10, -5], [18, 1, -7, 5, 7], [-28, 6, 3, 6, -9], [16, 6, 0, 10, -12], [32, 12, 0, -2, -2], [6, -1, -3, 5, 0], [-22, 4, 4, 1, -6], [0, -9, -9, -7, 2], [0, 10, 6, 6, -14], [-44, -4, 6, 0, -3], [-20, -6, -3, -10, -2], [-30, -4, -8, -4, 14], [-8, -3, 2, -1, 8], [-18, 18, 4, -1, 1], [44, -8, -4, -6, -2], [2, -4, 10, 4, -2], [32, -3, -8, 7, 6], [40, -23, -7, 3, 8], [-20, 15, -3, -6, 3], [58, -1, -4, -7, -2], [2, 0, -4, 1, -9], [-8, -9, 1, -3, -13], [10, 10, 2, -6, -8], [-16, 7, -5, 2, 9], [10, 5, 2, -3, -4], [14, 8, 4, -6, -12], [28, -13, -1, -11, 9], [-48, 11, -1, 2, -1], [14, -10, -3, 6, -5], [12, -2, 0, -4, -18], [38, -9, 3, -11, 5], [-38, 0, -3, -4, -17], [6, 13, 5, -10, -11], [-10, 26, 2, -3, -15], [-8, -5, -5, -1, -5], [6, -10, -7, -3, 2], [4, 0, 7, 2, 15], [-14, 4, -6, -8, -1], [-6, -2, -4, 4, 6], [-18, 7, 1, 1, -6], [-38, -7, -3, 1, 11], [64, 0, 0, 0, -8], [-44, 0, -6, 6, -3], [38, 10, 0, -10, -4], [20, -11, -3, -1, -1], [44, -2, -2, -4, -10], [12, -7, 3, 12, 9], [22, 7, -5, 0, 9], [56, -4, -4, 4, 4], [10, 12, 4, 4, 8], [-18, 9, -9, 3, -1], [56, -2, -6, 4, -1], [-2, 24, 4, -4, -18], [-12, 2, -6, -6, -16], [-22, -6, 0, 15, 3], [20, 2, 4, -6, 4], [-16, -9, -5, 1, 12], [18, 18, 0, 1, -9], [-12, -11, -5, -5, -3], [8, -4, 7, 4, 1], [-8, 0, 4, 0, -12], [10, -4, -8, -7, -9], [-44, 6, 1, 14, -6], [-62, 6, -7, -2, 3], [32, -2, 2, 3, -1], [6, 12, 2, 13, -5], [12, 4, -4, 10, 4], [18, -1, -4, -3, -8], [-2, -20, -5, 13, 12], [18, 12, 4, 12, 2], [-8, 15, 0, -3, -6], [12, -22, 2, 2, 12], [-22, 6, 0, -6, -18], [22, 3, 5, -9, 7], [-42, -4, -4, 2, 6], [2, 10, 1, 0, 7], [-32, 2, 2, 17, 1], [26, 1, 1, 7, 13], [-12, -13, 1, -1, 10], [4, 0, -7, 12, -5], [0, 0, -8, 7, 4], [-2, 21, -1, 5, 1], [30, 1, -4, 11, -2], [30, 1, 3, 3, -6], [-12, 7, 2, 0, 16], [-38, 11, 8, -11, -12], [-42, -5, 5, 17, 7], [-24, 16, -4, 2, -8], [12, 6, -5, 2, -1]];
    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_co();".
// 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_co(:prec:=5)
    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_co();".
// 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_co( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9576_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![64, 64, -32, -20, 2, 1]>,<11,R![128, 160, -40, -28, 2, 1]>,<13,R![-64, -352, 208, -12, -8, 1]>,<17,R![-5504, 1984, 192, -92, -2, 1]>],Snew);
    return Vf;
end function;