
// Make newform 4002.2.a.bf in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4002_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_4002_2_a_bf();".
// 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_4002_2_a_bf();".
// 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 := [-4, -4, 17, 8, -8, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 9, -3, -3, 1, 0], [-6, -9, 5, 3, -1, 0], [2, 7, 4, -6, -2, 1], [-2, 8, 11, -5, -3, 1], [-8, 4, 13, -5, -3, 1]];
    Rf_basisdens := [1, 2, 2, 2, 2, 2];
    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_4002_a();"
function MakeCharacter_4002_a()
    N := 4002;
    order := 1;
    char_gens := [2669, 3133, 553];
    v := [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_4002_a_Hecke(Kf)
    return MakeCharacter_4002_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 := [[-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, -1], [1, 1, 0, 0, 0, 0], [1, 0, -1, 0, -1, 1], [2, -1, 0, 1, 0, -1], [-2, 1, 1, 0, 2, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, -1, 0, 2, 0, 0], [1, 1, 1, -1, -1, 0], [3, -3, -1, -1, -1, 0], [0, -1, 1, 0, 0, 0], [4, 0, -1, -1, -1, 0], [0, 2, 2, 0, 2, 0], [4, -2, 0, -2, -3, 0], [0, 1, 2, 1, 1, -3], [2, -2, -1, 1, -2, 2], [1, -1, 2, 0, 2, -2], [-2, 2, 0, -2, 2, 0], [-1, -1, -1, -3, 0, 4], [4, 0, -1, 1, -2, 1], [2, 3, -1, 0, 1, -1], [0, -3, 2, -1, 1, -2], [4, -2, -1, -1, -2, 2], [-2, -4, 1, -1, -1, -1], [5, 0, 2, -1, 0, 0], [2, -2, 3, -1, 0, -1], [4, -1, 2, 1, 2, -1], [0, 2, 1, 3, 2, 2], [5, 1, -1, -1, -2, 0], [2, 1, -1, 2, 1, 3], [2, -3, -3, 0, -7, 3], [0, 2, 4, 2, 1, 0], [9, -4, -1, 2, -4, 1], [-1, 5, -2, -2, -3, 2], [1, 0, -2, -3, 0, 3], [6, 2, 1, 1, -4, 4], [5, -1, 4, 0, 1, -4], [-4, -2, 6, 4, 0, 0], [3, -2, -1, 2, -5, -2], [9, 1, -2, 0, -3, 1], [4, -4, 2, 4, -4, -2], [7, 2, 1, -2, -6, -1], [0, -2, -4, -2, -2, -3], [7, 2, 0, -1, -2, -1], [4, -2, 4, 6, -2, -6], [-1, 2, 0, -3, -2, 6], [-11, 3, -1, -5, 3, 0], [4, 0, -2, -4, 0, 0], [0, 3, -4, 3, -1, 5], [-1, -2, -3, 2, -4, -3], [6, 1, -4, -1, 1, 1], [-3, -2, 6, 1, 5, -1], [5, -1, 1, -3, -3, 1], [11, 2, -1, 4, -5, 5], [6, -4, -1, -3, 0, -2], [-9, -2, -1, 0, -3, 5], [-2, -4, 3, -1, 4, -1], [1, 8, -1, 0, 5, 3], [14, -3, -8, -3, -3, 0], [2, -2, 5, 5, -2, -6], [6, -2, -3, 1, 5, -2], [-7, 4, 3, 4, 0, -1], [-1, 2, -5, 0, -3, 5], [1, 2, -1, 0, -6, 1], [10, -1, 3, 4, 1, -4], [2, 8, -3, -1, 3, 2], [6, 8, 1, 1, 2, 4], [4, 0, -6, 0, -2, -2], [-3, 3, 3, -1, 3, 1], [-1, 3, -3, -1, 0, 4], [-6, 4, 1, 3, 2, -1], [-6, -5, 0, 1, 3, -6], [9, 2, -2, -1, 3, 3], [6, -3, 2, 3, -1, 2], [-1, -4, 4, -1, 1, 1], [-2, 4, -6, 2, -2, 3], [3, 2, 2, 7, 1, 3], [5, -4, -4, -3, 0, -2], [0, -3, -4, -5, 1, -3], [5, 0, 0, 3, -1, 3], [-11, -3, -2, 2, -6, -5], [5, -6, 3, 4, 2, -5], [8, -3, 5, 2, -7, -1], [-1, 5, 5, 9, 9, 0], [15, 2, -1, -4, -2, -1], [3, 4, -3, 4, -5, -1], [0, 4, 8, 0, 10, -4], [9, -3, 0, 4, 2, 2], [-4, -7, -3, -4, -1, -2], [7, -2, -5, 2, -12, 5], [-11, 3, 5, -1, 6, -2], [7, 3, -1, -5, -6, 2], [9, 3, -3, -3, 1, -3], [-11, -1, -2, 2, 7, 0], [-5, 6, 7, 0, 11, -8], [-6, 10, -3, -5, 8, -2], [4, -1, 2, -5, -2, 9], [8, 5, -3, -2, -1, 5], [12, 4, -2, 2, -7, 0], [-6, -11, 0, 3, -9, 1], [-2, 1, -6, -3, 8, -1], [-19, 6, -3, 4, 7, -1], [7, 0, -2, 1, -9, 1], [10, -4, -5, -3, 1, -6], [4, -4, 0, 6, 0, -2], [-4, -2, 4, -2, 0, -12], [-10, -2, 6, 4, 8, 2], [-1, 3, 7, -1, 0, -2], [10, -2, -1, -1, 2, -7], [4, -13, 0, -5, -4, -1], [-14, 1, 3, 6, 14, 0], [4, 8, 5, 1, 7, -11], [0, -5, 0, 3, 6, -1], [-13, 4, -4, -9, -1, 5], [-3, 5, 6, 6, 2, -12], [-22, 6, 3, 1, 4, -2], [18, -3, 3, 6, -1, -1], [-10, 6, 3, -1, -8, 1], [-8, 2, -7, -7, 3, 8], [-2, 1, -2, -1, 5, 10], [-12, 6, -2, -4, 5, 2], [5, 0, -4, 3, -1, -3], [2, -4, -4, -2, -7, -2], [-21, 6, 7, 2, 5, 0], [-9, -2, -5, 2, -1, -3], [-14, -3, 3, 4, 9, 1], [-8, 1, 2, 9, 3, -1], [4, -12, -3, -3, -2, -4], [5, -1, -2, 2, 3, -7], [-14, 5, 3, 4, 5, -7], [13, -3, -5, -7, -5, 6], [-13, 2, 2, -1, 11, -5], [-15, -1, 7, 5, 2, -3], [3, -5, -4, 2, -12, 13], [3, 2, -7, 6, -1, 7], [7, 3, -8, -6, 2, 3], [-10, -1, -8, -9, -7, 5], [-14, 7, -5, 4, 5, -3], [-14, -3, 5, 2, -7, -1], [-11, 10, -5, -4, -3, 7], [-8, 5, 0, 1, -1, 1], [-2, 6, -1, -3, -9, 6], [-5, 5, 0, 0, -9, 3], [16, 2, -7, 5, -9, 4], [-17, 4, -4, -9, 7, 3], [3, 6, -5, -2, 6, 5], [7, -8, -3, -2, -11, -1], [-1, -4, -4, 3, 3, 7], [-20, 11, 7, 6, 7, -3], [-19, 0, -4, 3, -1, 5], [0, -2, 6, 4, 4, -8], [10, -7, 2, -3, 9, -2], [8, 1, -5, 2, -13, 2], [-6, 6, -5, -9, -7, 5], [-6, -8, -2, 2, -4, 4], [-27, -2, 3, -6, 4, -3], [-6, 10, 7, -1, 6, 0], [-23, 11, -3, -5, 8, -2], [-14, -2, -1, 1, 0, 5], [7, -4, -6, -1, 1, -1], [-23, -1, 4, -2, 14, -2], [-9, 8, 4, -1, -11, 4], [-4, -5, -3, -2, -5, -2], [-3, -2, -11, -6, -4, 13], [20, -10, -1, -1, -17, 8], [-9, 4, -9, -6, -12, 11], [-21, -8, 6, 1, 5, 3], [-4, 8, 5, -9, 4, 1], [-26, -4, 4, 0, -2, -2], [-30, -8, 2, 4, 4, -2], [17, -3, 4, 0, 4, -5], [-9, 4, 5, 2, -10, 9], [1, 2, 8, -5, 1, -6], [-2, 6, 12, 0, 12, -4], [-2, -10, -4, -2, 0, 10], [-9, 5, -11, -11, 2, 8], [-22, 8, -1, 1, 4, -5], [9, -11, 2, 10, 0, -2], [6, 11, -8, -3, -1, -1], [15, -6, -11, 6, -11, 9], [-10, -4, 10, 2, 10, -6], [7, 6, 2, 1, -9, 3], [-20, 8, -5, -1, -2, 6], [12, 17, 0, -1, 2, -5], [-8, 3, 1, 0, 4, 0], [-5, 0, -2, -7, -1, -1], [10, 3, -3, -10, -5, -1], [2, 4, -2, 6, 3, -8], [31, 3, -3, -3, -4, 4], [-4, 7, 10, 5, 9, -10], [-7, 4, 12, -3, 7, -5], [11, 13, -2, 0, -4, 2], [-21, 4, -11, 4, 3, 0], [-2, 5, 5, 0, -1, 10], [-5, -6, 5, -4, 5, 0], [10, 2, -3, -1, 6, 1], [0, -7, -4, 9, 7, -5], [-10, -15, 0, 5, -11, -8], [3, -6, 3, 2, 4, -9], [-8, 0, 9, 3, 13, -6], [-28, 9, -12, -3, 9, 0], [15, -1, 3, -5, -20, 2], [15, 0, 1, 4, 4, -3], [17, 5, -3, -5, 0, 4], [-24, 6, -4, -2, 9, 4], [-12, 6, 2, 2, 2, 6], [10, -10, -5, -5, -14, 2], [-22, 8, 0, -4, 8, 4], [-19, 0, -7, 2, 4, 11], [13, -11, -5, 1, -12, -2], [-13, 1, -5, -3, -6, 6], [-2, 6, 3, 3, 3, 4], [-8, 2, -8, 4, 10, 4], [35, 0, -7, -2, -2, -5], [-12, -8, -1, -3, -8, 2], [13, -6, 6, -1, -5, 3], [13, -16, -7, -2, -11, 3], [2, -1, 7, 2, 5, -5], [-8, -8, -10, -14, -12, 1], [-11, 0, -2, 7, 2, -1], [-4, -7, -9, -10, -9, 11], [4, -12, 4, 8, -14, -6], [20, -13, -10, -5, 0, 1], [-4, -1, 6, -7, 6, -11], [32, 6, 0, -2, -4, -4], [1, 1, -11, -9, 8, 12], [0, -12, 1, 5, -12, -6], [-12, 1, 7, -6, 3, 3], [-15, 6, 4, -1, 13, -7], [0, 7, -1, -4, 13, 9], [-21, -8, 9, 0, 6, -7], [-5, 8, -1, -6, 11, -1], [25, 1, 7, 17, 1, -7], [-8, 9, -1, 0, -3, -4], [21, -5, -12, -8, -8, -5], [-2, 2, -2, -2, 1, 0], [18, 10, 3, 3, 3, -2], [1, -7, 5, 1, -12, -6], [18, 14, 0, -4, 2, 0], [-8, -10, 4, 2, -4, -4], [24, 8, 3, 1, -7, -2], [-16, 14, -2, -10, 0, 6], [17, -8, 4, 7, 1, 1], [24, 8, -4, -4, 0, 6], [2, 2, 3, -3, 0, -3], [-4, 0, 7, -3, -10, 4], [2, 7, -3, 0, 11, -10], [18, -3, 7, 12, 9, -3], [24, -2, 4, 12, -8, 1], [-1, 11, 9, -1, 13, -3], [-14, -7, 1, 0, 7, -8], [10, 4, 4, 0, 8, -4], [-18, -2, 0, 0, 3, 12], [-11, 9, -8, -2, -1, 2], [22, 3, -7, -6, 6, 6], [8, -11, -5, -10, -5, -3], [8, 7, 0, 5, 13, -7], [-8, 11, 3, 8, 15, 1], [-22, 9, 17, 12, 9, -5], [-8, 6, 5, -11, 6, -6], [0, -11, -2, -3, 1, -13], [-9, 4, -7, 4, -2, 1], [2, 11, 5, 6, -10, 4], [-2, 9, 0, 5, 3, -4], [10, -9, -6, 3, -16, 3], [22, 1, -9, -2, -10, 4], [-5, 3, -4, -2, -15, 10], [13, -16, -1, -4, 5, -11], [5, 6, 2, 5, 3, 9], [-1, -4, -11, 4, -14, -3], [-20, 7, -3, -6, 5, -1], [5, 2, 10, -7, 7, -9], [7, -4, 12, 5, 3, -13], [31, -9, -1, -5, -4, 4], [13, 4, -4, -7, 3, -9], [1, -5, 5, 15, 6, -1], [-15, 16, 12, 1, 14, -12], [-14, -8, -5, -3, 10, 6], [30, 4, 2, 0, 1, -3], [50, 8, -10, -4, -8, -4], [23, 6, -3, 0, -11, -4], [17, 8, -9, 2, -2, -5], [10, -10, -11, -11, -7, -6], [19, -19, 10, -4, -2, -5], [12, -3, 4, 7, 9, -2], [21, -4, -13, 2, -8, 9], [23, -18, -10, -1, -25, 7], [24, 4, 8, 4, -8, 6], [24, 6, 7, -1, 6, 5], [-29, -16, 0, 7, -3, -5], [-9, -7, -3, 3, 6, 4], [16, 4, 2, 12, 8, -2], [-14, -10, -6, -6, 1, 6], [-7, -3, -7, 11, 10, -2], [-12, 14, 6, 10, 16, 4], [0, -6, -4, 6, 15, -14], [15, 10, -5, -6, -22, 1], [3, 5, 3, -7, 0, 7], [-20, 0, 2, -6, 9, 4], [-16, 7, -3, 2, 11, 5], [-10, 4, -10, 0, -10, 10], [-6, 3, -2, -5, 11, -2], [2, 22, 1, -5, 1, -2], [-2, 19, -3, 2, 11, -1], [-31, -2, 3, 4, 9, 12], [9, -6, 0, -3, 3, -3], [-16, 6, 6, 2, 14, -18], [12, -1, 8, -1, -1, 9], [2, 9, 4, 1, -11, 12], [-34, 2, 4, -10, 0, -3], [17, -11, 5, 21, 6, -10], [2, -12, -4, -10, 10, -2], [-1, -14, -11, -2, -14, 17], [-13, -4, 0, 1, -1, -9], [-8, 9, 8, 5, 13, -9], [1, -2, -1, -2, -16, 5], [21, 0, -4, 1, -12, 15], [-8, 1, -1, 4, -2, 12], [-33, 0, 15, -2, 14, -7], [29, 8, -7, 2, -12, 3], [-4, -9, 1, -4, 3, -1], [20, 10, -1, 3, -2, 7], [25, -15, -7, -3, -20, 3], [8, 0, -13, -7, -4, 12], [-4, 6, 10, 8, -2, 10], [26, -3, 13, -6, 1, -16], [-2, -10, 8, 4, -11, 4], [20, -12, -9, 1, -3, 6], [-19, 2, -5, 6, 18, -3], [35, -6, 5, 4, -14, -3], [15, -18, 6, 5, -13, -3], [42, -7, 0, -1, -23, 2], [24, 16, -3, -7, 7, 2], [-19, -6, -3, 2, -7, 8], [0, -7, -9, -8, -27, 7], [-1, -4, 2, -7, -9, -1], [8, -4, -9, -9, 9, -3], [6, -1, -10, -11, 6, 15], [-31, -2, 2, -11, 21, -2], [14, -8, -13, 3, -14, -4], [-2, 14, 13, -7, 11, 3], [-46, 10, 1, -19, 13, 0], [7, -20, 0, 13, -3, -5], [16, 0, -1, 5, 7, -22], [23, -8, 0, 9, -14, 1], [2, -7, -17, 4, -15, 8], [-2, -1, 0, 3, -3, 10], [7, 7, 15, 11, 14, -2], [22, -1, -2, -3, 9, 7], [20, 26, -3, -7, 4, 12], [10, 24, -8, -10, -2, 7], [-27, 9, 10, 4, -3, -6], [-11, 8, 6, -1, 24, 3], [38, -9, -9, -10, -17, 2], [13, 2, -2, 1, 14, -6], [22, 0, 7, 11, -8, 0], [11, -11, 6, 0, 0, -18], [26, 0, 11, -3, 8, -15], [-5, 8, 12, -5, 9, 7], [30, -10, 3, 7, -7, -6], [6, -12, 5, 13, 6, -9], [-7, 12, -4, -9, -6, 17], [37, 10, -14, 5, -27, -1], [-1, -4, -6, -7, 9, -13], [28, -4, -1, 11, -1, -5], [-11, 5, 14, -6, 10, -9], [10, 15, -7, -10, 5, 7], [-42, -11, -3, 0, -13, 3], [30, -13, 9, -2, -1, 3], [-6, 19, -14, -7, -11, 12], [-16, 2, 5, -9, 13, -7], [0, -18, -5, 11, 6, -4], [-15, -2, 4, 9, -11, -5], [41, -3, 4, 2, -15, 13], [7, -11, 21, 7, 8, -11], [-12, -6, -8, 4, -2, 1], [0, 7, 6, -1, 14, -11], [-24, 1, -4, -9, 9, 0], [-28, -6, 2, -10, 0, 3], [2, 17, 4, -5, 9, 11], [51, 4, -14, 5, -16, 5], [-26, 4, 14, 2, 4, -4], [8, -4, -19, -3, 6, 4], [44, -17, 1, 8, -21, 3], [-27, -10, 2, 1, 11, 11], [8, -1, 17, 8, -6, -16], [-16, 5, 13, 8, 3, 12], [11, -1, -3, -3, -12, 3], [-39, 6, 7, 6, 2, -3], [12, -3, 11, 4, -1, -9], [-19, -20, 0, 3, -8, -5], [-27, -1, 4, 4, 5, 5], [-13, 10, -11, -6, 2, 5], [-6, -10, -2, 2, -7, 11], [26, -23, -2, 9, 14, -9], [-36, 0, -3, 7, -6, -4], [-5, -6, -8, 5, 5, -5], [-6, -14, -9, -7, -10, 22], [-28, 3, 0, -11, 5, 11], [31, 9, 3, 7, -2, 8], [-36, 18, 6, 10, 24, -7], [15, -13, 6, 0, -17, -2], [16, 14, 16, 2, 32, -10], [32, -10, -15, 7, -14, -3], [33, 2, -2, -11, -8, 1], [-10, -1, 6, -1, 3, -10], [-14, 14, -2, -6, 9, 10], [10, -10, -9, 5, -10, -10], [-23, 7, 8, -4, -5, -8], [33, 8, -8, 9, -10, -6], [49, 6, 0, -13, -7, -3], [-15, 3, -5, -11, -8, 21], [-10, -8, -12, 8, -6, -6], [14, 6, 5, -11, 7, 14], [-25, 8, 23, 0, 12, 1], [-20, 5, 19, 8, 5, -7], [-3, 17, -6, -8, 1, -5], [-54, 6, 10, 2, -5, 4], [-40, 23, 15, -8, 5, -5], [-24, 0, -9, -17, -11, 6], [8, 7, -3, 4, -11, 9], [-12, -11, -8, 11, -13, -13], [-2, 2, 6, 12, 2, 6], [-5, 12, -4, -13, 10, -14], [-19, 8, 3, -12, -2, -9], [10, 9, 7, 10, -1, -5], [-8, 12, 11, 3, 21, -16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4002_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4002_2_a_bf();".
// 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_4002_2_a_bf(:prec:=6)
    chi := MakeCharacter_4002_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(2999) - 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_4002_2_a_bf();".
// 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_4002_2_a_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4002_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, -1, 1]>,<7,R![-208, -124, 128, 40, -23, -2, 1]>,<11,R![32, -48, -48, 38, 6, -7, 1]>],Snew);
    return Vf;
end function;