
// Make newform 4005.2.a.n in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4005_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_4005_2_a_n();".
// 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_4005_2_a_n();".
// 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, 7, 18, 2, -8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, -5, -1, 1, 0, 0], [-3, -1, 1, 0, 0, 0], [3, 5, -4, -2, 1, 0], [0, 10, 7, -6, -2, 1], [-3, 18, 14, -11, -4, 2]];
    Rf_basisdens := [1, 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_4005_a();"
function MakeCharacter_4005_a()
    N := 4005;
    order := 1;
    char_gens := [3116, 802, 181];
    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_4005_a_Hecke(Kf)
    return MakeCharacter_4005_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, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [3, 1, -1, 1, -1, 1], [2, 0, 0, 0, 0, 0], [3, 0, -1, 1, -1, 0], [0, 1, 0, 0, 0, -1], [2, 0, 0, 0, 2, 0], [-1, -1, 1, -1, 1, -1], [-1, -2, 1, -1, 2, -2], [-1, -1, 1, -1, 1, 1], [2, 0, 0, -2, 1, 0], [4, 0, -2, -2, -1, 2], [1, 1, -1, 1, -1, 0], [2, 0, 0, 0, -2, 2], [-1, 2, 1, 1, 1, -2], [2, 0, -2, -2, -2, 2], [-1, -4, 1, 3, 3, -2], [1, 2, 1, -1, 0, 0], [3, 0, -1, 1, 1, 0], [1, 0, 0, 0, 0, 0], [-5, 2, -3, -1, 1, 0], [4, 1, 2, -2, 0, 0], [0, 1, 0, 0, -2, 4], [1, -3, 5, -3, 5, -3], [-1, 1, -3, -1, -5, 2], [3, -2, -1, -3, 3, 0], [-1, -1, -7, -1, 1, 1], [2, 4, -2, -2, -4, 4], [10, -2, 2, 2, 0, 0], [1, -3, -3, -1, 3, 0], [-1, -2, 1, 3, 1, 0], [0, 0, -6, -2, 0, 0], [-5, -4, -7, -1, 1, 0], [-2, -1, 2, 2, 0, 0], [2, 3, 0, -2, -4, 0], [2, -3, 0, 0, 2, 0], [0, 0, -4, 0, 0, 4], [-6, -2, -4, -4, 0, -2], [1, -1, 7, 1, 3, -4], [2, 2, 2, 2, -1, 2], [6, 4, -6, -2, -2, 4], [2, 1, 0, 0, 0, -3], [-2, 2, -4, -4, 0, 2], [-1, 4, 1, -3, 3, -2], [2, -4, 4, 2, 3, -4], [4, 2, 6, 2, -4, 0], [2, -1, 0, 4, -2, 0], [-3, 0, -5, 1, 2, -4], [-10, 0, -4, -4, 2, -2], [0, -6, 4, 4, 2, -2], [-2, 0, -4, -4, 1, -2], [1, 1, -7, -7, 1, 1], [2, -2, 4, 4, -4, 0], [0, -4, 8, 0, -1, -4], [-3, -2, -5, -3, 3, 2], [13, 1, 1, 3, -3, -3], [10, 0, 0, 6, -4, 0], [-1, 2, -5, 1, -1, -2], [5, -2, -5, -1, -3, 0], [0, 0, 0, 0, 4, 2], [5, 3, 7, 1, -5, -1], [9, 1, -3, -1, -3, 1], [14, 3, -2, 2, 4, 0], [-6, 6, -2, -2, -1, -2], [2, 2, -4, 2, 2, -2], [2, -4, 8, 0, 4, -6], [-3, -4, 5, -5, 7, -6], [4, 6, -2, 2, -5, 4], [4, 0, -6, -4, 2, 2], [-1, 4, -3, 3, -5, 6], [-2, 4, -8, 0, -8, 4], [6, 4, 2, 6, -6, 2], [4, -4, -6, 2, -1, 2], [0, -2, 0, 0, 7, -2], [4, 0, 2, -2, -2, 4], [-17, -4, -7, -1, 0, 0], [-2, -7, -6, 2, 2, 1], [-6, 2, -4, -4, 2, -4], [5, 1, 9, -1, 5, 3], [2, 3, 6, -2, 2, 0], [0, 4, -10, 2, -2, 2], [2, 5, -8, 2, 0, 0], [4, 4, -6, 6, -8, 2], [-10, 0, 6, 6, -4, 2], [2, 4, 0, 0, 0, 4], [-4, -12, 6, 4, 12, -10], [0, 3, -10, -4, 2, 4], [-6, -4, -2, -2, 6, 0], [-5, 0, -3, -7, -3, 2], [3, 2, -11, -5, 3, 6], [-2, -6, 16, 0, 8, -4], [0, -6, 8, 0, 2, -4], [-10, 2, 0, 0, 0, -6], [0, 0, -2, -2, 7, 2], [10, 2, -4, -2, 2, 4], [6, 4, 4, 4, 2, 0], [0, -5, -8, 4, -2, -4], [5, 6, -3, 3, -5, 0], [7, -6, 7, 1, 5, 0], [-3, -5, -13, -7, 5, 8], [12, -2, 2, -2, 8, 0], [-12, 2, 4, -4, 3, 2], [6, -2, 4, 6, -2, 2], [5, -8, 1, -1, 11, -2], [4, -2, -6, 6, -7, 4], [-20, 1, 4, -4, 2, 1], [7, -2, 5, 1, -1, -8], [-7, -2, 3, 5, 1, -2], [-1, 0, 3, -3, 9, -2], [-3, 6, -11, 7, -4, 0], [-1, 3, 3, 5, 1, 0], [6, 2, 12, -4, 4, -2], [17, -2, 7, 7, -3, -2], [5, 0, 13, -1, -3, -4], [-3, -2, -3, -5, -3, 0], [-6, 6, 10, -2, 4, 0], [4, -4, 2, 6, -2, 2], [-5, 2, -7, -5, 7, 6], [0, -2, 0, -4, 8, -6], [5, -4, 9, -5, -1, 6], [11, -7, 7, 5, 5, -8], [-4, -6, 18, 6, 2, -8], [-14, 4, -8, 0, -6, -2], [-3, 5, -7, -1, 1, 3], [11, -3, 17, 3, -3, -5], [1, 0, 7, 1, -7, 0], [-6, -4, 0, 0, -6, 2], [-15, 2, -7, 3, 3, -4], [1, 1, -5, 5, 3, 3], [-3, 0, -5, 5, -7, -2], [-4, 8, -2, 2, -14, 6], [-4, -5, 4, -4, -6, -1], [3, 6, -5, 1, -11, 6], [0, -4, -8, -8, -5, 8], [-8, -1, -6, 6, 4, -8], [20, 2, 2, 6, 2, 0], [22, 7, 0, -4, -8, 7], [-12, 6, -2, -6, -4, 4], [-7, 2, 7, -5, 5, -10], [-9, -12, 3, 5, 13, -12], [14, -2, 4, 12, 0, -4], [-5, 0, 9, 3, 8, -12], [13, 3, 11, 9, -1, -1], [24, 2, 12, 4, 4, 6], [-2, 6, 8, 8, -12, 4], [-6, 8, -2, -2, -10, -2], [0, 4, 4, 0, 2, 10], [-8, 0, 6, 2, 2, 0], [-6, 0, -2, -6, -7, 0], [9, 10, -7, 3, -15, 8], [11, 0, -7, 1, 5, -8], [-8, -2, -8, -4, 6, 4], [-6, -2, -4, 12, -6, 0], [8, -4, 2, -10, -2, 4], [-9, -8, -11, -9, 5, 10], [-30, -9, -6, -2, 6, -3], [-2, 5, -12, -2, -8, 8], [13, 8, -7, -1, -7, 10], [2, 1, -2, -6, -8, 4], [8, -2, 12, 0, -4, 2], [-8, 0, -14, -2, -5, 2], [-5, -9, -1, 3, 3, -9], [16, -8, 14, 10, 0, 0], [11, 12, -15, 7, -15, 6], [-16, -8, 2, 6, 10, -4], [3, -5, 3, 9, -1, -1], [0, 0, 12, -8, 8, -6], [9, 0, -5, -11, -4, 8], [6, 1, 0, 12, 4, -4], [6, -4, 6, -10, 4, -2], [-10, 4, -8, -16, 6, -2], [-5, 6, 9, 3, 5, -12], [20, -4, -4, -8, 3, 8], [1, -3, 17, -5, 9, 1], [3, -8, 1, -7, 5, 2], [-6, -2, -16, -4, 3, 0], [10, 11, -6, -2, -12, 8], [-7, 2, -9, 9, -15, 10], [6, 2, 8, -4, 2, 4], [3, 2, 7, -3, -3, 0], [-21, -10, 11, 1, 9, -12], [5, 5, -9, 5, -7, 8], [-4, 0, -2, 6, 4, 2], [25, 8, -1, 3, 3, 4], [6, -2, 16, 0, -4, 2], [-6, -7, 2, 2, 2, 1], [3, -2, 5, -1, 13, 0], [5, -2, -3, 7, 5, -10], [24, 4, 6, 10, 0, 0], [-11, -10, 7, -7, -2, 2], [-5, -2, -1, -7, -3, 8], [8, 4, 14, -2, 6, 4], [-15, -12, 3, -3, 6, -12], [12, 0, -16, 0, -10, 6], [17, 4, 9, 3, 13, -2], [16, -8, 12, 0, 4, 0], [-8, -2, 10, -2, -6, 6], [16, 0, 6, 2, 0, -2], [10, 4, -4, -8, -8, 0], [-2, -6, -8, 0, -8, 0], [-4, -3, -4, -8, 4, 5], [-11, -4, 5, -1, -5, 2], [6, 4, 16, 16, -6, 0], [1, 2, -7, 15, -13, 4], [17, -3, 5, -1, 13, -7], [31, 3, -7, -5, -3, 5], [8, 2, -12, 0, 7, 6], [-32, 8, -10, 2, -12, 6], [13, -8, 3, 7, 5, -10], [-3, -2, 13, 3, 9, -12], [2, 6, -6, 6, -14, 8], [6, -2, 12, 4, -6, -6], [3, -2, -7, 9, -13, 2], [15, 9, -3, 7, -9, 0], [9, -1, 21, -3, 9, -9], [-3, 0, 9, -5, 11, -2], [-4, 12, -10, 2, -16, 12], [13, 10, 9, -5, 1, 4], [13, 1, -9, 5, 5, -4], [24, 10, 10, 0, -4, -2], [20, -2, 12, -8, -4, 4], [6, -6, 12, 8, 3, 0], [16, 8, 14, -6, -1, -6], [14, 16, 12, 0, -10, -2], [7, -7, -7, -5, -7, 11], [-2, -3, -4, 8, 0, 8], [-2, 6, -2, -6, -5, -2], [6, -2, 4, 12, 4, 2], [-24, -1, 6, -6, 6, -17], [20, -10, 14, 10, 2, -10], [34, -4, 2, 2, 1, 8], [9, 2, 17, 3, -3, 10], [18, 4, 12, 4, -2, 14], [18, -7, 8, 0, 6, -8], [6, -5, -12, 0, -8, 0], [-20, 10, 4, -4, -10, 6], [0, -5, 10, -4, -2, 8], [3, 10, -5, 5, -1, -4], [5, 7, -13, 5, -13, 13], [16, 13, -4, -4, -10, 17], [1, 5, 5, 11, 5, 1], [-3, -2, -17, -15, 5, -6], [11, 9, -7, -1, -7, 7], [2, 8, 0, 12, 2, -10], [-28, -2, 6, -6, 6, -4], [23, 4, 23, 1, 1, 8], [-12, -6, 10, 2, 12, -14], [12, 2, -6, 14, -10, 2], [28, 3, -10, -4, 2, 0], [12, 12, 14, 2, 0, 2], [-20, -4, -6, -14, 1, 2], [4, -2, 0, 12, -4, -2], [-24, -10, 18, -2, 14, -14], [29, 2, 19, -5, 3, -6], [-13, 4, 7, -7, 3, 4], [5, 6, 15, -3, 1, -2], [12, -3, 12, -4, -2, 4], [24, 12, 0, -4, -2, 0], [5, -8, 3, 5, 2, -4], [10, 2, 20, 0, -6, -8], [-29, -6, -9, -3, 5, 2], [-15, 2, 9, 11, -2, -4], [-6, -2, 0, 8, -4, -6], [22, -7, 18, -2, 10, -9], [-31, -4, -9, -9, -7, -6], [8, 1, 10, 14, 4, 4], [-9, -15, 1, 11, 19, -14], [-24, -8, -16, 4, 11, 0], [32, 14, 8, 8, -11, 6], [-16, 1, -10, -14, -6, 9], [-27, -9, 3, 5, 3, -5], [-18, -4, -14, -10, 2, -4], [10, 14, -10, 10, -10, -2], [-1, 7, 1, -5, 11, -10], [-5, -17, -3, 7, 15, -10], [1, -8, 3, -7, 14, -20], [19, 10, -17, -11, -7, 8], [28, -12, 18, -2, 14, -2], [18, 12, -8, -2, -5, 8], [-23, -1, -13, -7, -3, -6], [4, -12, 22, 0, 6, -2], [-12, 2, -16, 4, 2, -12], [7, -4, 9, 11, 9, 0], [13, -3, 9, 3, -11, -11], [1, -6, -7, 15, -10, 8], [-4, 0, -14, -2, 0, 10], [30, -6, 4, 4, -8, -4], [3, 0, -1, 1, -17, 0], [-14, -4, 12, 8, 8, -4], [-20, -2, -14, -10, 1, 0], [-1, 2, -3, -13, -13, 4], [-36, -2, -18, 2, -2, 2], [-18, 0, 4, -4, 2, 2], [9, -8, 9, 11, -5, -4], [22, 2, 6, -2, -4, 8], [1, -3, 15, -11, 15, -1], [-4, -14, -4, 12, 7, -2], [-18, -1, -4, -12, 10, -12], [-24, -10, 2, 6, -6, 2], [-7, 1, 17, 5, -3, -19], [-33, -2, -9, 1, -5, 0], [-27, 7, -9, -3, -5, 11], [5, -3, 5, -7, 5, -11], [7, -7, 15, 5, 25, -1], [-10, 12, -14, -2, -18, 2], [-4, -13, 16, -4, 20, -12], [12, -17, 18, 20, 2, -8], [-18, -6, -16, -16, 6, 6], [14, 2, -16, -8, -20, 10], [7, -2, -3, 1, -13, 2], [2, -4, -4, 6, 9, 8], [12, -3, 4, -4, 18, -15], [27, -1, -25, -7, -9, 13], [-26, 0, -2, -6, 8, -6], [18, -10, -10, -10, 23, -2], [23, 6, -3, -21, -4, 6], [20, 1, 2, -2, -6, 9], [-47, -12, 19, 1, 13, -12], [-26, -8, -12, -12, 4, 4], [16, 8, -4, 0, 4, 8], [-6, 4, -8, 0, -13, -6], [3, 11, -11, -1, -17, 10], [-38, -12, -12, -4, 4, 2], [-26, -12, 24, 2, 4, -12], [-8, -8, -6, -8, 11, -4], [-7, -2, -3, -5, 5, -12], [-7, -4, 11, -3, 2, 8], [-18, 6, -10, -14, 10, 0], [18, 2, 0, 16, -12, 4], [24, 5, 14, 12, 2, 0], [24, 2, -2, 2, 4, -8], [2, 6, 24, 10, -14, -10], [6, 3, -26, -14, 4, -1], [30, 8, -22, 6, -18, 16], [-8, 4, 10, -2, 6, -8], [10, 0, 14, -14, 2, -12], [20, 8, 14, 6, -12, 10], [5, -1, -15, -1, 13, 7], [-22, 4, -28, -4, -2, -2], [-42, -2, -16, 0, 2, -2], [18, 10, 26, 2, -6, -4], [10, 0, 22, 2, 14, -8], [30, 0, 24, -8, 0, 0], [20, -3, 0, 16, 2, 0], [-39, -12, 3, 1, 10, -4], [-3, -4, -35, -17, 11, 0], [-6, -3, 12, 4, 16, -7], [-4, -6, 0, -8, -1, -2], [-18, -4, -8, 0, -14, 6], [4, -8, -10, -4, 6, -10], [-10, 12, -20, -28, 6, 2], [17, -9, 5, -5, 13, 7], [14, 10, -14, 2, -6, 6], [8, 6, -4, -4, 2, 2], [14, -10, -10, 2, 6, 8], [14, -10, 16, -6, 6, -14], [-4, 10, -2, -2, 4, -10], [24, 8, -2, 6, -13, 2], [-17, -3, -9, 3, 7, 5], [45, -4, 15, 5, -2, 0], [12, 8, 18, 24, -10, -6], [22, -4, -26, 2, -16, -2], [-11, 4, -3, 3, -19, -2], [24, 6, 6, 0, -2, 10], [-22, 6, 0, 0, 2, 6], [-20, 14, -4, -8, -26, 8], [3, 2, 7, 5, -20, -4], [-32, -18, 8, 4, 4, -8], [-8, 4, 6, 2, -14, 2], [-18, 6, 4, -4, 0, 2], [-10, -3, 14, 2, -2, 8], [-6, -12, 2, 10, 12, -14], [-52, 4, -8, 8, -14, 10], [-9, 1, 13, 23, -7, 5], [-15, -21, 15, 5, 21, -16], [7, -8, 17, -5, 7, 4], [-50, -2, 16, 8, 0, -2], [-1, 4, -13, -15, -3, 14], [6, 4, -2, 10, -18, 8], [46, 6, 10, 18, -4, 8], [31, 11, -27, 3, -23, 17], [39, 0, 23, -3, -9, -6], [15, 10, 7, -7, 2, 4], [-2, -6, 6, -14, 19, -2], [-7, 1, 35, 5, 11, -13], [-28, -18, 2, -4, 16, -18], [28, 2, -10, 10, -14, 10], [13, -6, -11, 3, -1, 6], [-1, -4, -27, -9, 6, -4], [16, 9, -8, -20, -14, 8], [-10, -13, 6, -2, -2, -1], [-6, 7, -10, -14, -6, 20], [-17, -15, -13, -3, 13, -8], [-27, 5, 7, -3, -15, 12], [-4, -8, 18, 6, 12, -8], [-2, 4, -12, 12, 2, -12], [5, -13, 13, -1, 23, 1], [29, 2, 5, 23, -13, -6], [-30, 6, -8, 10, -20, 12], [-20, 11, -2, -10, -4, 4], [12, -5, 18, 14, 0, 0], [-3, -7, 25, 7, -13, -4], [-10, -4, 2, -18, -4, 14], [4, -4, -2, -6, 20, -14], [-8, -8, -10, -4, 4, 2], [35, 8, -11, -1, -4, 8], [-28, -2, 4, 12, 11, -2], [-7, 7, -15, 11, 5, 11], [32, 0, -18, 4, -9, 12], [6, 14, 12, 4, 0, -14], [-19, -4, 11, 9, 11, -12], [-23, 3, 7, 5, -19, 8], [-10, 1, 8, 4, 8, 4], [-24, -20, 22, -8, 26, -10], [-57, -9, -9, -7, 7, -1], [-12, 18, -14, -4, -20, 22], [-6, -11, -2, -6, 10, -1], [-2, -14, 0, 4, 2, -12], [1, 7, -3, -19, -3, 3], [-22, -12, -14, -14, 8, 14], [17, -6, 19, 1, 5, -2], [-31, 1, 1, -5, 1, -11], [-11, 2, 13, -1, 5, 8], [12, 8, 2, -18, -2, 4], [-41, -5, 5, 11, 13, -11], [-11, -5, -21, 5, -5, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4005_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4005_2_a_n();".
// 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_4005_2_a_n(:prec:=6)
    chi := MakeCharacter_4005_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_4005_2_a_n();".
// 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_4005_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4005_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, -11, -13, 21, -2, -4, 1]>,<7,R![-128, -16, 96, 7, -20, -1, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;