
// Make newform 4018.2.a.bn in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4018_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_4018_2_a_bn();".
// 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_4018_2_a_bn();".
// 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 := [1, -18, 24, 9, -10, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0], [0, -5, 0, 1, 0, 0], [3, -5, -6, 1, 1, 0], [-2, 8, -5, -7, 1, 1]];
    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_4018_a();"
function MakeCharacter_4018_a()
    N := 4018;
    order := 1;
    char_gens := [493, 785];
    v := [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_4018_a_Hecke(Kf)
    return MakeCharacter_4018_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], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [-1, -1, -1, 0, 1, 0], [0, 0, 0, 1, 0, -1], [0, 0, 0, 1, 1, -2], [3, 0, -1, 0, -1, 0], [1, 1, 1, -1, 0, 0], [0, 0, 0, 1, 1, 1], [1, 0, 3, -1, 0, 0], [-1, 0, 0, 0, 0, 0], [3, -2, 0, 0, 0, 2], [3, -2, -1, 1, 1, 0], [3, -2, -1, 3, 1, 1], [-4, 2, -3, -1, 0, 2], [4, 0, 0, 2, 0, 1], [-1, 0, 3, 2, -2, -2], [2, -5, 0, 4, 2, -4], [1, 2, 2, 1, 0, 1], [2, 2, -2, 3, 0, -1], [-2, 2, 1, -2, 0, 2], [4, -2, 2, 2, 1, 0], [-1, 2, 1, 0, 1, -3], [-4, -4, -4, 7, 0, -3], [3, 0, -1, 1, 1, -2], [6, 4, 1, -1, 1, -1], [-3, -3, 3, 1, 1, -3], [6, 4, -1, -3, 1, 1], [5, -2, 1, -2, 2, 3], [3, -6, -1, 3, 4, -3], [-2, 0, 2, -1, 0, -4], [3, -2, -1, 2, 1, 4], [8, -1, 2, 5, 0, -1], [-3, 5, -1, -2, -6, 2], [-2, -3, 4, 2, -1, -2], [-1, -4, 4, 0, 3, -4], [3, -5, -1, 6, -1, -5], [2, 4, -2, -1, 1, 1], [4, -6, -2, 2, -2, -1], [-2, 2, -4, 5, 0, -3], [4, 1, -2, -3, -2, -1], [8, 4, 0, 2, -5, 5], [7, 4, 3, -2, -4, 1], [0, -5, -4, -1, -1, 1], [-1, -4, -3, -1, 1, -3], [-2, -2, -2, 0, 4, -2], [3, 4, 1, -5, 3, 0], [-7, 7, 1, -2, 0, -2], [13, -4, -3, 4, 2, -1], [3, -4, 1, 8, 2, -3], [-3, -8, 2, 1, 2, 2], [3, -4, 0, 4, -1, -8], [-2, -4, 2, 0, -3, -3], [15, 5, 3, -7, 0, 0], [-3, 4, 1, -5, -7, -3], [-3, -2, 5, -3, -1, -6], [2, -4, -2, 6, -3, -1], [9, -4, 3, -1, -6, -6], [2, -2, -3, -9, 2, 2], [2, -3, -2, -3, -4, 5], [3, 2, 2, -10, 5, 5], [9, -1, 3, 2, 0, -3], [3, 0, -7, 1, -2, 1], [6, 2, 0, -2, -3, 4], [-5, -6, 1, -2, 8, -5], [-3, 2, 4, -9, -1, 9], [10, -4, 0, 9, 7, -6], [3, -4, 1, -4, 2, 2], [13, -2, -6, -1, 1, 6], [10, 8, 6, -4, -2, 1], [-3, 0, -7, -5, 2, 1], [-1, -4, -3, 12, 4, -7], [-3, 14, 4, -5, -5, 2], [9, -5, 5, -4, 2, 8], [6, -6, 4, 0, 8, -6], [-11, 4, -1, 1, 5, -3], [7, -2, 2, 5, -4, 0], [14, 0, -1, -4, 1, 2], [-7, -2, 0, -4, -2, -2], [-7, -10, -5, 0, 10, -1], [11, -2, 1, -1, -1, 10], [-6, -12, 0, 2, 4, -6], [-4, -5, 2, -5, -5, 4], [13, -2, 6, -2, -3, 1], [-5, -12, -6, 2, 6, -1], [2, -2, -6, 5, 3, -1], [-14, -2, 0, 5, 3, 3], [13, 3, 5, -1, -1, 0], [3, 6, 3, -8, -3, 2], [-7, -1, 11, -8, 2, -3], [3, 8, 1, 0, -2, 6], [-2, 0, 3, -8, 7, 6], [1, 14, 1, -2, -2, 0], [-10, 2, -4, 7, 6, -6], [-2, -5, -4, -2, 3, 4], [8, 0, 2, -6, 5, 2], [4, -4, 0, 3, 1, 5], [-8, 0, 0, 5, -2, 0], [3, -2, 3, -4, -3, -6], [18, 1, -2, 7, -1, -3], [-8, 8, -6, 0, -1, 3], [12, -2, 3, 12, 2, -2], [-7, -2, 1, 3, -10, -2], [19, 0, 3, -3, -9, 3], [0, -2, 2, -11, 3, 0], [-11, 0, -1, 9, -2, -14], [7, 4, 5, -14, -5, -2], [-6, 16, 2, -3, -5, 0], [-18, 6, 2, -2, -5, -2], [-9, -8, 5, -4, 4, -6], [0, -16, -3, -3, 2, -1], [9, -4, -2, -4, 6, 2], [-5, 12, -3, -4, -7, 3], [-8, 4, 0, 5, -15, 0], [-4, -4, -6, 9, -6, 0], [6, 0, -4, 8, -3, -11], [-6, 17, 8, 2, -3, 0], [-2, 2, 2, 4, -5, 2], [22, 0, 0, 0, 2, -6], [-14, -4, -4, -9, 4, -2], [-3, 12, -3, -5, -2, 8], [-4, 2, -8, -1, 2, 8], [9, 8, 1, 2, -6, 8], [-14, 2, 0, -11, -2, 8], [-17, 1, -3, 12, 1, 0], [-12, -1, 6, 1, -2, -5], [0, 1, 10, -9, 3, -3], [-7, 0, 3, -3, -7, 1], [-11, -6, -2, 1, 12, 4], [4, 10, 0, 5, 6, -3], [6, -3, -6, 7, 7, 9], [-10, 3, 8, 2, 1, -11], [-9, -6, -9, 13, 4, -10], [7, 10, 3, -14, -3, 9], [20, 0, 12, -6, -5, 8], [5, -4, 5, -2, -9, -1], [-13, 0, 3, -8, 2, 1], [5, -6, 3, 6, 6, -4], [6, -10, -7, -3, 1, 1], [-8, 2, 8, -15, 2, 9], [9, -2, -1, 3, 7, -1], [-16, 6, -10, 0, -5, -1], [-7, 8, -3, 7, 1, -10], [-8, -7, -2, 1, 18, -1], [-9, -2, -5, 9, 6, 0], [18, -14, -1, 0, 7, -5], [-4, -6, -7, 3, -1, -5], [8, -2, 0, -1, -4, 1], [-11, -8, 3, -14, -5, 2], [17, -10, -6, 4, 7, 3], [1, 0, -5, 4, 6, 4], [-2, -7, 10, -2, 6, -7], [4, 10, -3, 6, 6, 3], [-8, 2, -4, 0, -7, 0], [1, -9, -3, -10, 3, 3], [4, 6, 6, -16, -2, -7], [12, -10, -8, -4, -1, -4], [-17, 6, -3, -10, 1, 5], [2, 0, -3, 11, 12, 3], [-9, 2, -4, 6, -8, -7], [-9, 3, -5, 8, -12, -4], [19, -4, -3, 10, 6, -1], [0, 10, -2, 3, -8, 16], [-12, 2, 4, -8, 2, -5], [15, -9, 9, 1, -4, -6], [20, 1, -6, 5, -2, 3], [-6, -4, 7, -13, -1, 1], [-2, 14, -2, -4, -11, 13], [-7, 12, 1, 0, 5, -4], [12, 12, 0, 4, -7, -2], [-10, -4, 8, -12, 3, -8], [-17, -10, 2, 12, -1, -11], [-6, -10, 0, -5, 0, -6], [-1, -8, 3, -1, 4, -5], [-11, -2, -1, 9, -5, 3], [-25, 8, 1, -8, 1, 3], [4, -5, 2, 5, -12, -4], [3, -10, 9, 5, 1, -10], [-6, 0, 8, 9, -5, -5], [-11, -2, -4, 8, 6, -12], [3, 3, 3, -2, 15, 3], [10, -10, 9, -5, -7, -7], [-18, 0, 8, -7, -1, -2], [13, 0, 5, -2, 2, -6], [17, -4, -3, -7, 1, 12], [-9, -4, 3, 8, -7, -7], [-21, 10, -3, -8, 1, 13], [2, 8, 6, -12, -3, 7], [16, 2, 3, -2, 2, -5], [20, 4, 10, -10, -17, 1], [9, 6, 4, -4, -3, 5], [1, -14, -1, 2, 5, -3], [-3, 0, -10, -6, -6, 3], [16, -18, 1, 8, 8, 2], [-23, 4, 1, 12, -8, -3], [-20, 2, -2, -2, -13, 6], [15, 12, 3, -10, 1, -5], [16, 13, -6, 12, -5, 8], [16, 4, 2, -23, 7, 16], [0, -2, 10, 2, -1, 7], [-26, -11, 8, -4, 0, 5], [-2, -2, 2, 1, -3, -3], [-22, 6, 6, -6, -4, 8], [-16, -13, 2, -1, -1, 5], [-8, -8, -2, 0, -2, 7], [12, -8, 2, 13, -5, -6], [7, 0, 2, 12, -10, 1], [24, 18, 19, -23, -2, 4], [-4, 6, 0, 6, -12, -6], [-25, 11, -7, -10, -7, 5], [0, -2, 0, -5, 12, -8], [1, 0, 13, -10, 9, -7], [-5, 3, -15, 3, -8, 4], [2, -22, 1, 1, 8, -7], [7, 5, 5, -19, -2, -7], [-19, -14, 1, -7, 6, -2], [26, 5, 4, -16, 1, 2], [14, 6, 0, 3, -4, -1], [-1, 22, -1, 0, -16, 10], [31, 8, -3, 11, -1, -3], [-12, -2, 4, -24, -9, 10], [0, 13, 4, 5, -10, 3], [-9, 6, -8, 0, -8, 3], [13, 6, -7, 4, 6, 4], [0, -8, -5, 8, 1, 5], [24, 6, -2, 3, 0, 3], [-7, 8, -1, -1, -7, -2], [15, -6, 0, -17, 8, 5], [-4, 10, -2, -7, -10, 5], [8, 4, -6, -3, -15, 11], [13, -2, -1, 1, -5, 5], [-13, -6, 10, 1, 1, -5], [-28, 4, -2, 7, -17, 4], [-17, -5, -5, 12, 11, -10], [3, -6, -5, 10, 5, -2], [-2, -5, 6, -13, 7, -3], [4, 10, 5, -7, -4, 3], [6, -13, 10, 1, 6, 1], [4, 2, -14, -3, 0, 0], [-3, 8, 1, 0, 3, -8], [-3, 3, -1, 16, -7, 1], [-1, 14, -9, 2, -11, 0], [-8, 0, 10, -7, 5, -1], [-15, 10, -9, -8, -6, 9], [16, -14, 2, -11, -11, 6], [-32, 4, 2, 15, -2, -13], [19, 12, 6, -4, -7, -3], [16, 2, 4, -10, 4, -11], [2, 6, -2, 2, 15, -11], [10, 10, 1, -17, 2, 16], [5, 4, -1, 26, -6, -11], [-23, -16, -11, 20, 3, 0], [-16, 13, 14, -3, 2, 3], [-2, -4, 7, -14, 3, -13], [1, -1, -7, -3, 13, 13], [2, 8, -6, -1, 2, 7], [-36, 11, 4, 3, -6, -10], [-27, 12, 1, -21, -8, 5], [10, -6, 4, 19, 1, 7], [12, -10, -8, 0, 2, 2], [17, -3, 5, -4, -9, -7], [17, 3, 5, -5, -11, -6], [1, -12, -2, 1, 9, -9], [-23, 22, 0, -4, -3, -1], [-1, 9, -1, 15, -18, -7], [-11, 6, -10, -3, -12, 13], [17, 14, 14, -11, -1, -2], [2, 3, -10, 13, -10, -14], [-2, -1, 10, -2, -4, 13], [10, -22, -5, 12, 11, 4], [-12, 8, -12, -1, -5, 2], [14, 6, 2, -6, 1, 0], [12, -28, 2, 11, 10, -7], [9, 16, -1, 2, -13, 1], [-15, 18, 3, -8, 3, 8], [4, -9, -10, 20, 3, -9], [-11, -2, -7, 0, -4, 0], [6, -12, -10, -5, 13, 5], [15, 5, 9, 9, -8, 11], [-3, 4, -13, 3, 11, 8], [15, -8, -9, 13, -2, -9], [32, -10, 10, 19, 2, -10], [-10, -2, -8, 14, -13, -3], [8, 22, 0, 1, -4, 11], [-11, -12, 15, 1, -3, -8], [-5, -10, -5, -1, 16, -2], [-37, -10, 1, 5, 7, -5], [-2, 12, 2, -13, 5, 11], [5, 4, 9, -27, 1, 24], [-4, -12, 6, 11, -1, 6], [34, 8, 12, -12, -4, -6], [-11, 4, 10, -2, -1, 7], [10, 3, 12, -20, -9, 7], [14, -3, 2, -6, -11, 20], [11, 6, 9, -6, 8, -2], [11, -2, -7, 20, -3, 1], [20, 6, 4, -6, -18, 20], [7, -8, 5, -8, 2, -18], [-25, 2, 0, 2, 11, -2], [-8, 1, 2, 5, -8, 6], [24, -2, -2, 5, 8, 0], [38, 6, -4, -1, 1, 3], [-15, -6, 1, -4, 3, -16], [22, 18, -4, 5, -7, 6], [-21, 6, -1, 8, -15, 16], [47, -18, -1, 13, 2, -15], [-4, 17, -16, -7, -5, 26], [-5, 10, -7, -10, 10, 7], [19, -3, 1, 27, 9, -17], [8, -4, 10, -16, 15, 5], [6, 22, -6, 2, -2, 2], [25, 10, -2, -9, -9, 18], [19, -30, -7, 3, 8, 1], [19, 8, 4, 3, -18, 2], [-5, 24, -5, 1, -12, 0], [-17, -20, -1, 4, -6, -2], [22, 2, -4, -22, 14, 0], [-1, -2, 1, 2, 22, -8], [-19, -24, -12, 26, 9, -13], [-27, -4, 9, -4, -3, -2], [-30, -6, -6, -7, 0, -2], [35, 4, 11, -4, 2, 2], [27, 14, -8, 26, 8, -9], [-28, 4, -14, 11, 11, -9], [13, -24, -3, 4, -1, 10], [-11, 10, 7, 8, 2, -10], [-8, 17, -10, 8, -12, -11], [-27, 2, -19, 6, 2, -4], [-22, 17, 0, -10, -18, 15], [-22, -6, -4, 1, -4, 7], [38, 2, 2, 5, 4, 4], [-13, -4, -9, 13, 17, -6], [-21, 8, 1, -16, -3, -2], [-5, -16, 11, -5, 14, -10], [6, 17, 10, -15, -17, -3], [7, -6, 6, -1, -13, -4], [13, -14, -3, -17, 5, 3], [-34, 0, 0, -3, 12, 10], [8, -11, 0, 1, 9, -12], [-5, -10, 5, 9, 8, -21], [6, -12, 0, 2, 9, -23], [17, -13, -1, 9, -1, -13], [12, -18, -6, -13, 12, 10], [16, 8, -1, 7, -1, -4], [-27, 18, -9, -14, 2, 10], [16, 4, 6, -15, 7, -6], [52, 15, 0, 1, -17, 9], [24, 2, -10, 6, 0, 2], [-3, 0, -5, 23, -8, -6], [22, -16, 1, 9, 17, -11], [-24, 1, 4, -1, -5, 17], [27, -14, 1, 19, 18, -19], [2, 13, 8, -7, 2, -10], [10, -14, 0, 29, 0, -12], [3, 2, 13, -12, -3, -7], [7, 0, 18, -12, 1, 19], [27, -6, -2, -3, 10, 18], [-29, -6, -1, -1, 7, 2], [3, -10, -19, 6, -9, 12], [15, 4, 11, -3, 5, -11], [13, -14, 9, 10, -10, 1], [29, 8, 7, -11, -14, 15], [-27, 24, 5, -6, -3, 8], [-8, 4, 4, 9, 18, -9], [-19, 15, 11, 1, 0, 10], [21, 6, -7, -12, 6, 1], [20, 14, 8, -3, -13, 5], [3, 4, 13, -8, 8, 12], [-7, -10, 3, 16, -8, -13], [-12, -12, -4, 22, 8, -22], [-12, 6, -6, 3, -2, -3], [-9, 6, -5, 8, 1, 9], [-10, -17, 2, -10, 4, -1], [51, 10, 6, -3, -20, 10], [16, -10, 10, -1, 8, 1], [-29, -10, -9, 3, -8, 2], [-7, 8, -7, 8, 8, 14], [-34, -20, 6, -12, 6, 2], [-35, -12, -11, 17, 9, -16], [-22, -6, -13, 3, 1, -7], [5, -4, -11, 17, 3, -15], [3, 4, 11, -28, -7, -4], [-17, -2, -21, 7, 8, 9], [-9, 0, -13, 8, -3, 10], [-28, -9, -6, 0, 13, -21], [27, -6, 1, 10, -2, 13], [28, -20, 3, 17, 11, 1], [29, 0, 1, 8, 4, 0], [8, 6, -2, 25, -17, -13], [22, 21, 22, -7, -10, 5], [-2, -12, 6, -9, 3, -5], [-26, -4, 9, -7, 4, 0], [-8, 6, 2, 21, 2, -5], [30, -8, 0, 8, 0, 6], [-10, 1, -12, 5, -6, -3], [-6, -6, 4, 5, -15, -2], [-15, -2, -13, -6, -1, 5], [-42, -6, -8, 20, 5, -8], [15, -2, -7, -17, 23, 5], [3, 18, 5, -2, -13, -10], [34, 14, -6, -5, -3, 9], [-37, 8, -13, -9, -27, 19], [-8, -12, 2, -6, -9, -2], [2, 27, 12, -10, -11, 8], [-7, -26, -1, 0, 18, -8], [-7, 0, 9, -4, -6, 12], [12, -4, -4, 11, 20, -10], [46, -2, -2, -16, 0, 27], [14, 32, -9, -3, -10, -3], [6, 6, 6, -11, 17, -4], [4, 4, 12, -12, -4, -10], [3, -10, 9, -4, 5, -8], [-9, 16, 7, -5, -19, -8], [30, 16, 8, -4, 6, 2], [33, -22, 11, 14, 16, -9], [-4, -15, -4, -10, -15, 8], [17, 2, 13, 6, -6, -11], [7, 24, 3, -15, -17, 22], [-14, -10, -2, -14, 19, 5], [-17, -7, 7, 4, 8, 19], [12, 8, 16, 0, -10, 5], [30, -18, 4, 3, 12, 0], [-25, -4, 8, -15, -9, 18], [45, -8, -1, 1, -11, -12], [17, -18, 3, 5, -8, -3], [-9, 12, 11, 12, 8, -3], [38, -2, 4, -4, -19, -6], [-23, 1, 1, 12, 12, -22]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4018_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4018_2_a_bn();".
// 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_4018_2_a_bn(:prec:=6)
    chi := MakeCharacter_4018_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_4018_2_a_bn();".
// 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_4018_2_a_bn( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -4, 30, -5, -11, 1, 1]>,<5,R![-12, 12, 56, -1, -15, 0, 1]>,<11,R![-684, -48, 320, 7, -38, -1, 1]>],Snew);
    return Vf;
end function;