
// Make newform 4018.2.a.br in Magma, downloaded from the LMFDB on 20 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_br();".
// 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_br();".
// 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 := [14, -24, -104, 260, -69, -174, 75, 36, -17, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-330, -18, 2442, -1896, -1610, 1201, 330, -234, -18, 13], [2842, 230, -20714, 15490, 14188, -10109, -3005, 2023, 167, -114], [-2996, -358, 21804, -16220, -14673, 10513, 3064, -2086, -169, 117], [594, 52, -4344, 3266, 2975, -2130, -630, 426, 35, -24], [6230, 614, -45404, 34004, 30938, -22157, -6528, 4424, 362, -249], [6496, 356, -47216, 35916, 32206, -23290, -6799, 4641, 377, -261], [6832, 558, -49632, 37378, 33605, -24201, -7053, 4809, 390, -270], [-1018, -84, 7416, -5552, -5087, 3628, 1079, -727, -60, 41], [-8050, -474, 58592, -44840, -39401, 28835, 8222, -5698, -453, 319]];
    Rf_basisdens := [1, 2, 14, 14, 2, 14, 14, 14, 2, 14];
    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, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 1, 0, 0, 1, 0], [-1, 1, 1, 1, 0, -1, -1, 1, -1, 0], [-3, 1, 1, 0, -1, 0, 0, 2, 1, 0], [0, -1, 0, 0, -1, -1, -1, 0, -1, -1], [1, 0, -1, 0, 0, 1, 1, -1, 1, 1], [1, -2, -2, 0, 2, 1, 1, -1, -1, 1], [1, -1, 0, 0, -1, 1, 2, 0, 0, 1], [-3, 0, 1, -1, -1, 0, 0, 1, 0, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -2, 1, 0, 3, 0, 0, -1, 0, 0], [-1, -2, -1, -1, 0, 1, 3, -4, 2, 1], [-2, 2, 1, -2, -3, -1, 1, 1, 2, 0], [-2, 3, 2, -1, 0, 0, -1, 2, 2, -1], [-1, 1, 0, 0, 1, -2, -2, -1, 0, 0], [2, -1, 0, 1, -1, 1, 0, 1, -2, -1], [1, 0, 0, 1, 4, 0, 1, -1, 0, 1], [1, 1, 3, -2, -1, -2, 0, 1, -2, -2], [2, 0, 1, 2, -2, -1, -4, 1, -4, -1], [-6, 0, 2, 0, 0, 1, -2, 1, 0, -1], [-3, 0, 1, -2, 2, -1, -3, 0, -2, -3], [3, -5, -3, -2, -2, 3, 5, -3, 1, 0], [0, 2, 2, 0, 1, -1, -1, 1, -2, -2], [-6, 4, 1, 0, 0, -1, -2, 3, 2, -1], [1, -2, 1, 0, -1, 0, 2, -5, 0, 2], [3, -1, -3, -2, 3, 2, 1, -3, 0, 1], [-2, 1, -2, 0, 4, 0, -4, 4, -2, -1], [4, -2, -2, 1, 1, -1, 1, -4, -1, 2], [-3, -1, -1, 0, 3, 1, 2, -4, -2, 1], [-1, 4, -1, 1, 0, -3, -3, 2, -2, 0], [2, 0, 0, 0, -1, 1, 2, 1, 2, -3], [-5, 4, 2, -2, 0, -3, -5, 3, 1, -1], [2, 2, 1, -3, 3, -2, 1, -3, 3, 0], [-2, 4, 4, -4, -3, -3, 3, -1, 4, 0], [-4, 4, 0, 2, -2, 2, 0, 2, 4, 0], [-14, 1, 2, 0, 0, -2, 0, 1, 1, 3], [-12, -1, 3, -3, -1, -3, 1, 1, 2, 0], [-2, -2, -2, 4, -1, 2, -3, 0, 0, 1], [3, 2, 1, 3, 2, 3, -1, 0, 2, -1], [6, -6, -7, 0, -2, 7, 2, -3, 0, 1], [-8, 1, 0, 0, -1, -2, -2, 3, 1, 1], [6, 0, -2, -2, 0, 2, 4, 0, 0, -2], [0, -8, -4, 4, 1, 3, 1, -1, -2, 2], [-4, -5, 0, -4, -4, 0, 3, 1, -1, 2], [-5, -2, 0, -3, 0, -2, 1, 1, 0, -1], [-4, -1, 0, 3, 4, 0, 0, -5, -3, 3], [3, 4, 1, -3, 4, -1, 3, 0, 2, -1], [4, 3, 2, -1, 2, 1, 1, 1, -2, -2], [8, 0, 0, 3, 1, -3, 1, -4, -3, -1], [-8, -1, 0, -1, 0, 3, -1, 5, 4, 2], [-5, 2, 1, -3, 2, 4, 3, -1, 2, 1], [-1, -2, -5, 2, -2, 1, -1, 0, -2, -1], [1, 0, 1, 1, -2, -3, -3, 0, 0, -4], [-2, 0, -2, -2, 1, -1, 6, -5, 0, 1], [-2, 3, 2, 7, 7, -1, -7, 1, -2, 2], [0, 0, 2, 0, -2, 0, 4, 2, 2, -2], [-5, 2, 3, 3, 2, -3, -1, -2, 2, 0], [3, -4, -1, -1, -1, 0, 3, -3, -2, -3], [2, -2, -6, 2, 1, -1, 3, -5, -2, 0], [-1, -4, -3, -1, 3, 0, 5, -9, -4, 3], [-4, 2, -2, 0, -1, 3, -1, 1, 4, 2], [-4, 4, 0, -2, 1, -3, -5, 3, -6, 0], [0, -3, -2, 2, 4, 0, 1, -1, 3, 4], [2, -4, 2, 0, -3, 2, 5, -4, 0, -1], [5, 1, 1, 12, -6, 1, -3, 1, -1, 0], [4, 1, 4, 0, 2, 2, -1, 1, 1, -6], [10, -8, -8, -2, -3, 3, 4, -5, 0, 1], [-8, 6, 4, 4, 4, -4, -4, 0, -6, -2], [4, -1, -2, 4, 1, -3, 1, -2, -1, -2], [-9, -1, 2, 0, -1, 5, 2, 4, 2, -3], [-2, -5, 2, -4, 3, 5, 3, 2, 5, 0], [12, -5, -4, 3, -5, 3, 1, 2, -3, -1], [-15, 1, 3, 5, -9, 0, 0, 2, -1, 0], [-2, -2, 0, 6, -4, 2, -2, 0, -2, 2], [9, -2, -3, -1, -4, 1, 3, -6, -2, 5], [-6, 1, 2, 0, 4, -2, -6, 0, 0, -3], [3, 5, 3, 4, -4, 2, 2, 1, 2, -2], [-4, 7, 2, 5, 5, -4, -7, 2, -2, -3], [-5, 12, 4, 2, 6, -3, -13, 7, -3, -3], [-7, -6, -1, 0, -8, -1, 3, 1, -3, 3], [-1, 2, 1, -1, 0, -1, -5, 2, 6, -2], [-8, 4, 6, -6, 7, -3, -3, 5, -2, -2], [-1, 0, 5, -4, -5, -2, 2, 3, 2, -2], [2, -2, -4, 1, 2, -2, -2, -2, 0, 1], [2, -1, -2, 2, 3, -4, 0, -1, 1, -1], [-5, 2, 2, 3, -2, -5, -1, 0, -2, 4], [-1, 0, 3, 5, -2, 1, -1, 0, 2, 0], [-7, 3, 1, 6, -6, -3, 0, 2, 2, -3], [-11, 4, 1, -5, 2, -3, -5, 6, 0, -7], [-6, -3, -2, 4, -1, 1, 3, 0, 3, 6], [1, 6, 1, 2, 9, -6, -10, 1, -4, 0], [7, -11, -5, -2, -1, 4, 9, -3, 0, 3], [-8, -4, -6, 4, 6, 6, 0, -2, 0, 0], [0, 0, 4, 6, 7, -1, 9, -5, 0, 2], [-4, 0, 2, -2, 1, -1, -1, 3, -6, 4], [-17, 7, 9, 0, -6, -7, 0, 4, 2, -3], [-8, 14, 4, -6, 6, -4, 0, 4, 4, 2], [5, -11, -7, 1, -6, 7, 10, -7, 1, 0], [-14, 10, 5, 2, -1, -5, -5, 1, 2, -2], [-10, -2, 2, -4, -3, 1, -1, 5, 6, -2], [-4, -3, -2, -2, 14, 0, 4, -6, 0, 5], [-5, 3, 7, 0, -5, -2, -3, 3, 0, 3], [-10, 4, 8, -4, -5, 5, 1, 7, 2, -2], [14, -3, -6, 7, 6, 2, 0, -5, -7, -3], [-9, -3, 3, -4, 2, -1, 7, -9, 5, 4], [-16, 5, 0, 2, -1, -1, 1, 0, 7, 6], [9, -3, 1, 2, 7, 6, 6, -8, 5, 6], [2, 5, 5, 3, 5, -12, -7, -4, -4, -3], [6, -1, -4, 0, -3, 5, -1, 4, 3, 0], [-3, 0, 3, -2, -10, -1, 3, 1, 3, -3], [10, 12, 12, 6, -2, -1, -8, 7, 0, -3], [-3, 5, 3, 0, -3, -6, 2, 1, -6, 1], [-3, 9, 3, 1, -1, 1, 3, 1, 1, 1], [8, -3, -8, -11, 2, 4, 0, 1, 7, -3], [-11, -2, 6, -1, 0, 2, 1, -1, 0, 1], [6, 0, -9, -5, 10, -2, -6, 3, -1, -1], [-4, 1, 6, -3, -1, 3, 8, 3, 4, -1], [-3, 0, 5, 5, 1, -4, -7, 3, -8, 7], [6, -3, 6, -3, -8, 3, 7, 1, -2, 0], [-9, -1, -3, -4, 2, 3, -2, 4, -6, 1], [15, -5, -3, 6, -5, 2, -1, -7, -2, -3], [10, -5, -2, -5, -4, 0, 6, -1, -1, -1], [5, -1, -5, -3, 4, -7, -5, -3, -1, 3], [10, -8, -7, -4, 11, 3, 3, -7, 6, 2], [-9, 1, -1, -5, 1, 2, 0, 2, -1, -2], [7, 3, 3, 3, -7, 4, 2, 0, 3, 0], [7, 10, 0, 7, -8, -3, -9, 10, -6, -2], [11, -4, -7, -2, 1, 0, 0, -3, -2, 0], [-10, 8, 2, 0, 14, -2, -14, 4, -8, -6], [3, -6, -11, 6, 1, -2, 0, -4, -5, 1], [4, 4, -1, 1, -14, -4, 0, -5, -1, -3], [9, -2, -3, -3, 1, 3, 1, -4, 8, 4], [8, -3, 0, 7, -1, 3, -5, 7, 0, 0], [-7, 13, 7, 1, 16, -3, -7, 7, 3, -4], [-4, 3, -4, -3, 6, 0, -7, 6, -8, -1], [0, -7, -7, 1, 7, -3, 3, -1, -8, 4], [2, 1, -2, 4, -11, 4, 0, 5, -1, 1], [18, -9, 0, 3, 1, 2, 5, -6, -2, 3], [4, 5, 12, -2, -5, -5, -3, 8, 1, -4], [-6, -6, -9, -4, 0, -5, 0, -7, 0, 1], [9, -5, 1, 12, 3, 2, 3, -1, 2, 5], [3, 4, -2, -5, -10, -1, 1, -2, 0, -4], [2, 0, -2, 8, 1, 7, -7, 3, -2, 2], [18, -5, -9, 7, 5, 9, -5, 3, -6, -2], [-13, 12, 5, 7, 0, -3, -3, 10, 4, -2], [12, 3, 2, 5, -9, -4, -1, 2, -6, 7], [-2, 13, 6, 0, 5, -9, -9, 6, 1, -6], [-14, -2, 4, -2, 16, -2, 2, 0, 0, 2], [6, 9, 4, -3, -2, -3, -5, 9, 4, 0], [12, -3, -12, -6, 6, 4, 5, -7, -1, -2], [6, -6, -12, -8, -5, -1, 3, -3, 4, 2], [-11, -4, -1, -2, -1, -2, -2, -3, -2, -10], [11, -16, -11, -8, 2, 17, 13, -3, 9, 5], [-3, 6, 0, 1, 8, -2, -11, 3, -6, 3], [8, 3, 4, 6, -12, -6, -6, 1, -3, -3], [9, 5, 7, 0, -8, -1, 1, 1, 3, -4], [13, -2, 5, 9, -1, -2, 3, -5, 2, 5], [6, 0, 0, 14, -4, 6, 6, -2, 4, 2], [18, 5, 10, 5, 4, 0, 0, 4, 6, 0], [-7, 2, 6, 10, -17, 1, -4, 7, 5, 2], [0, 6, -2, -11, -4, 2, 2, -2, 6, -2], [0, 6, 8, 2, -6, 0, -2, 14, -4, -4], [-6, 1, 5, 11, -15, 0, -7, 8, -8, -5], [7, -2, -7, -1, 14, -5, -1, -6, -2, 6], [-6, -1, 4, 0, 3, 3, -7, 4, 7, 1], [1, 1, 1, 0, -16, 1, 1, 1, -1, -4], [-18, 2, 6, -8, 9, -3, 3, 1, 4, 2], [-7, 4, 5, 4, -8, -4, 5, 0, 3, 4], [27, -2, -2, -7, 2, -3, 11, -4, 0, 2], [14, -10, -12, 2, -2, 4, 6, -16, -2, 6], [15, -10, -5, 5, -2, 1, -3, -13, -5, 8], [3, -7, -9, 9, 5, 2, 4, -4, 1, 2], [7, 15, 5, 0, -5, -8, -14, 8, -5, -4], [3, -7, -5, 4, 13, 6, 1, -1, 0, -3], [0, -3, 4, 4, -13, -1, -3, -2, -3, 4], [-17, 6, 9, -5, -12, -11, 1, 2, -2, -3], [19, 3, 7, 1, 10, 5, 1, -1, 5, 0], [-2, -3, -4, 6, -1, -5, -5, -4, -7, 0], [-21, 1, 5, 4, -9, -1, 8, 4, 6, 3], [-15, 17, 13, 0, -3, -10, -5, 9, -6, -3], [0, -9, -2, -1, -6, 1, 1, -3, -2, 2], [3, -16, -13, 1, 1, 4, 8, -17, -8, 5], [8, -4, -9, 2, 6, -4, 4, -10, 6, 2], [18, 0, -10, 6, 6, -8, -6, -2, -2, 2], [4, 3, -6, 15, 7, 0, -5, -10, -2, 5], [0, 4, 2, 4, 4, -8, 0, 0, 2, 0], [19, -3, -7, 5, -9, 0, 0, -4, -3, -2], [4, -4, 4, -2, -2, 4, 14, -2, 8, 6], [-23, 7, 11, 4, 11, 0, -11, 15, -4, -5], [9, 9, -1, 3, -4, -5, -5, 5, 3, -5], [9, -9, 0, 8, -15, 2, 10, 5, 0, 2], [17, -6, -9, 0, 3, 14, 8, -5, 4, 8], [2, -10, 2, -4, -13, 8, 4, -2, -4, -2], [-14, 11, 14, 2, 1, -12, -6, 13, -9, -3], [4, -5, -6, -7, -1, -3, 8, -7, -2, -1], [-13, 9, -1, -6, 7, 2, 2, 6, 9, 8], [-13, 10, -1, -9, 8, -3, -5, -4, -2, -1], [11, -6, -7, 13, -3, 6, 0, -11, 2, 5], [-11, 2, 3, 5, -2, -1, -3, 2, -10, 2], [1, -3, 5, -9, 0, 1, 5, 9, 5, -4], [12, -20, -6, -6, 4, 2, 14, -14, -2, 8], [8, 2, 2, -7, -14, 6, 4, 0, 4, -4], [-10, 12, -2, 10, -1, -7, -13, 9, 2, 0], [-18, 2, 5, -4, -20, -3, 6, 7, 12, 7], [13, -20, -11, -1, -5, 9, 8, -18, -2, 9], [-11, 16, 7, -4, -2, -15, -11, 7, -1, -3], [6, -8, 2, 8, 2, -3, -2, -1, -2, 1], [-14, 0, -6, 9, 9, -5, -5, 0, -1, -3], [7, 5, 1, 15, -10, -5, -11, 3, -11, 4], [-8, -2, -8, -16, -4, -6, 2, -6, 4, 2], [-8, -11, 0, -1, -15, 6, 3, -6, -4, 1], [5, -14, -17, 3, -3, 14, 8, -17, -2, 7], [28, -19, -14, -7, 7, 3, 9, -11, -2, 0], [-11, 2, -8, -1, 20, 2, -3, -3, 8, -3], [0, 13, 2, 3, 10, -5, -7, -3, -4, 2], [-21, 6, 6, -3, -2, -6, 1, 3, 0, 1], [-11, -4, -11, -9, 13, 8, 8, -9, 4, 1], [-20, 4, 11, 6, 3, 1, -11, 17, -2, -2], [-28, 15, 11, -3, 5, 0, -9, 12, 2, 1], [-5, -4, 3, -3, 4, -1, 17, -10, 4, 4], [15, 6, -5, 0, 7, 2, -2, 2, -7, 3], [-13, 10, 11, 5, 5, -8, -11, 9, -10, -3], [-6, -10, -5, -6, 7, 7, 9, -11, -2, 12], [4, -18, -4, -2, -6, 8, 6, -18, -4, 0], [-8, 7, 12, -4, -15, -11, 3, 8, 1, 2], [7, -8, -15, -5, -6, 7, 3, -10, 0, 7], [-5, -1, 5, -3, -10, 3, -3, 1, -3, 3], [-4, 8, 10, 6, 10, -2, -8, 10, -4, 4], [-30, 11, 4, -9, -6, -10, -6, 5, 9, 1], [-22, 5, -4, 7, -6, -6, -10, 6, -10, 2], [-5, -12, -17, -15, -2, 9, 9, -11, 5, 0], [-24, -4, -8, -4, -1, 1, 3, -7, 4, 8], [-10, -2, 2, 0, 14, 2, 2, -8, -2, 10], [21, 11, 7, -4, -11, 0, 2, 2, 3, -2], [15, 1, -3, -3, -8, -5, 0, 7, -1, -8], [-8, -1, 4, -7, -5, 3, -1, 16, 7, -1], [5, -6, -3, -3, -15, 6, -2, -1, 0, 1], [3, 2, 1, -20, -6, -2, 3, 2, 1, 4], [16, -6, -6, 9, -20, 4, 0, -2, -8, -6], [-24, 20, 4, -6, -9, -5, -7, 15, 2, -8], [0, -9, -6, -4, -9, -1, 5, -12, -1, 0], [-6, -6, 2, -2, 14, 6, -2, 2, -4, 0], [-16, -9, -3, 9, -7, -2, -5, 2, -6, 7], [-12, -1, -4, 3, 5, 7, 1, 9, 0, -4], [18, 6, -2, -9, -18, -6, -6, 10, 2, -8], [-14, 5, 8, -3, 1, -3, -1, 8, -3, -3], [-11, -5, -3, -12, 15, 0, -4, 1, -6, -3], [-8, 7, 0, 2, -17, -1, -11, 4, -3, -2], [-25, 13, 5, 12, 7, -8, -16, 11, 0, -4], [9, -3, -17, 2, 19, 6, -2, -1, 4, 1], [-3, 8, -3, -5, 7, -5, -1, 4, 8, 0], [-10, -1, -6, 10, 11, -7, -7, -8, -17, -1], [-20, 13, 16, 3, -9, 3, -11, 18, 1, -5], [-3, 0, -2, -11, -1, 8, 16, 5, 16, -2], [10, 2, -10, -4, 1, -7, 7, -5, 0, 2], [9, 3, 3, 1, -14, 3, -1, 7, -9, -10], [13, -4, 1, 4, -4, -5, 5, 2, 16, -5], [3, 6, -7, -1, 11, 2, 4, 3, 6, 5], [-20, 12, 14, -12, 1, -8, -1, 14, 2, -7], [-5, 4, -3, -1, 8, 3, -5, 6, -2, -11], [-26, 15, 4, -12, -3, 1, -1, 8, 19, -3], [15, 2, 11, 17, 3, -1, 3, 0, -2, 2], [-16, -10, -2, -18, -5, 5, 4, -9, -2, 1], [5, -10, -10, 13, -3, 0, -2, -9, -18, 0], [9, 5, 7, -2, 4, -2, 4, 7, -2, -10], [13, -3, -13, 7, 8, -5, 5, -15, -9, 11], [6, -7, -12, 4, -6, 0, 5, -7, 3, 8], [15, 2, 1, 9, 4, 1, -9, 12, -10, 5], [0, -9, -14, -17, 15, 3, 13, -12, 1, 5], [9, 11, 15, -8, 6, -6, -2, 1, -4, -16], [-14, 6, -4, -5, -3, 7, 9, 2, 3, -8], [-23, -7, -1, 2, 17, 4, 6, -8, -3, 6], [9, -3, -3, 14, -17, -2, -4, 1, -12, -8], [-13, 1, -9, 12, 7, -2, -11, -3, -6, 1], [-25, 11, 13, -4, 21, -6, -7, 23, 2, -5], [15, -1, -7, 7, -1, 11, 3, 1, 9, 3], [-2, -2, -4, -2, 19, -1, -1, -7, 10, 4], [-25, 4, 3, -1, 4, 3, 5, 8, 2, 3], [8, -20, -13, -6, -4, 3, 10, -7, 6, 5], [-18, -1, -1, -5, -17, 10, 7, 8, -2, -7], [11, 5, 1, 6, -2, 5, -2, 14, 2, -9], [18, -7, -2, 7, 0, 2, -14, -9, -3, 3], [6, 4, 5, -1, -15, 0, 3, -1, 7, -10], [-17, 6, 9, -15, 16, -11, -13, 14, 6, -8], [18, -10, -14, -6, 4, 8, -4, -8, -6, -8], [-3, -2, -9, 13, -1, 6, -12, 1, -8, 5], [-15, -1, 11, -10, -17, -2, 4, 6, 5, -2], [3, -18, -14, -3, 15, 0, 2, -21, -8, -4], [-16, 7, 4, -7, 6, 6, -3, 6, -2, -3], [-19, 1, 1, -1, 1, -9, -13, 5, -9, -3], [-5, -6, -9, 1, 1, 10, 3, -5, 14, 11], [20, -5, -2, 0, -10, 8, 13, -1, 19, 6], [12, -10, -4, -2, -1, -3, -1, -5, -10, -2], [-31, 20, 9, -9, 14, -9, -11, 22, 10, 1], [-14, -13, -14, -15, 0, 2, -3, -8, -8, -9], [-14, -2, -10, 9, -8, 0, -4, -4, -6, 6], [-18, 20, 12, 4, -21, 1, -1, 11, 2, 0], [9, -15, -17, 4, 5, 2, -2, -19, -2, -8], [6, -5, 4, -12, 0, 8, -1, 7, 5, -4], [-3, -15, -21, -8, 17, 10, 16, -5, -2, 5], [1, 0, 11, 12, -6, -12, -9, 8, -13, -6], [-18, 9, 0, -9, -7, 5, -7, 16, 5, -1], [-13, 0, 5, 15, -30, 3, 3, 5, -7, 6], [14, 10, 10, 0, -2, -8, -4, 8, -10, -4], [-28, 1, 10, 7, 4, -10, -13, 0, -6, 1], [5, 0, -2, -6, -5, 11, 8, 7, 19, 4], [0, -4, -11, -13, -8, 6, 10, 3, 1, -1], [-25, 7, -7, -9, 7, 0, -12, 10, 3, 0], [-14, 4, 13, -3, -22, -4, 6, 7, 9, 1], [3, -24, -7, 3, -20, 7, 11, -19, -11, 4], [-11, 4, 9, -4, 1, -2, -10, 15, 6, 2], [-26, -8, 0, -8, -2, -4, 4, -8, -8, 4], [-5, -3, -3, 12, 15, 0, -12, 3, -4, 0], [0, -4, -8, 2, 3, 9, 8, -3, 0, 9], [-6, 11, -4, -5, 8, -2, -16, 12, -2, -8], [-5, -6, 9, 6, -10, -3, -7, 2, -10, -3], [36, -5, -4, 3, 6, 11, 5, -5, 2, 0], [22, -5, 10, 13, -18, -4, 5, -10, -10, 3], [-30, 7, 4, 3, 6, 2, -6, 8, 2, -2], [-11, -1, -9, 6, 8, 11, -2, -2, 14, 5], [5, 6, 0, -7, 8, -6, 9, -9, -6, 1], [-8, 19, 12, 16, 2, 0, -12, 12, 4, -7], [16, 2, 4, 0, 9, -13, -11, 7, -8, 0], [4, 12, 4, 12, -1, -6, -25, 12, -10, -1], [2, -11, -8, -9, 18, -2, 12, -19, 9, 7], [9, 6, 4, -14, -5, -5, -8, 5, 1, -2], [-1, 0, 5, -7, -1, -10, -2, -3, -2, 11], [36, 0, 5, 17, -8, 4, 8, -17, 3, 5], [20, -14, -6, 2, 0, 4, 4, -8, -2, 0], [-7, -7, -16, 8, 5, 1, 2, -14, -6, -3], [16, -10, -2, -1, -4, 14, 10, -6, 14, 0], [-22, 1, 0, 5, 13, -5, -17, 4, -1, -1], [-16, 3, 2, -10, -5, 1, 5, 10, 13, -3], [-19, 0, -2, -11, -15, 2, -2, 3, 8, 4], [-7, 4, -13, -7, -11, 13, -5, 8, 6, 2], [21, -17, -9, -16, 9, 0, 18, -18, 3, 4], [-19, 6, -1, -13, 2, -3, -9, 2, 0, -8], [19, 8, 0, -7, 1, 2, 8, -9, 6, -2], [12, -4, -2, -1, 8, 2, -2, 0, 0, 3], [4, 3, 2, -10, 5, -15, 3, -4, 11, -1], [-16, 8, 15, -14, 2, -1, 8, 5, 10, -7], [-13, 4, 11, 13, 3, -14, -10, 1, -26, -3], [-9, -10, -3, -6, 13, 8, -2, 5, -6, -2], [-3, 7, 3, 6, 2, -17, -16, -6, -14, 5], [13, 5, 3, 1, -8, 1, -7, 13, -15, -3], [-4, 6, 4, -2, 6, -6, -2, 12, 16, -6], [-8, 7, 8, 3, -26, 0, 5, 0, 8, 5], [-2, -6, 0, 3, 12, 6, 0, -2, -8, 12], [-5, -1, 7, 4, 12, -11, 9, -3, 9, 0], [8, 2, -11, -20, 9, 1, 11, -7, 14, 6], [-5, 12, 9, -11, -4, -1, 3, 12, 2, -4], [8, -6, 3, -3, -4, 8, -14, 1, 5, -1], [-20, 5, 8, 12, -12, -4, -4, 3, 9, -9], [25, -4, 0, -7, 10, 2, 7, -1, -8, -3], [19, -6, 1, 19, -7, 10, 9, -5, 10, 7], [8, -4, -12, 0, 0, -6, 14, -2, 4, 0], [33, 6, -5, 5, 1, -4, -2, 5, 0, -5], [16, -15, -8, -4, 5, 11, 9, -12, -9, 10], [5, -3, 1, -10, -9, 12, 20, -2, 9, -2], [5, 6, -3, 2, -11, -6, 12, -12, 1, 3], [7, 2, 17, 2, -7, 8, 4, 5, 2, -10], [6, 4, -10, 3, -18, 4, -2, 10, -8, -8], [-24, 11, 10, -12, 5, -6, -14, 1, -1, -5], [12, 5, 10, 0, -17, -3, -7, -2, 1, 3], [-11, 16, 13, 2, 10, -13, -21, 9, -13, -9], [-18, 0, 0, 1, 6, -6, 2, -14, -8, -7], [-2, 3, 14, -15, -5, -7, 10, 1, -2, -1], [4, -21, -16, -9, 8, 16, 18, -7, -1, 15], [-7, 3, -5, -16, 5, 0, -4, 13, 4, 5], [24, 17, 10, -5, 11, -5, -4, 3, 6, -5], [-3, -9, -1, -8, -7, 4, -4, -13, 0, 6], [-11, -2, 14, -17, -7, 2, 18, 3, 4, -4], [4, -4, 4, 2, -5, -1, 4, 7, 8, 1], [6, 3, 14, 0, -3, -7, -13, 22, -7, -14], [-34, 14, 8, -2, -14, -10, -4, -2, 12, -6], [2, -9, -6, -18, 4, 12, 16, -17, 11, 1], [-3, 8, 7, -2, 4, 1, -3, 14, 4, -13], [21, 4, 3, -7, -2, -7, 7, -6, 4, 11], [19, -25, -5, -8, -16, 3, 19, -9, 5, 6], [3, 5, -5, 8, -25, 14, 4, 9, 4, -1], [17, -1, -13, 10, -6, -3, 1, -9, -5, 4], [-26, 12, 10, 9, -6, 4, -10, 2, -4, 8], [-14, 14, 19, -6, -12, -15, -6, 9, 10, -9], [16, 2, 17, 4, -10, 1, -2, 15, 2, -9], [17, -7, -27, -19, 12, 5, 3, 1, 9, 6], [9, -14, -17, 2, -5, 2, 4, -7, -2, 4], [12, -24, -14, 0, -1, 11, 11, -9, 8, 12], [0, 10, 2, 3, 13, 9, -7, 10, 5, -9], [-28, 24, 16, -4, -23, -3, -3, 23, 18, -6], [-4, 8, 16, -1, -20, -7, 12, 6, 7, 8], [-3, 36, 15, 7, 8, -14, -17, 15, 6, -11], [-4, -7, -16, 12, 15, 17, -1, -8, 5, 6], [21, -6, 1, -5, -6, 1, 1, -2, 4, -2], [-20, -9, -4, -16, 10, -2, 4, -17, -11, -5], [-2, -17, -2, -9, 2, 11, 5, -3, -14, 6], [7, -8, -9, -6, 21, 0, -4, -15, -8, -10], [-16, 18, 7, 10, 3, -17, -25, 17, -16, -6], [-4, -10, -1, -6, 12, 18, 8, 2, 12, 4], [-1, 8, 3, 1, 18, 3, -5, 10, 16, 8], [-18, -15, 2, -5, -16, -4, 24, -10, 6, 2], [-26, -4, 8, -4, 13, -3, 1, 11, 2, 0], [-13, -2, -4, 11, 2, -3, -9, -6, -2, 0], [-17, -14, -11, 9, 24, 3, 15, -22, -2, 11], [0, 3, -2, -14, -3, 7, 15, 10, 21, -2], [-1, 2, -3, 16, 19, -4, 10, -24, -9, 5], [14, 6, 2, 11, 10, 0, 6, -6, 0, 0], [13, 16, 3, 13, 6, 2, 9, -5, 10, 11], [-10, 1, -6, -5, 21, -9, -7, 11, -10, -4], [-18, 13, 14, -4, 7, -13, -5, 10, 11, -2], [-37, 10, 19, 0, 18, -10, -11, 10, -3, -8], [-3, 6, -9, 7, 9, 19, 0, -2, 8, 3], [0, 0, -6, -18, -18, 2, 10, -2, 6, -6], [-24, 5, 7, -9, -5, 3, -7, 19, 10, 4], [5, -14, -9, -3, 11, -14, 0, -9, -8, -5], [-1, 0, -13, -7, 3, 16, 8, -5, 14, -9], [17, -6, -13, 6, 20, -9, 3, 2, -10, -1], [-23, 22, 14, 3, 14, -12, -25, 9, -4, -15], [19, -5, -3, 15, -14, -1, -5, 7, -5, -9], [-3, 13, 5, -5, 10, -9, -7, -1, 11, 0], [12, -11, -10, 15, 2, 14, 2, -20, -10, 16], [28, -6, -4, -2, -1, -5, 3, 7, -8, -4], [-28, 11, 4, -5, -8, 3, -7, 11, 18, -8], [13, 0, -11, -1, 19, 2, -17, -5, -2, -1], [-19, -27, -5, -3, -6, 11, 16, -9, 3, 14], [15, -15, -13, 2, 8, 10, 14, -3, -12, 14], [28, -5, -4, -6, 27, 1, -9, -2, 3, 1], [-3, -18, -9, -5, 6, 9, 13, -22, 4, -5]];
    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_br();".
// 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_br(:prec:=10)
    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_br();".
// 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_br( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![36, 120, -148, -444, -2, 264, 41, -56, -12, 4, 1]>,<5,R![2, 16, -76, -380, -160, 340, 101, -76, -20, 4, 1]>,<11,R![-2104, 6960, 26288, 9456, -8350, -2468, 1010, 184, -54, -4, 1]>],Snew);
    return Vf;
end function;