
// Make newform 4018.2.a.bu in Magma, downloaded from the LMFDB on 25 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_bu();".
// 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_bu();".
// 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 := [-12, -204, 608, 231, -579, -109, 181, 19, -23, -1, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [4512, -20680, -26574, 17781, 15220, -4759, -2594, 513, 132, -19], [-16484, 56056, -41760, -33721, 29562, 4525, -5712, 235, 314, -39], [-124, 56056, -45032, -33721, 29562, 4525, -5712, 235, 314, -39], [-7936, 3108, 6310, 2603, 1161, -3244, -740, 725, 55, -42], [-4516, -35880, -3192, 30673, 7846, -11521, -2208, 1909, 142, -101], [-1788, -6120, -20264, 18748, 16381, -8003, -3334, 1238, 187, -61], [-256, 38256, 10732, -44154, -6401, 15345, 1058, -2048, -51, 91], [-19612, 1408, 92388, -30817, -53980, 18823, 10104, -3521, -552, 191]];
    Rf_basisdens := [1, 1, 1636, 3272, 3272, 1636, 3272, 1636, 1636, 3272];
    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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 0, -1, 0, 0, 1, 1, 0, 0], [2, 0, 0, 0, 0, 0, 0, -1, 0, -1], [-1, 0, 0, 0, 0, -1, 1, 0, 0, -1], [0, 1, -1, -1, 0, 0, 1, 0, 0, -1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, -1, 0, 0, 0, -1, 0, -1, -1], [-2, 1, -1, -1, 0, 0, 0, 0, -2, -1], [3, -1, 0, 0, 2, 0, 1, -1, 0, 0], [-2, 0, 0, -1, 0, 0, 0, 0, 1, 0], [0, 1, 0, 0, 0, 1, 0, -1, -1, -1], [0, 1, 0, -1, 0, 0, 0, 1, 0, 0], [0, -2, 0, -2, 2, 1, 0, 0, 0, 0], [1, -1, 1, 3, -2, 0, -3, -2, 1, 1], [-1, 1, 1, -2, 2, -1, 3, 1, 0, 0], [2, 0, 1, 0, -2, -1, -1, -1, 1, 0], [0, 0, 0, 1, -2, -1, 1, 0, 2, 1], [1, -1, -1, -1, 0, 1, 0, 2, 0, -2], [1, -2, 0, 1, 0, 2, 0, 0, 0, 0], [2, -1, 1, 0, 2, 1, 0, 0, 0, 2], [3, 0, -1, 2, -2, 0, -2, 1, 1, 1], [1, 0, 1, -1, 0, 0, 0, 2, 0, 2], [7, 0, 1, 1, 1, 1, -1, -1, -1, 0], [1, 1, 1, -2, -2, -2, 1, 2, 0, 1], [-1, 1, 0, 1, -4, -1, 0, 1, 0, 1], [4, 0, 1, 1, 2, -1, 1, -1, -2, 0], [4, -3, 0, 1, 0, 0, -1, 1, 0, 2], [1, 1, -1, 0, -2, 2, -3, -1, 0, 0], [-4, 0, -1, -2, 2, 0, 1, 2, -2, -2], [-4, -3, 1, 2, -2, 1, 2, 1, 2, -1], [4, 1, -1, 3, -2, -1, -2, 0, -1, 0], [-1, -2, 0, -3, 0, -2, 1, 1, 0, 1], [-4, -3, -1, -2, 0, 1, 2, 2, -1, -2], [1, -1, 0, -1, 0, 0, 0, 2, 2, 2], [9, -4, 2, 3, 0, 0, 0, -2, 0, 2], [-1, 1, 1, -1, 2, -1, 1, -1, 2, -1], [-1, -4, -1, 1, -2, 0, 0, -1, -2, -1], [8, 0, 2, 2, 2, 2, 0, -2, -1, 2], [-5, 1, 1, -2, 0, 0, 2, 1, 2, 0], [3, -4, 1, 1, 2, -1, 0, 0, 2, 0], [-2, 0, 0, -4, 4, -2, 4, -2, 0, 0], [0, -2, 2, 0, -2, -1, -1, -3, 0, 1], [-4, -2, 1, -3, 2, -1, 3, 0, 0, -1], [1, -1, 1, 0, 0, -4, 1, 0, 0, 1], [-3, 0, -2, -2, 0, -2, 2, 3, -2, -2], [-12, -3, 0, -1, 2, 1, 0, -1, -3, -1], [0, 1, -1, -1, -2, -1, 2, 0, -1, 0], [5, -4, 3, 1, 2, 2, -2, -5, 0, 1], [2, 0, 0, 3, 0, 3, 0, -1, 0, 0], [2, -2, -1, 2, -2, 0, 0, 1, 1, 1], [2, 1, 1, 0, 2, 1, -2, 2, 0, 0], [-1, -2, 1, 0, -1, 1, -1, 1, 2, 2], [2, 3, -3, 1, -2, 2, -1, 0, 2, -1], [-2, -2, 2, 1, 0, -2, 0, 0, -1, 0], [3, -1, 1, 0, 2, 0, 1, 1, 0, -2], [-5, -1, 2, -3, 6, -2, 3, 3, -1, 0], [3, -2, 0, 0, 2, 3, -2, -3, 0, 0], [5, -3, -2, 1, -4, -1, 0, 1, -2, 1], [2, -2, 0, -1, -2, -3, 1, 0, 0, 1], [5, 1, 3, 5, 0, 1, -3, -4, 0, 2], [7, -1, 1, 1, 3, 0, -1, -2, 1, 1], [4, 1, 0, 2, -4, -2, -3, -2, -2, -2], [4, 5, 0, -2, 2, 1, 0, -1, 0, -3], [-2, 1, 0, -1, 3, 1, 0, 1, 1, -3], [-9, 0, -1, -2, 0, -1, 3, -1, -2, -2], [-5, -1, -2, 3, -4, 1, -2, -3, 0, -1], [4, -4, 2, 2, -2, 0, 4, -2, 1, 0], [-8, 3, -2, -2, 4, 2, 1, 1, -1, 0], [0, 1, -2, 2, -6, 0, -4, -1, -2, -1], [1, 1, 3, -4, 0, -4, 3, -2, -1, 3], [5, -1, 0, -2, -2, -2, -1, 1, 2, 0], [2, 1, 0, -2, -3, 0, -3, -1, -1, 0], [-8, 2, -1, -2, 1, 1, 1, 3, -1, 2], [-2, 5, -3, 0, 0, 4, -1, 0, -3, -3], [-1, 3, 5, 0, -6, -4, 1, 0, 0, 1], [-2, -1, 1, 2, -2, -1, -2, 0, 2, 2], [-3, 3, -1, -2, 4, -4, 3, 2, -1, 1], [4, 6, -1, 2, 0, -1, 2, -1, 0, -3], [-5, 5, 0, -5, 2, 0, 1, 3, 1, 0], [-4, 6, -1, 2, 0, -1, -3, -1, 0, 2], [5, -6, 0, 0, 0, 1, -3, 2, 0, -1], [-11, -2, -2, -3, 3, 0, 0, 2, 0, -4], [-8, -3, -1, 4, -4, 2, -4, -2, 2, -2], [-6, 3, -2, 1, -4, -2, 5, -1, 0, -2], [11, -3, -1, -3, 8, 0, 3, 0, 0, 0], [-10, 5, -2, -4, -2, -1, 4, 3, 0, -3], [-2, -4, 1, 1, 4, 0, 6, -1, -1, -1], [-2, 5, -4, 1, 0, 0, -4, -1, 0, 2], [-4, 1, 0, 1, -4, 1, 2, 0, 2, 1], [2, -1, 1, 4, -6, -1, -6, 1, 2, 3], [8, -4, 2, 1, -4, -1, 1, 0, 4, 1], [-5, 2, 0, -2, 4, 5, -1, 0, 0, -3], [-2, 2, -1, -3, 2, -5, 9, 3, 0, 2], [2, -1, 2, 2, 2, -1, 5, 5, 2, -1], [1, 7, 1, -3, 0, -5, 2, 1, 4, -1], [19, 1, 2, 2, -6, -5, 3, -2, 0, 1], [12, -2, 1, 6, -6, 1, -3, -1, -1, 2], [-2, -4, 0, 4, -4, 1, 0, -5, 0, 3], [-5, 3, 0, -2, 2, -2, -1, 3, 0, 2], [-12, 1, 0, 2, -6, 0, -5, -2, 0, -2], [0, 1, 1, -3, 4, -2, 5, 2, 0, -1], [6, -3, -2, -1, 4, 0, -1, 3, -2, -2], [-8, -2, -3, 0, -2, 0, 2, -1, -2, -3], [2, 4, 0, -3, 2, -3, 1, 2, -2, 3], [0, 3, 2, 1, -1, 2, -1, 1, 6, 4], [10, -5, 0, 1, 4, -3, 0, 1, 2, 1], [-16, 1, -3, -8, 4, -2, 1, 4, -1, -3], [-7, -9, -1, 1, -4, 1, 0, 2, 0, 4], [8, -4, 1, 6, 2, 0, 0, 1, 2, 1], [2, 1, 2, -2, 2, 0, -2, 0, -4, 2], [15, 2, 1, 1, -2, 2, -2, 5, -2, -1], [14, -3, 3, 4, -2, -3, 2, -1, 2, 1], [-4, -4, 2, 2, 0, -2, -1, 0, 0, -2], [2, -8, -2, 3, -1, -1, 1, 0, -4, 1], [8, 2, -1, -1, 0, 1, -7, -1, -2, 0], [4, -1, 3, 0, 1, 1, 2, 0, 0, 0], [12, 4, -1, 2, -8, 0, -1, 1, -2, 1], [6, 0, 1, -1, -4, -4, -1, -2, 2, 1], [-2, -4, -3, -2, -3, -4, 0, -5, -4, -1], [2, -2, 1, 6, -6, 0, -6, -6, 2, 0], [-15, 8, -2, -2, 2, 0, -4, -2, 0, -1], [13, -5, -1, 2, -4, -2, 2, -1, 0, 2], [-5, -3, 2, 4, -2, 0, -1, -1, 6, 2], [0, 1, 0, -1, 2, -3, 2, -5, 1, -3], [3, 6, 0, 1, -2, 0, 0, 4, 2, 2], [-5, -6, 2, 1, 6, 3, -1, -4, -2, 2], [13, -5, 2, 2, 2, 0, 5, -4, 0, 1], [0, -1, -3, -3, -4, 2, -3, 2, -6, -3], [-4, 8, -2, -1, -4, -1, 3, 2, -1, 1], [6, 4, 0, -3, -2, -1, 1, 0, -4, 1], [1, -3, 5, -1, 4, 1, 2, -4, 2, 2], [-11, 3, -1, 2, -2, 6, -9, -2, -2, 1], [0, -5, 2, -2, -2, 3, -2, -3, 0, -1], [-5, 2, -5, -4, -2, -4, 4, 1, 1, -1], [11, 2, 0, 4, -3, 3, -5, -2, 4, 1], [-11, -4, -1, 2, 4, 0, 0, -2, -4, -1], [1, 4, -1, 0, -4, -1, 0, 3, 4, 0], [-2, -1, 1, 2, 2, 1, 2, -2, -4, -2], [7, 4, -2, 1, -2, -3, -4, 2, -4, 6], [5, -8, -4, -3, 0, 0, 4, 2, -3, 0], [5, 6, 4, -4, -1, -4, 4, 0, 5, 0], [-11, 8, 1, 4, -2, -2, 0, 1, 3, -1], [-13, 0, 0, 1, -2, 0, 2, -4, 2, 2], [3, -2, 1, -3, 4, 4, 0, 3, -1, 3], [-12, 0, -3, 5, -6, 0, -4, -1, 2, -1], [-2, -1, -2, 3, 0, 2, -6, 1, -2, 0], [-7, -2, 3, -3, -6, -5, 7, 5, 4, 3], [-1, 4, 0, -2, 6, -4, 6, 0, 1, 0], [-4, -4, -2, 5, -6, -3, -3, -2, -2, -3], [-21, 11, 1, -4, 2, -5, 2, 0, 0, -1], [-17, 4, -1, -2, 2, -1, 1, 11, 0, 4], [10, 2, -2, -3, 2, 5, -3, 0, 2, -3], [-4, -8, 4, -3, 0, -5, 5, 0, -1, -1], [9, -4, 1, -1, -4, 3, 1, 3, 1, 2], [3, 9, -4, 4, 0, 1, -6, 1, -1, 1], [4, 3, 2, 1, 0, 1, 4, -1, 0, 4], [9, 3, -3, 1, -8, -1, -3, 2, -4, 0], [2, -6, -1, -3, 0, 1, 1, 1, -2, 2], [1, 2, 1, 0, -4, 3, -3, 3, 2, 2], [-16, -1, -1, 6, 0, 2, -5, -8, 0, 0], [-7, -1, 1, 0, -2, -4, 5, 5, 4, 6], [5, 0, 3, -2, -2, 0, -2, -1, -1, 1], [15, 2, 3, 5, 0, 6, -8, -3, 0, 1], [8, -2, -5, -4, 2, 3, 0, -2, 0, -2], [-6, -6, 8, 1, 2, 1, 3, 0, 2, 9], [22, -3, -3, 2, 0, 2, 0, -3, 0, -3], [-10, -2, 1, -1, 6, 4, 0, 5, 4, 3], [14, 5, 4, 4, 4, 4, -4, -4, -2, 0], [-5, 13, 0, -7, 4, -1, 0, 3, 4, -1], [-5, 2, -3, 0, 6, 4, -2, -2, -2, -1], [4, -4, 0, -3, 0, -1, -3, -6, -3, -3], [2, -2, -3, -4, 10, -1, 7, -5, -2, -7], [-2, 5, 1, -2, -4, -3, 6, 4, 2, -2], [-7, 2, 4, -2, -4, -1, 3, 2, 2, 3], [-24, -7, -1, -2, 0, -4, 5, 2, 1, -1], [3, -6, -2, 2, -2, -1, 3, -1, 0, -2], [7, -6, -3, 8, -2, 4, -4, -3, -3, -3], [10, 4, -3, -2, 8, 4, -2, 1, -4, -3], [7, 8, -3, -6, 6, 1, -3, -1, -5, 0], [6, 9, -1, 2, -2, -5, 4, 2, 2, 2], [-9, 1, 3, 1, 0, 3, 2, 4, 2, 2], [3, 1, 1, 7, -4, 1, -6, -10, 2, 4], [1, -3, -4, -10, 10, 1, 7, 4, -2, -1], [-3, 6, 2, 2, -2, 6, -2, 2, 3, 4], [1, 5, 2, 0, -2, -1, -5, -2, -4, 1], [3, -5, 2, -5, 6, 2, -5, 1, -1, 0], [19, 1, -1, 0, 1, -2, -1, -6, -2, -1], [17, -1, 0, -2, -4, -1, 2, -1, -3, -3], [0, -6, 3, 0, 6, 1, 5, -3, 5, 0], [11, -9, 3, -4, 6, 2, -1, 2, 4, 5], [2, -6, 4, 7, -4, 5, -3, 4, 2, 1], [-3, -5, 1, 3, 8, -1, 6, -6, 4, 4], [-30, -2, 2, -7, 2, -2, 3, 0, 0, -1], [18, 6, 1, 7, -11, -1, -3, 1, 2, 4], [9, -2, 3, 7, -6, 2, -6, -3, 2, 3], [1, -5, -2, -3, -4, -5, -2, 2, -2, 0], [17, -4, 4, 2, 0, 5, -1, 0, -2, 7], [0, 12, -4, -1, -2, -1, -2, 0, 2, -3], [-3, -3, -3, 7, -2, 5, -6, -3, 0, -1], [-3, -2, 3, 0, 4, -1, 3, 1, 8, 6], [4, -9, 0, -1, -2, 1, 7, 2, 0, 1], [4, -4, 5, 0, 8, -5, 3, -1, 6, 4], [16, -10, 4, 11, -4, 8, -10, -8, 3, 0], [6, -2, 2, -6, 6, 2, 4, -2, 4, -2], [19, -10, 4, 2, 6, 1, 1, 1, 2, 6], [-8, -12, -1, -1, 2, 3, 3, -3, -4, 0], [-21, -11, -5, -5, 4, -1, 10, 6, -2, -6], [12, -3, 5, 10, 2, 0, -9, -4, 2, 4], [14, 3, 0, 5, -12, -1, -8, -5, 0, 1], [5, 11, -2, 0, 2, -3, 0, 3, 0, 0], [-8, -3, -3, 3, -8, -2, -1, 0, -2, 1], [16, -5, -3, 5, -2, 2, -6, -9, -2, -6], [1, -6, 0, -7, 6, -4, 4, 12, -4, -2], [6, 5, -4, -2, 4, 5, 2, -1, 0, 1], [-1, 9, 6, -2, 2, -2, -3, -1, 0, 0], [6, 12, -2, -5, 0, -3, 7, 2, 0, 1], [-9, -2, -8, 2, -6, -2, -8, 0, -6, -3], [11, -15, 4, 8, 4, 7, -10, -11, 1, 3], [8, 3, 2, 0, 0, -5, 6, -1, -4, -3], [5, 6, 0, -4, 9, 6, -2, -2, 3, 0], [-7, -6, -2, 5, -10, 0, -8, -6, 0, 4], [10, -12, 2, 0, 2, -1, 1, -7, -4, -1], [9, 7, -3, -5, 2, -2, 1, 2, -5, 3], [-9, 2, 2, 0, -13, -3, -5, 0, 2, -3], [21, -4, 0, 1, 6, 2, -5, 0, -4, 2], [-17, -1, 0, 6, -6, 2, -9, -5, 0, 0], [9, -3, 1, -3, -7, 0, 3, -4, -3, 1], [12, 1, 2, 3, -8, -1, -1, -2, 2, 5], [4, 1, 2, 0, 8, -4, 3, -5, -4, -3], [-8, 5, 2, -5, -6, -4, 7, 1, 4, 0], [11, 11, -3, 1, 2, 5, 0, -1, 2, -1], [-9, 0, -7, 3, 0, -1, 3, 9, -5, 0], [-5, -11, -3, 5, -2, -1, -1, -3, -4, -3], [0, 0, -1, 0, 8, 8, -7, 1, 0, -3], [6, -2, 7, 1, 2, 4, 2, -2, 4, 3], [-15, -7, -5, -8, 4, -3, 6, 4, 0, -5], [20, -13, 4, 5, -4, 4, -3, -3, 6, -2], [-35, 8, -2, -2, 0, 3, 3, 4, -2, -3], [31, 3, 5, 4, 2, 4, -3, -10, 2, 3], [18, -2, 1, 4, 8, 6, 0, -1, 0, 5], [-17, -1, 1, -8, -4, -5, 7, -3, 0, 0], [-11, -7, 0, -7, -2, -2, 7, 9, -1, -4], [7, -4, 7, 2, 4, 3, -3, -7, 6, 0], [-4, 3, -4, -7, 2, -4, 1, 1, -3, -4], [-3, 3, -5, -5, 0, 0, 5, 8, -4, -3], [-1, 7, -5, -2, 2, -8, 3, 0, 2, -5], [0, -1, 0, -4, 1, 3, -4, 1, -2, -1], [-23, 3, 5, 4, 2, -4, 1, 1, 4, 6], [-5, 6, 0, 5, -8, -1, -1, -2, 5, 1], [11, 6, -1, -1, 0, 10, -4, 0, -6, 0], [-1, -2, -2, -4, 0, -1, 10, 6, 4, 1], [4, 5, 0, -10, -2, -7, 2, 6, 0, 4], [7, 1, 0, -1, 6, 5, 2, -5, -2, -3], [-11, 0, 0, -2, 4, -3, 1, -2, 2, 3], [12, 3, 3, 4, -10, 0, -3, -1, 0, 1], [-1, 4, 2, -8, -4, -1, 5, -1, 2, -4], [10, 10, 2, 9, -8, 3, -9, -11, 4, 4], [-12, 10, -4, 2, -4, 7, -7, 4, 8, -1], [-3, 7, 4, 5, -6, 2, 1, 3, 3, 2], [-7, -6, -3, 4, 2, 1, -7, -4, -6, -3], [-6, -6, 0, -1, 6, -4, 8, 4, 0, 2], [0, 9, 0, 0, 4, -2, 3, 3, -3, 2], [8, 7, 2, 2, -8, 3, -2, -2, 0, 8], [23, -3, 2, 3, 0, 3, -4, 2, 4, 2], [10, -4, 1, -5, 4, 8, -4, 3, -1, 5], [-10, 4, -1, -3, -3, 1, 5, 1, -4, 0], [-12, 6, -4, 7, 2, 5, -5, -10, 2, -5], [-15, -6, 4, 11, -14, -2, -4, -2, 4, 2], [-9, 7, 1, 9, -8, -3, 0, -2, 4, 2], [12, 5, -2, -4, -2, -1, -1, 7, 0, -1], [9, 7, 0, 6, 6, 4, -7, -2, -6, 3], [20, -9, 2, 2, 2, -7, 4, 1, -6, 1], [15, -5, 5, 6, 0, 5, -4, 3, 6, 6], [3, 0, -1, -5, 0, -6, 7, 0, -4, 2], [-14, 12, 4, -4, 0, 1, 3, 1, 6, -1], [-19, -1, 4, -3, 8, 3, 4, -3, 0, -5], [-16, -6, 1, 2, -3, 8, -4, -3, -4, -1], [3, 6, -1, 2, -4, -3, 0, -3, -4, -2], [5, -5, 7, -3, 2, -4, -3, 1, 0, 7], [14, -5, -7, 0, 6, -3, 0, 2, -4, -2], [22, -13, 3, 2, -1, 9, -6, -6, 0, -2], [-1, -3, 6, 9, -4, 5, -15, -11, 2, -1], [7, 10, 2, -2, 8, 9, -3, 2, 2, -1], [-2, 13, 5, -4, 6, 1, 6, 2, 4, 0], [-4, -10, 1, -4, 4, 4, 0, 1, -4, 1], [8, 4, -6, 6, -10, -4, -4, 6, -4, 2], [11, 3, -6, 1, -2, 0, -1, -3, -3, -4], [17, -3, 4, 6, -6, -2, 1, -4, 6, 5], [10, 7, -3, -6, 2, -1, 6, -1, -2, 1], [26, -8, -7, -7, 2, 0, 3, 2, -6, 1], [-6, -2, -4, -3, 8, -1, 5, 6, -3, -5], [-2, 3, -1, -6, 10, 4, 5, 11, -4, 1], [22, 8, 1, -10, 4, -3, -1, 0, 0, 6], [-16, 7, 1, 0, -3, -10, 5, 2, 3, -5], [2, -5, 4, -6, 6, -5, 2, -6, 2, 4], [8, -10, 1, -3, 10, 3, 1, -5, -6, -2], [5, -4, 0, -7, 10, 2, -1, -2, 0, 0], [-16, -3, 4, 1, 4, -2, 9, 3, 0, -8], [22, 8, 3, 0, 8, -2, 2, -3, 2, -1], [16, 3, -4, 3, 4, 6, -5, -7, -4, -4], [-15, -9, 0, -2, 1, 0, -5, -3, -2, -2], [22, 8, 1, -3, 0, 6, -6, 1, -6, 0], [-7, 3, 1, 0, -4, -2, -1, 0, 6, 3], [15, 2, -1, 4, 6, -3, -10, -2, -2, 3], [3, 0, -5, 9, -6, 2, 2, -3, 4, -1], [15, -7, 5, 2, 4, -2, -3, 0, 1, 7], [0, 5, 2, 4, 4, -6, -3, -3, 1, 4], [23, 3, 2, 17, -12, -3, -8, -9, -2, 1], [-22, 7, 2, 4, -5, 2, 1, -3, 1, 2], [-28, 1, -7, 7, -8, -2, -3, 2, 2, -5], [10, 7, 6, 0, -6, 1, 0, 7, 2, 3], [-4, 8, 2, -2, 0, 0, -8, 8, 2, 8], [12, 1, 6, 0, 8, 3, 0, -1, 0, 7], [-25, 7, -4, -11, 2, -2, 3, -5, -1, -4], [-13, -12, -5, 3, -8, 2, -6, 3, 3, 3], [3, 11, 0, 7, 2, -1, -4, -5, 1, -1], [-26, 6, 1, -3, -4, -2, 5, -4, 6, 1], [11, -1, -4, 6, -4, 1, -10, -13, -3, 1], [25, 2, -2, -8, 8, -1, 3, 2, 2, 1], [12, 2, 5, 10, -8, 3, -10, -6, 2, 8], [6, -2, -3, -8, 8, -5, 11, 2, 4, -6], [27, -18, 3, 2, -4, 2, -3, -7, 0, -6], [18, -1, -2, 5, 3, 2, 5, 7, 0, 2], [29, 5, 0, -7, 4, 1, 2, 2, -4, 0], [37, 2, 6, 3, 2, -2, -2, -10, 0, 6], [-20, 0, -1, 1, -2, 7, -5, 3, 0, 0], [16, -6, -4, -9, 6, -6, 3, 11, -6, -2], [4, 11, -6, 3, -4, -2, 3, -5, 2, -4], [-4, 3, -5, -4, 10, -3, 3, 8, 0, 0], [11, -3, 1, 1, 10, 7, -4, -5, 2, -1], [-22, 3, 4, -4, -4, 2, 5, -3, 7, -6], [-5, 7, 4, -4, 10, -8, 9, -3, -4, 2], [-2, 6, 1, 1, 2, 1, 1, 9, 2, 6], [-30, 1, -2, -1, 6, 0, 3, -2, -2, -11], [10, 2, -3, 1, 0, 5, -15, -5, -1, 10], [-13, 10, 5, 1, 2, -8, 2, 1, 8, 3], [-31, 9, -4, -9, 6, -4, 7, 11, -4, 1], [-10, 2, 1, 1, -2, -7, 9, 11, 0, -4], [16, -4, -4, -4, 10, 7, -9, 0, -9, 1], [0, -1, 2, 7, -7, -8, -5, -5, -2, 4], [24, -2, 3, -4, 4, 0, -2, 3, 4, 7], [-41, 1, 3, -2, 0, 0, 0, 7, 2, 6], [-18, 6, -4, 6, 0, 4, 0, 0, -6, 2], [-4, -19, 1, 2, 6, -1, 3, 0, 6, 0], [-13, 10, -3, -4, -10, -4, 2, -1, -1, -5], [-5, 7, -1, -11, 16, 5, 0, 3, 2, -5], [-13, 5, -4, -5, 0, 7, 0, 1, 6, 1], [2, -15, 0, -4, 2, 7, 0, 0, -6, 8], [36, -6, 7, 11, -4, 1, -7, -7, 4, 10], [-5, 5, 3, -3, -4, -3, 4, 6, 0, 2], [11, -1, 0, 1, 6, 0, -5, -7, -1, 2], [36, -9, 0, -3, 6, -5, 2, -1, 1, 3], [6, 1, -7, -4, -14, -5, 6, 0, 0, -2], [-5, 4, -1, -7, -4, 2, -2, 5, -3, 1], [19, -3, -4, 4, 10, 2, -7, 5, -2, -2], [29, 4, 6, 2, -2, -2, 10, 3, -2, 0], [11, 3, -8, -1, 4, -13, 4, -1, -2, -5], [0, -9, 0, 4, 10, 7, 3, 1, -2, 3], [-11, -8, 4, -3, -2, 8, -4, -4, -4, -4], [18, -10, 2, -10, 8, -2, 4, 5, 2, 1], [19, -3, 5, 8, 2, 2, -3, -6, 4, 9], [18, 17, 5, 4, -4, 5, -6, -2, 7, 10], [-28, -18, -3, -1, -10, -4, 3, 6, 2, 5], [7, -6, 2, 7, 2, 1, -4, -11, 0, 5], [18, -4, 4, 0, -8, 2, -4, -8, 4, -2], [3, 4, 6, -11, -2, 4, -4, -2, 2, -2], [-5, -7, -3, 1, -4, -1, 10, 4, 2, 2], [-13, 11, -2, -11, 6, -5, 2, 8, 2, 2], [26, 1, -2, 11, -10, 3, -8, -9, -5, -3], [-5, 10, -4, 3, 6, 4, -2, 0, -4, -2], [-7, 6, -10, 3, -4, 4, -2, 2, -3, -2], [-2, -7, 2, 0, 2, 7, -4, -3, 2, 1], [2, 14, 6, -6, 4, 6, 4, 8, 8, 4], [18, -3, -3, -2, 12, 3, -8, 6, 0, 2], [20, 1, -2, 0, 2, 6, -3, 1, -2, 2], [-13, -1, -2, 2, 2, 6, -15, -7, -2, 0], [8, 9, 0, -6, -5, -1, -4, -1, 6, -1], [21, -2, 3, 7, -2, 7, 3, 10, 8, 4], [-22, 1, -2, 1, -8, 8, -3, 3, 0, -6], [-31, 4, 3, 5, 6, 7, -4, -1, -2, -1], [3, -13, -1, -4, 0, 4, -3, 2, -2, -1], [1, -4, 9, -2, 10, 4, -6, -7, 1, -1], [-34, 8, -5, -9, -2, 0, -5, 2, -4, 1], [11, 6, 0, -6, 2, -5, 7, -6, 0, 1], [-9, -1, 1, 0, -8, 12, -7, 5, 0, 2], [1, -13, 7, 1, 6, 2, 6, -1, 4, 1], [-2, 0, -2, -7, 0, 4, -8, 8, -1, -4], [1, -6, 4, 11, -14, -2, -4, -8, -4, 4], [-30, 14, -4, 6, -6, 0, -2, 6, 2, 0], [11, 0, 2, -5, 8, -5, 2, 2, -6, -4], [-4, 12, 3, 5, 11, 9, 3, -1, 2, 2], [1, 11, 1, 1, -12, -5, -4, 8, 0, 2], [-14, 6, -6, -15, -6, -5, 7, 6, -4, 1], [1, 1, 2, -9, 19, 1, 12, 3, -2, 1], [-4, 1, 0, -13, 8, 2, 1, 3, 2, 0], [-29, 2, -10, 2, 4, 1, -1, 12, 0, 1], [3, 11, -2, 2, -10, -6, -1, 1, 4, -4], [-8, 8, 6, -9, 6, 5, -1, 4, 2, -5], [-2, -7, -5, 5, -10, 2, -4, -3, 2, -6], [-26, -5, 2, -11, 15, 4, -3, -1, 2, 0], [-14, 7, -6, 8, -4, 1, -8, -5, 0, 3], [46, 3, 2, 0, 4, -2, -2, -6, -4, 0], [-3, -4, -3, 0, 0, -1, 9, 7, 2, -2], [10, -1, 0, -1, -8, -9, 0, 5, 0, -1], [2, -14, 1, 1, 13, 3, -5, -1, -4, -2], [-6, -3, -3, -10, 4, 0, 15, 3, 2, -3], [-4, -10, -1, 2, 2, 7, 0, 0, 10, -4], [-18, -3, -3, -2, 6, -9, 8, 0, -2, -4], [-20, 4, 2, -6, 8, 0, 4, 2, 0, -2], [9, -7, -1, -4, -6, -6, 17, 0, -2, -5], [-10, -4, 2, 5, -12, 2, -3, -10, 0, -9], [-11, -5, -3, 0, -4, -8, -3, -6, -2, -1], [26, 9, -1, -2, -6, -5, -4, -6, 0, -2], [15, -12, -11, -1, 10, 2, 2, 3, -6, -3], [3, -10, 2, 7, 4, 0, -6, 5, 2, -1], [-5, 2, -5, 0, -8, -5, -9, -3, -6, -6], [-22, 9, -10, 0, -10, 2, 0, 6, -4, -2], [-4, 1, 6, -5, 11, -1, 10, 9, 5, -1], [25, 2, 2, -8, -2, -1, -1, 10, -7, -1], [21, 6, -7, 8, 4, 0, -1, 4, 0, -1], [-3, 3, -7, -10, -8, -10, 11, 0, -2, -1], [-10, -5, 1, -11, 4, -9, 2, 1, 4, -2], [-8, -5, -6, -6, -8, -8, 4, 5, -4, -7]];
    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_bu();".
// 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_bu(: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_bu();".
// 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_bu( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-12, -204, 608, 231, -579, -109, 181, 19, -23, -1, 1]>,<5,R![84, 740, -742, -2963, -1137, 808, 373, -71, -35, 2, 1]>,<11,R![-2268, 38592, -54918, 8589, 14070, -4429, -845, 415, -6, -11, 1]>],Snew);
    return Vf;
end function;