
// Make newform 2009.2.a.u in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2009_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_2009_2_a_u();".
// 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_2009_2_a_u();".
// 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 := [49, 42, -1179, -1972, 5690, 9524, -13352, -18446, 17721, 18080, -13797, -9852, 6449, 3090, -1820, -552, 302, 52, -27, -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, 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, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6860, 3957, -132531, -249231, 280878, 878553, -21401, -1196150, -380011, 789606, 366927, -277250, -148929, 53542, 30507, -5497, -3106, 257, 125, -3], [207375, -57319, -3052103, -1618184, 11384027, 3390237, -19463382, -243673, 17565750, -3497403, -9030010, 3116367, 2725642, -1207269, -475649, 241298, 44215, -24289, -1689, 974], [-1923, 32529, 26849, -515722, -454807, 1847879, 1386290, -2860951, -1741594, 2298507, 1102030, -1051789, -379058, 285307, 71783, -45386, -7023, 3905, 277, -140], [196049, -119417, -4323244, -661121, 20985660, -24455, -41409619, 5354233, 40772796, -8390942, -22147850, 5578466, 6908584, -1944145, -1227388, 370173, 115070, -36373, -4408, 1441], [397047, -562859, -8439072, 3569523, 43597061, -17218799, -90288507, 36775414, 92211853, -37398186, -51465599, 20472398, 16368355, -6387514, -2948266, 1133945, 279121, -106301, -10768, 4075], [136681, -212470, -2928332, 1569115, 14968615, -7680454, -30661461, 16374317, 31025651, -16781193, -17217267, 9276793, 5465175, -2915591, -985066, 519731, 93461, -48804, -3616, 1871], [2317504, -2806969, -49964329, 7693925, 242024086, -49298757, -479242301, 133111450, 474855711, -152204430, -259682035, 89221158, 81489167, -29080478, -14549501, 5315407, 1369512, -508397, -52631, 19767], [-3155999, 3014029, 69269979, 1044336, -330940935, 29783317, 647387306, -128646073, -634890138, 168574693, 344392654, -105260377, -107435598, 35551495, 19103225, -6643398, -1793047, 644715, 68773, -25318], [385549, -568756, -8303666, 3815525, 42599011, -18471832, -87751919, 39493437, 89377701, -40349501, -49846133, 22198929, 15862357, -6952139, -2860796, 1237033, 271265, -116106, -10482, 4453], [1715049, -2004129, -36808230, 3687413, 175629453, -28460643, -343888221, 86094140, 337661823, -103346572, -183389459, 62243884, 57277865, -20628888, -10195866, 3811197, 957937, -367103, -36774, 14341], [-2112117, 2678875, 45106927, -10391849, -220425769, 58149675, 439149488, -143314770, -436717672, 158073734, 239401956, -91075798, -75265998, 29394218, 13459641, -5338649, -1268655, 508339, 48813, -19699], [-6105596, 8035483, 129312481, -30707973, -626706506, 165911551, 1243768209, -406481732, -1234767991, 447924232, 676372607, -258032904, -212578283, 83273652, 38010317, -15123407, -3582608, 1439923, 137847, -55795], [6810118, -7823409, -144478337, 20554343, 693636784, -138161701, -1366475007, 380081640, 1349119441, -438652340, -735967357, 258680916, 230601293, -84645344, -41139285, 15510665, 3870922, -1485881, -148747, 57829], [6091414, -8635897, -129260161, 46618937, 640952560, -231088533, -1290502667, 518115762, 1293341169, -546335846, -713165485, 307049398, 225221581, -97569654, -40414977, 17535335, 3819554, -1657237, -147271, 63867], [-6892942, 9091823, 145600115, -40276409, -709488424, 211440319, 1412960229, -499167236, -1406264571, 539750428, 771777607, -307579248, -242911087, 98583464, 43481023, -17820791, -4101614, 1691191, 157917, -65375], [3591140, -4740983, -76082910, 23015472, 375432804, -120087963, -754399768, 277891143, 755681308, -296641877, -416625344, 167604135, 131533684, -53410665, -23592048, 9617310, 2228382, -910249, -85868, 35122], [-4729788, 6084097, 100201394, -26758406, -490465964, 143694241, 979627856, -342559345, -976724486, 371818545, 536630064, -212261757, -169013794, 68100795, 30265756, -12318296, -2855704, 1169521, 109964, -45224]];
    Rf_basisdens := [1, 1, 1, 844, 11816, 844, 2954, 5908, 1688, 11816, 11816, 1688, 5908, 5908, 11816, 11816, 11816, 11816, 5908, 5908];
    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_2009_a();"
function MakeCharacter_2009_a()
    N := 2009;
    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_2009_a_Hecke(Kf)
    return MakeCharacter_2009_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 := [[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 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, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0], [1, -1, 0, 0, 0, -1, 1, -1, 0, -1, -1, 0, 0, 0, -1, -1, -1, -1, 0, -1], [1, -2, 0, 0, 1, 0, -1, 0, -1, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1], [-1, -1, 0, 0, 1, 0, 0, 0, -1, -1, -1, 1, 0, 0, 0, -1, 0, 0, 1, 0], [-3, 1, 0, 0, 0, -1, -1, 2, -1, 1, 0, 0, 2, 2, 1, 1, 2, 0, 1, 1], [4, 1, 1, 0, -1, 1, 0, 1, 0, 1, 2, -1, 1, 0, 1, 2, 0, 1, 0, 1], [-1, 0, 0, -1, -1, 0, 0, 0, -1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, -1, 1, -1, 0, 0, 0, 0, 0, 1, -1, -1, -1, -1, 0, 0], [3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, -1, 0, 0, 1, 0, 1, -1, 0], [-1, -1, 0, 0, 0, 0, -1, 1, -1, 1, 0, 1, 0, 2, -1, 1, 0, 0, -2, 0], [-1, -1, -1, -2, 0, -2, -1, 1, -2, -1, -2, 0, 2, 1, -1, 0, 1, -1, 0, -1], [4, -1, 1, 0, 0, 1, 0, -1, 0, 1, 1, -1, -1, 0, -1, 0, -1, 0, -2, 0], [-1, -1, -1, 1, 0, -2, 0, 1, 0, 1, -2, 2, -1, 0, -1, 0, -1, 0, -1, -2], [3, -2, 1, 1, 1, -1, 0, -1, -1, 0, 0, -1, -1, -1, 0, -1, -2, 0, 0, -1], [3, 0, 0, -1, -1, 0, -1, 1, 0, 1, 1, 0, -1, -1, -1, 0, 0, 2, 0, -1], [0, -2, 1, -2, -1, 1, 1, -1, -1, -1, 0, 0, 0, 0, -1, -1, -1, -3, -2, 0], [2, -2, 0, 1, 0, 2, 0, -1, 0, -1, 1, -1, 0, 2, 1, 0, -1, 0, 0, 1], [3, 1, 1, -1, 0, 2, 1, -2, -1, 1, 3, -2, 2, -3, 1, 0, -2, 0, 2, -1], [0, 0, -2, 0, 0, 0, 0, 2, 2, 1, 0, 2, -2, 2, 0, 0, 0, -2, -2, 0], [1, 0, -2, 0, -1, 0, 1, -1, 0, -1, 1, -3, 1, 0, 0, 1, 1, 1, 2, 0], [2, 1, -1, 0, 1, -2, -1, 0, 0, 0, -2, 1, 0, 0, 0, 1, 1, 1, -1, 0], [0, 0, -2, -4, 1, -1, -1, -2, -1, -2, -2, 0, 0, -1, -3, 0, -1, -1, -1, -3], [1, 3, 1, 0, 0, 0, 1, 1, 2, 3, 2, 0, -1, 1, 1, 4, 1, 1, -3, 1], [1, 0, 0, 2, -1, -1, 1, -2, 0, -2, 2, -1, 2, -2, 1, 0, 0, 2, 3, -1], [-2, 1, -4, 0, -3, 0, 0, -1, 2, 0, 0, 1, -2, -1, -4, -1, -3, -1, -1, -1], [0, 0, 0, 2, 1, 1, -1, 3, -2, -1, -1, 0, 3, 2, 3, -1, 2, 1, 1, 0], [-1, -1, 1, 0, 3, -3, -1, -1, 0, 0, -3, 2, -1, 0, 0, -3, 1, -1, 1, 1], [7, 3, -1, 0, -1, 1, 0, -3, 0, -3, 0, 1, -1, -1, -1, 0, -2, 1, -1, -2], [-1, -1, -1, 0, 1, 4, 0, -2, 0, -2, -3, 2, -1, 1, 0, -2, -1, -3, -1, 1], [6, -3, 3, 0, 0, 2, 0, -1, 0, -2, 4, -1, 1, -2, 2, 1, -2, 3, -1, 0], [3, 1, 0, 2, -2, 1, 0, -1, 0, 0, 1, 1, 1, 0, 1, -1, -3, 0, -1, 1], [2, 1, 2, 1, 1, 1, -1, -1, -1, -2, 1, -1, 2, 1, 1, -1, -1, -1, 1, 0], [-2, 0, -1, 4, -3, 0, 1, -2, 0, -2, 1, 1, -1, -1, -1, -2, 0, 1, 2, 1], [-2, 1, -1, -3, 1, -2, -3, 4, 0, 2, -2, 1, -2, 2, -2, 1, 1, -1, -4, -2], [3, 0, -2, -1, 0, 1, 1, 0, 2, 1, 1, 1, -1, -2, -1, -2, -1, -1, 2, -2], [5, 1, 0, -1, 1, 0, -3, 6, 1, 1, 2, 2, -2, 2, 0, 4, 0, 2, -5, 0], [2, 3, -2, 1, -4, 2, 0, 0, 3, 0, 3, 0, 1, -2, 0, 3, -1, 3, 0, 1], [-1, 0, -5, -1, 0, -3, -2, -1, 1, -1, -4, 2, -2, 1, -5, 0, 0, 0, -3, 0], [-4, 2, 0, 0, 4, 0, -2, 5, 1, 3, -1, 0, -2, 3, 2, 2, 5, -1, -2, 3], [6, 7, -1, 2, 0, 1, -1, 3, 5, 4, 1, 3, -1, 1, 0, 4, 1, 0, -4, 2], [1, -3, 4, -2, 3, 2, -1, 1, -1, -2, -4, -1, -3, 2, 0, -2, 2, -2, 0, -1], [8, -3, -1, 4, 0, 4, 2, -5, 0, -4, -2, 0, -3, 0, -2, -3, -4, -3, -1, 2], [-7, 3, -5, 1, -2, -2, -2, 1, 0, 1, 1, 0, 4, 1, -1, 4, 3, 4, 1, 1], [7, 3, 4, 3, -1, 4, 3, -2, 1, -1, 2, -2, 2, -2, 4, -2, 0, 0, 5, 0], [2, -9, 2, 0, 3, 2, 0, -1, -1, -3, -2, 1, -4, 1, -1, -3, -1, -2, -3, -2], [-2, 2, 2, -2, 4, 1, -4, 2, -4, 2, -1, -2, 1, 1, 2, 4, 4, 2, 0, -1], [3, -1, -1, 0, -2, 0, 2, -4, 0, -1, 1, -1, -3, -1, -3, 0, -3, -2, -2, 0], [-6, 1, 1, -2, 3, -6, 0, 1, 0, 4, -3, 0, 1, 4, -2, 0, 3, -5, -1, 1], [2, -2, 3, 0, 1, -4, 1, -1, -3, -4, -3, 0, 2, -3, 1, -4, -1, 0, 2, -3], [5, 2, 3, 1, 1, 4, 1, -1, -1, -2, 3, -2, 0, -1, 4, -2, 0, 2, 5, 3], [4, -2, 4, 3, 2, -1, 1, 0, 4, 1, 0, 0, -3, 0, 1, 1, -1, -2, -4, 2], [7, 0, 0, -2, 3, 4, -2, 2, -2, -1, -2, 1, -4, 3, -1, 0, 2, -5, -1, 2], [2, -6, 2, 3, 0, -3, -1, 0, -2, -2, -1, -1, 2, -1, 3, -4, -1, 3, 2, -1], [0, 2, -3, 5, -5, 1, 3, -1, 3, -1, 2, 1, -1, 0, -3, 2, -1, 0, -3, 2], [-3, -2, 2, 2, 1, 1, 1, -2, -2, -4, -4, 1, 1, -1, 0, -7, -1, -3, 4, 2], [-7, -5, 3, -4, -1, 3, -1, -1, -6, -6, 0, -4, 4, -1, 0, -1, 1, -1, 3, 0], [0, -2, 0, 0, 2, -3, -3, 2, -2, 2, -3, -1, -1, 0, 2, -2, 2, 3, 3, -2], [4, -3, -2, -2, -3, -1, 1, 0, 2, -2, -2, 0, 0, 0, -3, 0, -2, 1, 1, -1], [3, 2, 2, 2, -3, 2, 3, 4, 3, 3, 7, 0, 4, 2, 4, 5, 0, 2, -2, 2], [4, 1, 1, 0, 0, 4, 1, -2, 0, 2, 3, -4, 2, -2, 2, 4, 0, 3, 2, 3], [4, 3, 0, -1, -2, -2, 0, 4, 3, 2, 3, 2, -2, -2, 0, 1, -3, 1, -6, -3], [1, -1, 0, 1, 2, 3, -1, 0, 1, 0, 0, -1, -2, 2, 1, -1, 0, 2, 0, 1], [7, -3, 2, -1, -1, -6, -1, 0, -1, -4, 1, -1, 3, 1, 2, 1, 0, 1, 0, -1], [-1, 6, -2, -2, -2, -2, -2, -3, -2, 0, 0, -6, 4, -3, 0, 0, 2, 2, 4, 0], [-2, -1, 1, -1, 2, 0, 1, -3, 0, 0, -1, 3, -4, -2, 0, 0, -2, 1, 1, -2], [-4, -4, -2, -3, 0, -2, -2, 1, 1, 0, -2, 1, 0, 1, -2, -2, 1, -1, -2, -1], [5, -4, 2, -1, -1, -4, 1, -1, 0, 1, -1, -4, 3, -3, 1, -1, -2, -1, 3, -3], [-7, 2, 0, -2, 3, -2, -2, 4, -1, 5, -2, 2, -2, 4, 0, -2, 4, -2, -3, 0], [-5, 7, -4, 0, -4, 5, 3, -1, 3, 2, 4, 1, 0, 0, -3, 7, -2, 2, -2, 2], [-5, -4, 0, 0, 3, -6, -1, 1, -2, -1, -5, 1, 1, 4, 0, 3, 3, -1, 2, 0], [3, 3, -2, -4, 3, -5, -5, 7, 0, 1, 0, 1, 2, 0, 3, 3, 4, 4, -3, -4], [-3, 1, -2, -2, 2, -1, -2, 2, -1, 3, -1, 0, 0, 4, 1, 3, 2, 1, 1, 3], [0, 0, 4, -2, 4, 2, 0, 2, -1, 0, -4, 1, 0, -1, 0, -4, 1, -3, 0, -2], [5, 1, 1, -3, -1, 0, 1, -2, 2, 1, 1, 0, -2, -5, -1, -2, 0, 2, 0, -1], [-1, 0, 0, 0, 0, -1, -1, 3, 2, 0, -3, 3, -5, 3, -2, 2, -2, -3, -6, 0], [2, -1, 2, 0, -3, 3, 3, -4, -1, -5, 0, 0, -1, -2, -3, -2, -5, 0, 0, 2], [3, -1, 2, -4, 2, 0, 0, 2, -1, -4, -1, 0, 0, 1, 2, -1, 1, -3, -2, 0], [4, 2, -4, -2, -1, 0, 0, 2, 4, 1, 0, 3, -2, -2, -1, 4, 0, 3, -4, -1], [4, -2, -5, -2, -3, 0, 3, -3, 1, 2, -3, 0, 0, -1, -5, -4, -1, -4, 2, -3], [-6, 2, 0, 1, -7, -5, 1, 2, -1, -1, 7, -3, 2, 1, -1, 6, 1, 3, 3, 5], [-1, -1, -2, -2, -2, -3, -1, -1, 3, 0, 2, -1, -2, -2, -3, 5, -4, 2, -6, -2], [6, -4, 0, 2, -3, 1, 0, 1, 4, 3, 0, 0, -1, -4, -3, -3, -4, 0, -2, -2], [-5, -1, 3, 1, 1, 4, 3, 3, 2, 3, 3, -2, 4, 1, 3, 3, 0, 3, 2, 2], [-7, -1, -4, -4, -1, 1, 0, 7, 1, 2, 1, 0, 1, 4, 2, 5, 4, 2, -1, -1], [-2, 1, 1, 0, -3, -1, 1, 6, -2, -2, 6, -3, 4, 2, 2, 5, 5, 3, 0, 4], [-5, 6, 1, 0, -2, 0, -2, 6, -3, 8, 4, -5, 5, 3, 1, 9, 4, 3, 2, 1], [0, 4, 0, 5, -1, 1, 3, 3, 7, 4, 4, 0, -2, -1, 3, 4, 1, 1, -1, 2], [0, 2, -2, 6, -5, 6, 2, -2, 4, -3, 2, 1, -2, -1, -2, 2, -5, 1, -1, 2], [2, 1, 0, 0, 0, -2, 3, -7, -1, -10, 0, -1, 1, -4, 0, -9, -5, -2, 7, -2], [-2, -3, 0, -8, -1, -3, -1, 1, 1, 0, -2, -3, -5, 2, -5, 6, 3, 0, -3, -1], [0, 0, 0, -3, 1, 7, -2, 1, -5, -5, 1, -2, 2, 0, 1, -1, 1, -1, 1, -3], [-2, -3, 2, 4, 1, 1, -2, 0, -2, 1, -4, 5, -7, 4, -1, 2, 0, -2, -4, 5], [1, -7, -1, -4, -2, -6, 0, -3, 0, -2, -2, -4, -1, -1, -6, 0, -2, 0, -2, -3], [2, -2, -2, 5, 2, 4, 2, 0, 3, -1, -3, 6, -8, 5, -2, -4, -3, -7, -6, 0], [5, -6, 2, -3, -4, 3, 5, -7, 4, -7, -2, -1, -3, -5, -5, -4, -5, -3, -2, -4], [6, 3, -5, 1, -1, -3, -1, -2, 2, -1, 0, -2, 1, 1, -1, -2, 0, 4, 4, -3], [-6, -7, 9, 6, 4, 1, -1, 1, -1, 2, -1, 1, -1, 3, 2, -2, 3, -2, -2, 4], [-10, -4, 0, -4, -2, 0, -2, -1, -7, -5, -3, -2, 4, -1, -2, -2, 3, 1, 6, 1], [1, -4, -2, 1, -1, -3, 0, -2, 0, -4, -1, -2, 1, -1, -2, 0, 1, -1, -1, -2], [-5, 4, -4, -7, 2, 3, 0, -3, -2, 0, 0, 1, -3, 2, -3, 2, 1, -2, 0, -1], [3, -2, -2, -2, -1, 0, -1, 6, 3, -1, 1, 2, -4, 2, 0, 5, 0, 0, -6, 0], [3, 2, 0, 3, -2, 3, 5, -6, 5, -1, 4, -1, -1, -6, -1, 2, -8, 4, -1, 1], [4, 2, -2, 1, 0, 4, -2, -1, 1, 0, 2, 5, -4, 1, -2, 0, -3, -5, -6, 7], [3, -5, 3, -5, -4, 2, 0, -2, -3, -4, 3, -3, 3, -1, -1, 6, -2, 1, 1, 1], [9, -6, -6, 2, -2, -1, 0, -7, -2, -4, -4, 4, 0, -3, -6, -9, -8, -3, -2, -4], [-1, -3, 2, -2, 1, 6, -2, -6, -1, -1, -1, -3, 0, -4, 0, -3, -2, 0, 1, 0], [3, 3, -6, 0, 1, 5, 1, -4, 4, -4, 1, 2, -4, 4, -1, 3, 0, -6, -7, 5], [19, 0, 2, 3, 3, 1, 0, -9, 2, -3, 0, -3, 2, -7, 1, -5, -4, 2, 6, -1], [-1, -3, 0, 0, 0, 2, -2, -2, 0, -6, 1, -3, 0, 0, 2, -3, 0, 6, 8, -2], [4, 7, 0, -5, -6, -7, 3, -2, 1, -2, 6, -2, 1, -6, -2, 4, -5, 5, -2, -5], [7, -4, -3, 2, -2, 0, -1, -3, -2, 3, -2, -1, -1, -1, -3, -7, -3, 3, 5, -3], [3, 0, -2, 1, 0, 2, 3, -9, 2, -2, -5, 1, -1, -5, -2, -10, -4, -5, 3, -2], [4, -4, 3, 4, 0, -1, 4, -1, 1, 0, 0, -3, 3, 3, 0, 3, 1, 1, 3, 0], [10, 4, 6, 3, 1, 11, 1, 2, 2, 3, 5, -1, 1, -2, 7, -1, 0, 3, 1, 1], [4, -5, -3, -1, -6, 8, 0, -2, 2, -4, 0, 0, -3, -4, 0, -6, -2, 4, 8, -1], [-6, 3, -6, -6, -4, -7, -2, 1, -3, -6, -3, -1, -1, 4, -4, 4, -1, -4, -5, 0], [-2, 3, -5, -1, -4, -5, 1, 3, 8, 10, 3, 3, -6, -2, -2, 11, -2, 6, -8, 0], [-7, -2, 0, -1, -2, 5, -1, -1, -2, -3, -4, 1, -4, 3, -1, 0, -1, -3, 0, 4], [11, 0, -3, -5, -1, 1, 2, 3, 4, 4, 1, 5, -5, -3, -2, 4, -1, 0, -7, -2], [5, 8, -2, 5, -6, 1, 7, -1, 5, 1, 8, -1, 2, -6, 3, 4, -4, 0, 2, 0], [-5, -9, -3, 0, 2, 0, -5, 3, -4, -1, 0, 0, 5, 3, 1, 4, 5, 3, -1, 1], [4, 7, 4, 0, -5, 9, 1, -5, 1, 5, 6, -7, -1, -4, 3, 2, 1, 2, 1, 5], [17, -7, -1, 2, -1, -3, 1, -4, 1, -3, 0, 3, -5, -1, -4, -5, -4, -2, -1, 2], [-8, -7, -1, -1, 3, -6, -2, 0, -4, -4, -7, 2, 3, 0, 0, -4, 1, -3, -1, -6], [4, -3, 0, 2, -2, 9, 1, -6, -2, -3, 7, -5, -2, -3, -1, 3, -3, 3, -2, 5], [6, -1, -4, -5, -3, -10, -2, -3, -2, -7, -5, -4, 2, -7, -6, -3, 1, 3, 3, -7], [0, 6, -1, -8, 1, -3, -2, 6, -3, 5, 3, -4, 5, -4, 3, 8, 1, 3, -2, -5], [0, 1, 1, -2, -1, -8, -2, 3, -2, 1, -6, 1, 1, 2, -3, -6, 4, 1, 9, -1], [11, 1, -1, -4, -2, 5, 2, -2, 2, -3, 1, -1, 1, -1, -1, -4, -3, -6, -4, 0], [12, 2, 0, -3, 1, 2, 1, -4, 2, -5, 1, 2, -1, -4, 1, -4, -4, -2, -4, -5], [-2, 4, 5, 1, 0, -1, -2, 4, 0, 1, 0, -3, 1, -1, 0, 1, 2, 2, 0, -2], [2, 7, -5, -1, -1, 0, 2, 2, 2, 3, 4, 5, -1, -2, 1, 1, -2, 0, -5, 0], [-12, 3, -5, 1, -1, 12, -3, 3, 2, 2, 2, 5, 2, 2, 0, 6, 1, 2, -2, 1], [-2, -1, -3, -1, -1, -4, 5, 3, 4, -1, 0, -3, 1, 3, 3, 2, 6, 0, 5, 4], [9, 1, -3, -4, -3, 1, 0, 9, 4, 3, 4, 1, -3, 1, -1, 5, 4, 2, -5, 0], [0, -5, 4, 3, -4, 5, 3, 6, -2, 2, 4, 0, 2, 1, 3, 7, 3, 3, 0, 1], [3, 4, -6, 5, -1, 9, 2, -7, 6, 2, -2, 6, -5, -2, -2, -2, -3, 1, -1, 0], [-11, 3, -3, 1, 1, -5, -5, 4, -2, -2, -6, 2, 5, 5, -2, -2, 3, 2, 3, -1], [9, -5, -3, -2, 2, 1, 2, -2, 2, -4, -2, 2, -3, -3, 0, -7, -4, 1, 4, 0], [10, 9, -5, -2, 0, -1, -2, 6, 6, 5, 5, 3, -7, -1, 0, 2, 0, -2, -7, 0], [-4, -7, 3, 4, 0, -2, 2, 0, 0, 4, 0, 2, 3, 6, -2, 1, -2, -1, 0, 5], [1, -5, -3, -5, 8, 3, -4, -1, 0, -2, -6, 4, -7, 1, -2, -6, 2, 0, -1, -3], [6, -1, -1, 2, 7, -5, -6, 5, -2, -1, -4, 3, 1, 4, 1, -6, 6, 1, 2, 1], [-8, 3, -6, 2, 3, -8, -6, 5, -1, 10, -2, 2, 4, 4, -2, 3, 2, 2, -5, -3], [4, 2, -4, -1, 1, -4, 0, -7, 8, 7, 0, 3, -4, -3, -8, 2, -5, -1, -11, 1], [-8, -9, -1, 6, -2, -9, 4, -2, -2, 2, -4, -2, 5, 0, 0, -3, 0, -1, 6, 1], [5, -9, -5, 3, 1, 6, 0, -9, 0, -4, -2, 3, 0, -4, 0, -9, -9, -4, 0, 1], [6, 2, -3, 4, 7, -6, -4, -6, 2, 3, -5, 0, -4, -1, -3, -2, -2, 2, -7, -1], [8, 9, -4, 4, 0, 3, 1, 0, 4, 3, -1, -3, 4, 1, 1, -5, 1, -1, 8, 5], [-6, 8, 0, -1, 1, 5, 3, -1, 1, 0, 6, -2, 0, -1, 5, 0, 3, -3, 3, 0], [11, 15, -5, -2, 2, -3, -3, 1, 4, 1, 3, 2, 0, -8, -1, 4, 1, 9, -5, -4], [-4, 7, -2, -5, -2, 0, -6, 3, -3, 1, 1, 1, -1, 2, -4, 5, 1, -3, -5, 2], [1, 5, -4, -2, -2, 1, 3, -7, 0, -7, 5, -8, 2, -8, 3, -7, -3, -1, 12, -1], [5, 11, -1, 0, 2, -4, -4, 0, 2, 9, 5, -1, 1, -3, 3, 4, 1, 6, -1, 0], [2, 8, 2, 8, 7, 0, 0, 2, 4, 3, -4, 5, -2, 3, 6, -2, 5, -3, 5, 6], [-16, 7, -6, 0, 0, -11, 3, -4, 6, 9, -3, 3, -2, 1, -3, 3, -1, -5, -8, 1], [6, 4, -5, -6, -1, -2, 0, 3, 1, 5, 3, -5, 2, -4, 3, 4, -1, -1, -1, -2], [-3, -4, 0, 0, -3, -4, 1, -4, -3, 5, -1, 0, 0, -4, 0, -3, -2, 0, 6, 0], [-1, 19, -5, 0, -5, -7, 2, 2, 12, 8, 6, 2, -1, -8, 0, 8, -1, 9, 3, 0], [-11, 0, -6, -6, 3, -1, -2, -3, -2, -1, -8, 5, 0, -1, -3, -9, 2, -8, -1, -1], [-12, 8, 4, 0, -4, 8, 3, -1, 2, 0, 3, -2, 4, -2, 6, -1, 2, -2, 8, 5], [-5, -3, -7, 0, -1, 5, -1, 5, 2, 3, 0, 2, 0, 1, 0, -3, 3, -1, 5, 4], [-9, 7, -10, 2, 0, -8, -2, 5, 0, -1, -2, 2, 0, 2, 2, 3, 8, 4, 8, -1], [-11, -8, 5, 0, 6, 4, -2, 1, -4, -6, -8, 0, 4, 7, 3, -9, 7, -2, 8, 3], [4, -5, -5, -3, -1, 0, 0, -3, -4, -1, -4, -4, -1, 2, -3, -1, 4, 0, 2, -1], [1, 5, -3, -1, 6, -4, -2, 2, 0, 9, -7, 5, -3, 3, -3, -2, 3, -8, -6, 4], [3, 5, 2, 5, -5, -2, 4, 0, 5, 8, 3, -1, 5, -2, -2, 4, 2, 1, 0, 2], [-3, 0, -2, -3, -3, 10, -2, 0, -1, -2, -1, 1, -2, -6, -2, -2, 0, 4, -1, -2], [11, 3, 0, -4, 2, -5, 2, 1, 2, 0, -1, -1, 3, 4, 1, -5, -1, -6, 3, -3], [-9, 0, -2, 1, 0, -17, -2, -1, 0, -3, -2, -1, -1, 2, -5, -2, -1, 2, 2, -3], [-2, 3, 6, 2, 6, -2, -1, 6, 3, 12, -1, 1, 1, 7, 4, 5, 7, -2, -9, 8], [-16, -14, 0, -7, 4, -7, -1, 0, -2, -4, -13, 5, -4, 9, -7, -10, -1, -9, -2, -3], [-4, 3, -13, -5, -1, -2, -4, 7, 1, -1, -4, -1, -1, 5, -3, 3, 11, 1, 0, 1], [-13, -5, -1, -4, 7, -5, -1, 3, -4, -4, -6, 6, -4, 3, -2, -1, 3, -3, -1, -4], [2, 1, 6, 7, -2, 5, 1, 3, 2, 3, 5, -5, 6, 7, 7, 3, 3, 1, 2, 4], [0, -3, -1, 1, -6, 2, 1, -4, -2, -3, 8, -2, 6, -4, 3, 2, -5, -1, 1, -1], [1, -2, 4, 1, 2, -9, -2, -4, -2, 7, 3, 1, 3, -1, 1, 3, -5, 1, 0, -1], [-1, 8, 0, -3, 2, 3, 3, 8, 4, -2, 3, 7, 5, 3, 2, 4, 2, -3, -4, -3], [6, 8, 4, -5, 1, 9, -7, 4, 1, 9, 5, -3, 0, -3, 3, 4, 5, 11, -1, 1], [-5, 3, -7, 4, -10, 4, 3, -5, 2, -1, 0, -2, 0, 1, -7, -2, -7, -3, 1, 5], [-24, -3, 6, -5, 5, -2, -6, 8, -4, 6, -8, -1, 4, 5, 4, 1, 11, -3, 5, 2], [1, -10, -8, 5, -5, 0, -3, -5, 2, 1, 3, 2, -7, 1, -5, 4, -6, 0, -11, 5], [1, -6, 0, -10, 1, 1, 1, 5, 2, 2, -2, 4, -6, 3, 2, 2, 1, 2, -5, -5], [6, -7, 4, -5, -4, -2, 7, -8, -8, -12, -5, -3, 3, -4, -2, -5, -6, -3, 8, -2], [-10, 7, 1, 12, -3, 0, 1, 4, 4, 11, 5, -2, 8, 4, 3, 13, 6, 7, 5, 10], [29, -14, -2, -2, 4, 12, 0, -9, 0, -9, -3, 0, -8, -3, -7, -12, -11, -6, -7, -8], [-10, -15, 2, -8, 4, -7, -3, 5, -8, 0, -2, -6, 2, 3, 1, -1, 5, -1, 6, -4], [0, 3, 5, -4, 0, -3, 8, -1, -2, 3, 5, -7, 5, 0, 5, -3, -1, 1, 10, -6], [-16, -1, 4, 0, 1, -4, 2, 3, 1, 8, 0, -2, 6, 4, 0, 3, 4, 0, 5, 3], [4, -4, -4, -2, -6, -5, 3, -7, 4, -2, -1, -1, -5, -2, -4, 2, -4, -1, 0, 1], [-2, -7, -4, 5, -4, 1, 3, 0, 1, -2, 4, 2, 3, 4, -2, 2, 1, -1, 4, 5], [-6, 6, -3, -3, 6, -5, -2, -5, -1, -2, -6, 1, -3, 1, -2, -6, -3, -12, -1, -2], [-5, 8, -3, -4, -2, 4, 6, -6, 1, 10, 2, -1, 1, -3, 3, -1, -4, -3, 6, 1], [-4, 2, 9, -2, 1, -6, 1, 1, 5, 5, 1, 0, 2, -5, 3, -2, 3, 0, 4, 1], [-15, 3, 10, 1, 2, 4, 4, 1, -5, -2, 3, 0, 10, 3, 8, -3, 3, -5, 4, -1], [15, -14, -2, 0, 1, -7, 0, -5, -4, -11, -10, 1, 2, -5, -1, -12, 0, 3, 9, -7], [-14, 3, -13, 1, -3, -8, 1, -5, 3, 4, -10, 6, -6, -1, -11, -5, -7, -8, -3, -3], [9, -9, 5, 3, 0, 2, 2, -2, 1, 0, 3, 3, -5, -3, -5, 0, -6, -1, -13, 1], [0, 10, 2, 9, 7, -7, -2, 9, 3, 7, -1, 8, 0, 6, 7, -3, 7, 1, 1, 7], [-5, 4, 7, 2, 4, -6, -4, 8, 1, 3, 2, 3, -1, 7, 3, 1, 2, 1, -2, 1], [12, 10, 4, -1, 2, 3, 1, 3, 0, 9, 12, -6, 7, 2, 9, 14, 9, 3, -3, 7], [9, -14, 4, -2, 5, -11, -3, 1, -2, 4, 0, 0, -3, -1, -2, 4, 1, 0, -7, 3], [-3, -1, 6, -1, -2, 9, 1, -2, -5, -4, 6, -7, 10, 0, 3, -1, -2, -6, 4, 1], [-2, -12, -2, -2, 7, -10, -1, -1, -2, 1, -11, 4, -5, 6, -3, -4, 6, 0, 5, 4], [-1, 7, 2, -3, 2, 0, -9, 4, -3, -1, -1, 4, -5, -1, -1, 4, 4, 7, 1, -2], [3, 4, -2, -4, 3, -2, -6, -4, 3, 8, 3, -1, -9, -2, -4, 5, -4, 1, -11, 0], [19, -1, 8, 0, -4, 4, 0, 6, 4, 4, 7, 3, -6, 2, 4, 5, -4, 4, -12, -2], [1, -10, 2, 5, 0, -8, 0, -5, -4, -8, -4, 2, 4, 1, -8, -4, -4, 0, -4, -2], [-5, 13, -10, 0, -4, -9, 2, 6, 5, 4, -1, 8, -4, 0, -5, 3, 0, -1, 1, 1], [8, -9, 3, -8, 1, -4, -6, 8, -4, 0, 1, -6, 9, -4, 4, 1, 5, 4, 4, -8], [5, -2, 1, -1, 0, 3, -4, 9, -9, -1, 0, -6, 8, 1, 5, 0, 12, 4, 7, -2], [-3, -2, 6, -1, 1, -6, 2, 2, -2, -4, 5, -8, 2, -1, 6, -5, -1, 2, 7, -2], [9, -9, 8, 0, 4, -5, -1, 2, -5, -11, -6, 7, -2, 0, -3, -7, -2, -6, -8, -1], [-1, -2, -10, -5, -2, -9, -3, 3, 0, 3, -6, 8, -4, -1, -5, -2, 1, -1, -8, -6], [3, -6, 6, 0, -1, 10, 1, 3, 4, 4, 9, 2, -7, 2, 6, 12, 3, 8, 3, 12], [15, 6, 2, 5, 0, -9, -2, -1, 0, 5, 7, 1, 2, 1, 3, 0, 1, 2, 0, 2], [-9, -3, -9, 3, -3, -2, -2, 0, 0, -6, -3, 7, -3, 1, -4, 5, 1, 2, -2, 5], [6, -4, 0, 2, 1, -5, 2, -1, 4, 1, 2, 5, -7, 2, 1, -3, 0, -6, -11, 2], [-13, 9, -7, 1, 1, 11, -3, 11, 4, 11, -1, 14, -8, 9, -1, 4, 6, -2, -9, 6], [15, -14, 2, 2, 3, -1, -4, -9, -4, -10, -2, -2, -2, -6, 0, -11, -5, 0, 1, 2], [-14, 7, 7, -3, 1, 2, -4, 12, -8, 9, -2, -3, 9, -2, 7, -2, 8, 5, 5, -6], [-14, 4, -8, 2, -2, -2, 2, -3, 1, 1, -7, 2, 0, -1, -6, 2, 1, -3, -6, -3], [10, -2, -6, -5, -3, 5, 1, -1, 5, 0, 3, -3, 1, -4, -3, 3, 3, 6, 4, -5], [-10, 9, -2, 6, -2, -9, 1, 6, 2, 13, 9, -1, 8, 3, 3, 9, 5, 11, 0, 3], [-16, 3, 0, 7, 7, 11, -5, 5, -1, 2, -7, 5, -8, 5, 1, 3, 3, 3, -3, -2], [-3, -8, 4, -3, -2, 15, -1, -4, -3, -13, -2, 1, 1, 2, 1, -6, -4, 0, -1, 5], [3, -5, 6, 3, -6, -1, 3, -10, -5, -4, 8, -9, 4, -10, -5, -1, -6, 4, 0, -3], [10, -5, -4, 3, 2, -14, 2, -8, -7, -11, -8, -3, 4, -5, -2, -15, -3, -7, 10, -6], [-14, 7, 4, -5, 0, 10, -2, 17, 1, 14, 5, -3, 1, 8, 8, 19, 9, 3, 3, 6], [10, -4, 1, 9, 0, 5, 7, 0, 4, -4, 4, 4, 0, -3, 4, -5, 0, -5, -1, 2], [11, 8, 4, -2, -1, 0, 0, -4, 5, 6, 3, -5, -5, -8, 2, 1, -6, 1, -1, 0], [13, 12, 5, 5, -7, 6, 2, 7, 3, 6, 15, -3, 8, -4, 10, 5, -6, 4, -2, 2], [-11, 0, 2, -8, -1, 12, 2, -5, -5, -2, 1, -5, 2, -2, 0, 4, 0, -2, 5, 3], [23, -6, 1, -5, -7, -7, 3, -7, -4, -7, -3, 0, -6, -5, -8, -8, -13, -5, -8, -12], [-3, 0, 6, 2, -3, 5, 4, -5, 6, 2, 0, -1, 0, -2, 0, -3, -5, -2, 2, 2], [5, -4, -6, 6, -1, 2, 1, -3, 6, 9, -3, 9, -5, 0, -6, 1, -9, -7, -12, -2], [-10, -9, -6, -7, -2, -2, 0, -1, -3, -7, -11, 1, 1, 2, -4, -5, 1, -3, 7, -8], [-11, 11, 0, 0, 1, 2, -6, 8, -3, 3, -3, -3, 8, -2, 8, -1, 10, 10, 13, 0], [-12, 4, 8, 0, -2, 11, -2, 0, -4, -4, 2, 2, 0, 2, 6, 4, 4, 0, 0, 12], [-9, -10, -2, 0, 9, 2, -2, -3, -4, -9, -12, 7, 0, 3, -1, -14, 8, -5, 4, 7], [11, -3, -1, 8, 2, -3, 2, 2, 4, 0, 0, 0, -1, 5, -4, 7, 2, -3, -1, 4], [9, 12, 2, -5, 0, 1, 2, -3, 4, 8, 6, 1, 3, -6, 3, 8, -1, 8, -2, -5], [12, 7, 0, 5, -8, 5, -1, -4, 2, 6, 10, -6, -2, -5, 1, 9, -1, 5, -6, 1], [1, -12, -6, -9, 3, -6, -2, -2, 0, -2, -7, 6, -12, 1, -8, -8, -5, -1, -5, -6], [-6, -4, 4, 4, -1, -1, -3, 2, -5, 4, 4, -6, 6, -3, 5, 2, 1, 9, 3, -5], [-20, 19, 5, -1, 3, 6, -3, 10, -4, 3, 5, 2, 8, 2, 7, 14, 10, 8, 2, 8], [2, -7, 2, -2, 7, -6, 2, -12, -2, 9, -7, 0, -4, -5, -8, -11, -9, -9, -9, -10], [4, 5, -7, 7, -2, 16, 1, 1, 4, -2, -3, 7, -4, -2, 2, 0, 4, 3, 2, -2], [-15, -4, -5, -1, 1, -11, 2, -5, -6, -8, -7, 1, 3, 7, -4, -2, -3, -6, -3, -2], [3, -4, -10, 9, -1, -4, 2, -15, -7, -15, 2, 4, 8, -6, -4, -12, -6, 2, 3, -3], [-1, 5, -11, 2, -2, -2, -4, -4, 0, -3, -3, 4, 1, -3, -7, 1, -1, 1, -9, -2], [-8, 7, 11, 3, 9, 4, -5, 6, 0, 10, -4, 5, -7, 9, 1, -3, 4, -5, -5, 3], [-6, 5, -1, 9, -9, 0, 9, -14, 6, 0, 6, -5, 3, -13, -1, -3, -10, -1, 1, 5], [-17, 0, 8, -1, -10, 7, 0, 4, -9, -2, 10, -10, 8, 6, 3, 12, 8, 3, 6, 17], [-1, -12, 4, 4, -3, 5, 5, 4, 8, 2, 4, -1, -8, 0, -3, 2, -4, 8, -3, 1], [-7, -6, -5, -1, -1, 1, 2, 4, 3, -4, -7, 0, -1, -2, -4, -5, -2, -2, 5, 1], [0, 2, -6, 5, 1, 1, 5, -3, 5, 2, 1, -5, 1, -4, -3, 5, -1, 0, -2, 1], [5, -1, 4, 3, -5, -2, 7, -1, 4, -7, 10, -3, 5, -8, 6, -4, -3, 9, 15, -2], [1, 14, -2, 3, 0, 5, 7, -5, 0, -5, 2, -2, 5, -7, 3, -8, 1, 1, 11, -2], [11, 1, 3, 0, -3, -4, 3, 1, 6, 5, 7, -6, 2, -1, -1, 5, -2, 3, 0, -6], [-6, -5, -8, -2, -1, 4, 2, -1, 1, 5, -10, 3, -5, 3, -5, -1, -3, -6, -9, -4], [-1, -5, -3, 10, 3, 5, -2, -3, 0, 1, -2, 3, -3, 3, -3, -1, -4, 2, -5, -4], [0, 5, 6, 7, -2, -1, 11, 0, 0, 0, 10, -10, 8, 3, 5, 11, 1, -1, 0, 7], [9, 1, 3, -10, -3, 1, -4, 2, 6, 4, 12, -9, -7, -4, 0, 8, -5, 5, 0, 0], [17, -12, 8, -2, -7, 6, 1, -3, -4, -8, 3, -9, 3, 0, 4, 1, -7, 3, 0, -4], [2, -4, 2, 0, 10, -16, 0, 2, -7, -2, -8, 7, 10, 9, 2, -4, 7, -7, -2, -4], [8, -1, 5, 8, -4, 9, -1, 4, 10, -1, 2, 8, -10, 4, 1, 4, -3, 1, -16, 3], [-10, -10, -4, 5, -6, -6, 10, -7, 3, -2, -2, 5, -2, 3, -6, -12, -9, -11, 0, -3], [-15, 10, 3, 8, -10, 18, 3, -3, 0, -1, 10, -11, 5, -1, 5, 5, -5, 9, 9, 5], [3, 12, 2, 8, 5, 11, 1, -1, 2, 6, 2, 1, 11, 3, 6, 3, 3, -1, 9, 3], [-1, 9, -6, 11, -2, -1, 1, -4, 5, 3, 0, 0, 1, 2, 3, -4, -2, 5, 7, -1], [-15, 3, 0, 3, -7, 1, 4, 7, 5, -2, 5, 6, 0, 4, 2, 6, 4, 3, -1, 11], [-4, -5, -2, -11, 1, -1, -10, 13, -9, -3, -9, 6, 1, 8, -3, 8, 3, -7, -12, 1], [3, 9, -5, 6, 7, 7, -1, 3, 4, 7, 10, 2, -2, 3, 8, -1, 3, -1, -1, 8], [-6, 5, -1, -4, -4, 3, -4, -1, 5, 1, 3, 2, -8, -9, 2, 0, -5, 3, 1, -8], [0, 1, 12, -3, -1, 8, 2, -2, -5, 7, 5, -9, 3, -8, 7, -8, -1, 5, 13, -10], [-3, 2, -4, -4, 4, -16, 0, 5, -4, 1, -5, 3, 0, -1, -1, -10, -1, -4, -1, -12], [-8, -17, 3, -6, -6, 5, 2, 7, 4, -2, 4, -2, -3, 8, 2, 7, 2, -5, -7, 8], [11, 5, -5, 10, -1, 3, -2, -2, 9, 2, 2, 6, -6, -3, -4, -7, -10, 7, -2, -6], [-2, 5, -2, 3, 2, 4, -2, 1, -1, 6, 8, 1, 0, -3, 4, -1, -1, 1, 1, 1], [-20, -9, -6, 5, -3, 0, 7, -4, -3, -7, -5, 1, 7, 5, -7, 5, 1, -3, 5, 3], [-15, -8, -2, 8, -1, 0, 6, -17, 2, -4, -7, 4, -4, -3, -2, -12, -3, -3, 13, 1], [35, -10, 2, 1, 6, -5, 0, -7, 2, 0, 4, 1, 1, 0, -1, -6, -7, 2, -4, -1], [-9, 15, 8, -1, -7, -4, -6, 13, -2, 8, 18, -12, 12, 4, 10, 16, 9, 14, -2, 4], [6, -9, -4, -5, 6, -9, -6, 3, -13, 0, -5, 0, 4, 2, 0, -7, 5, 1, 0, 0], [-7, -1, 7, -3, 0, 4, 2, 1, -2, 0, -4, -4, -1, -1, 4, 0, 2, -4, 6, 1], [1, 5, 4, -1, 0, 2, 5, -2, 1, 5, 7, -6, -1, 1, 1, 8, 4, -1, -5, 6], [-12, 4, 4, 5, 10, 7, -1, -3, -6, 1, -2, 2, 7, 0, 5, -1, 3, -4, 4, -3], [-15, -8, 0, 6, -1, -11, 4, -5, 2, 2, -4, 9, -6, -2, 0, -5, 1, -4, -2, 2], [3, 0, 5, 9, 1, 7, -3, -3, -4, -9, 3, -6, 8, -7, 4, -8, 1, 9, 6, 4], [2, -9, -3, 1, -3, 2, -6, 7, -3, -5, -6, 7, 1, 5, -5, 1, 1, 3, -4, -3], [-15, -2, -4, 8, 7, -9, 3, -4, -4, -4, -6, 3, 7, -1, 2, -14, 1, -12, 4, -2], [-4, 11, 0, 1, 4, -6, 5, -8, -2, -4, -1, 1, -3, -2, 0, -1, -4, -11, 2, 4], [-3, -5, 5, -4, -3, -5, -3, -3, -6, -7, 6, -10, 10, -7, 4, -3, 1, 9, 5, -6], [-7, -8, 12, 4, 8, 1, 0, 3, 0, -2, -4, 0, -10, 1, 4, -2, -2, 2, -7, -4], [-7, 5, 6, -4, -5, 4, 0, 7, -9, -6, 6, -13, 3, 3, 6, 7, 12, 0, 10, 4], [-9, -3, 0, 0, 3, -1, -1, 12, -4, 0, -7, 12, -2, 12, 1, 7, 2, -6, -9, 5], [9, 1, 2, -3, 1, 16, -6, 11, -9, -4, 4, -3, 11, -3, 10, -2, 8, 9, 16, -6], [10, -9, -4, -11, -3, -4, -4, -8, -6, -2, -14, -1, 0, -7, -6, -13, -7, 3, 7, -12], [-3, 0, -4, -3, -5, -11, 3, -3, 6, -2, -2, -1, -2, 2, -9, 10, 0, -8, -11, 2], [-18, 18, -2, -4, -7, -7, -1, 4, 3, 4, 0, -2, 0, 1, -3, 14, 3, 3, 3, 1], [-12, 13, -7, -11, 4, -8, 0, 0, 6, 12, -2, 4, -3, -4, -6, -6, -2, -4, 6, -9], [-7, 2, -4, -1, -9, -10, -2, 5, 0, 12, 5, -13, 6, -1, 2, 8, 9, 11, 6, 3], [-17, -1, -8, 3, 2, 8, -2, 3, -3, 1, -6, -3, 0, 1, 3, -11, 2, -1, 14, -1], [-5, 16, -3, 6, -8, 4, 2, -4, 3, -2, 8, -1, 5, -7, 1, 3, -2, 1, -2, 1], [13, -1, -9, 4, 3, 4, 0, -2, -2, -10, -3, 5, -5, -3, -4, -2, 3, -3, 0, 5], [-3, 0, 2, -4, -1, 10, 0, 2, -1, -7, 8, -2, -6, -2, -2, 6, -6, -6, -9, 6], [-11, -14, 8, 0, 6, 7, -5, -4, -11, -1, -6, 1, 10, -6, 7, -10, 0, 2, 10, -5], [-7, -3, 0, 5, 8, -20, 0, 2, 4, 1, -8, 4, -2, 0, 2, -1, 6, 2, 6, 2], [12, -1, 11, 12, 6, 0, 6, -2, -4, 3, 9, -1, 5, 0, 13, -8, 1, 0, 2, 3], [-12, 17, -1, -2, 4, 5, -9, 8, -4, -1, -6, 2, 0, 4, 5, 3, 15, 4, 2, 11], [1, -2, -10, -3, 0, 8, -6, 2, -2, 3, 3, 5, 0, 4, -2, -4, 4, 0, -8, 4], [7, -5, -1, -2, 3, -1, 1, 0, -1, 1, 2, -9, 5, 5, 4, 1, 0, 4, -1, 0], [-10, 0, -3, -3, 8, -3, 0, -2, -9, -4, -7, 1, 7, 4, -4, 0, -1, -4, 3, -4], [-5, -10, 0, 3, -4, 5, -2, 2, -8, 1, 5, -5, 6, 7, 3, 8, 7, 6, 0, 13], [-3, 6, -6, 3, 2, 3, 3, -8, -5, -6, -3, 6, 2, 0, -3, -2, -2, -6, 4, -5], [-1, 8, -4, 10, 1, 15, 4, -3, 14, 9, 8, -1, -10, -3, 5, 4, -4, 1, -5, 9], [-1, 5, 3, -12, 6, -12, 0, 1, 4, 3, -2, 1, -1, -4, 1, -1, -3, -7, 2, -6], [2, -3, 4, 0, -1, 4, 4, 3, -4, -8, 2, 9, 2, 3, 6, 1, -1, -1, -7, 5], [-7, 4, 3, -8, 4, 4, -6, -2, -5, -7, -6, -3, 9, -1, 1, -3, 2, 1, 4, 1], [-15, -14, -5, -2, 3, -4, -6, -3, -13, -8, -11, 0, 7, 6, -4, -6, 0, -3, 5, -4], [-3, -5, -8, -7, 1, 2, -3, -1, 8, 3, -14, 9, -11, 2, -6, -6, -1, -7, -1, 0], [13, 2, 0, -3, 3, 5, -1, -3, 2, -6, -4, -4, 2, -1, 6, -5, -3, 1, 3, -5], [8, 4, 6, 3, 4, 10, -8, 0, -3, 1, 9, 0, -6, -5, 0, 0, -3, 9, -4, -2], [-6, 4, 1, 8, 4, -1, -2, -5, 3, -4, -6, 9, -3, 5, 0, -5, -1, -1, -11, 6], [-18, -1, -6, 5, -9, -10, 0, -6, -3, -4, 2, -2, 0, 4, -10, 9, -4, -2, -7, 8], [17, -8, 4, -5, 7, -10, -6, -1, -6, -1, -10, 3, -4, -1, -3, -7, 0, -8, -11, -5], [-26, 26, -3, -5, 0, -5, -2, 16, 6, 7, 0, -5, 1, 5, 6, 15, 16, 10, 8, 0], [-3, -12, 0, -2, 3, -9, -5, 6, -6, -8, -6, -3, 0, 2, 1, -10, 4, 0, -1, 5], [3, -3, 10, -11, -3, -4, -1, -10, -9, -11, -6, -4, 2, -6, -8, -12, -6, -2, 9, -6], [0, -3, 11, -4, 9, -1, 2, 4, -2, 1, 8, -4, 4, 4, 9, -2, 6, -1, 8, 6], [-15, 5, -5, 7, 8, -8, -8, 0, -6, -3, -5, 7, 5, 5, 1, -7, 7, 1, 7, 2], [7, -2, 14, 11, 2, 5, 7, -2, -1, -3, 6, -3, 5, -2, 9, 0, 0, 0, 9, 1], [1, -2, 14, 0, 2, -8, 2, -1, -4, 6, 8, -12, 4, -3, 2, 1, 2, 3, -1, 0], [-19, -5, -5, -3, 3, 5, 0, -2, -8, -4, -16, 1, -3, 6, -2, -7, 11, -2, 10, 4], [14, -15, 2, -6, -2, -4, -4, -9, -1, -15, -8, 5, 0, -1, -8, -7, -7, -5, -12, -3], [-8, 8, -5, -3, 4, -5, 4, 5, 7, 8, 0, 6, 2, 7, -4, -1, 3, -11, -14, -2], [13, 2, -3, -2, -5, -1, 4, -2, 3, 2, 9, -3, -2, 2, -2, 4, -2, -3, -4, -1], [2, -3, 6, 4, -6, -14, 4, 7, 5, 7, 4, 1, -4, 1, 0, 3, -5, 9, -8, -1], [-13, 3, -6, -9, -3, -7, -7, 10, -4, 5, -6, -1, 0, 10, -3, 7, 10, 0, 3, 3], [2, 7, -7, 7, -1, 1, 5, -8, 2, -6, -6, 9, -2, -6, -2, -7, -3, -9, 4, -4], [1, 12, -2, 9, -8, 10, 4, 10, 12, 5, 11, 1, -2, 4, 2, 18, -2, 4, -4, 6], [10, -3, -7, -9, 5, 8, 4, 3, 5, 5, -8, 5, -11, 9, -3, 5, 1, -11, -6, 3], [1, -11, 3, 7, 7, 0, -2, -11, -8, -7, -14, 1, -7, -1, -1, -9, -8, -6, 3, -10], [15, -8, 4, 1, 4, -5, 2, 0, -1, -12, -14, 4, -4, 4, -5, -16, -4, -9, 2, -3], [-4, 3, -2, 10, 4, 9, 1, -11, 8, -6, -4, 4, -8, -11, 3, -15, -11, -1, -2, -2], [10, 4, -7, 4, -7, -4, 4, -3, 9, -1, 1, -4, -4, -4, -11, 6, -9, 1, -10, -4], [20, -1, -3, -7, 1, 13, 8, -2, 8, -5, 4, 5, -7, 0, -3, 3, -6, -6, -12, 0], [9, -10, 6, -4, 1, 5, 1, -9, -4, -20, 0, 2, -1, -5, 0, -4, -7, 4, 5, -3], [1, 19, -5, -2, 5, -19, -3, 1, 6, 7, 0, 4, 2, 3, 2, 5, 3, -5, -1, 6], [8, -4, 4, 1, 9, -1, -6, 2, -4, -4, -9, -4, 0, 8, 2, -3, 13, 0, 7, 11], [-17, -11, 5, 1, 4, -10, -2, 4, 2, -2, -16, 2, -7, 7, -4, -10, 0, -8, -3, -6], [-6, -6, 0, -3, -9, 7, 5, -3, 5, -8, 4, -2, 2, -3, -5, -2, -5, 5, 3, -4], [-10, -6, -4, 2, -3, 3, 1, 0, -7, -8, -6, -2, 4, -3, 1, -22, 3, -1, 23, -3], [-18, 8, -2, -5, -3, -11, 1, 14, 13, 5, -3, 1, 6, 3, 5, 6, 5, 3, 9, -5], [14, -8, -2, -2, 0, -8, 0, 3, 9, 3, -4, 3, -7, 0, -8, 6, -1, 1, -14, 2], [16, -8, 3, -3, -1, 28, 3, 2, 5, -7, 6, 0, -5, -2, -1, -4, -5, -2, -6, -5], [-10, 1, -2, 1, 1, -9, 3, -9, -1, -6, 1, -7, 2, -7, -3, -3, -9, -3, -7, -6], [13, -7, 1, -6, -3, -13, -1, -13, -6, -7, -4, -4, 4, -5, -8, -11, -9, -1, -1, -14], [-24, 10, -14, -4, -4, -6, -6, 4, 6, -2, -4, 3, 0, -2, -8, 8, 6, 0, 1, 2], [-19, -2, -1, -2, 2, -14, 0, -2, -9, -6, -12, 7, 3, 7, -7, -3, 0, -15, -6, 1], [27, -16, 0, 0, 3, 4, 0, -14, -5, 1, -4, 0, -8, -8, -8, -8, -8, -4, -7, 0], [3, -9, 0, 4, -6, -4, 2, -2, 0, 0, 1, -7, -12, -2, -6, -9, 0, -2, 6, 2], [7, -1, 0, -6, 4, 5, -4, 10, 3, 17, 10, -7, -2, 13, 5, 14, 2, -2, -8, 8], [9, -10, 6, -16, -1, 6, -3, -5, -6, -7, -5, -9, -7, 0, -2, -3, -5, -9, 0, 2], [8, -18, 4, -8, -6, -7, -9, -11, -12, -20, -9, -7, 0, -3, -9, -4, -7, 1, 2, 0], [-10, 30, -1, -11, -6, 8, -5, 12, 1, 2, 4, 2, 0, 2, 0, 17, 9, 4, -8, 4], [16, -5, -1, -7, 0, -6, 2, -9, -4, -13, -9, 1, 5, -10, -5, -22, -7, -2, 25, -12], [-4, 13, -3, 3, -9, 11, 8, 0, 14, 3, 12, -2, -1, 0, 3, 13, 4, 2, -4, 12], [-12, 14, -12, 3, -3, 1, 7, 3, 6, 6, 0, 9, 1, 2, -2, 3, -3, -1, -7, 0], [-6, 2, -8, 5, -10, -16, 4, -3, 16, 10, 4, 0, -2, -8, -8, 2, -10, -2, -6, -3], [0, 9, -7, 6, 2, 2, -3, 9, 4, 3, 10, 4, -8, 6, -1, 10, 5, 5, -5, 10], [3, 0, 7, 8, 6, -6, -2, -1, 2, -6, -2, 0, -2, 3, 3, 3, 5, 2, 2, 1], [-24, 3, 5, 3, -5, 12, -3, 14, 2, 8, 2, 1, 7, 5, 3, 7, 6, 3, 5, 7], [-1, 6, -10, 6, -1, -6, 1, -8, 13, 15, 1, 2, -4, -6, -8, 1, -6, 0, -10, -6], [26, -3, 14, -2, 2, 3, 7, -3, 2, -12, 2, -6, -6, -11, 5, -13, -9, 1, -4, -8], [13, -20, 16, 6, 14, -5, -5, -2, -6, -5, 0, 2, -4, 1, 5, -10, -1, 5, -2, 3], [-24, 1, 0, -1, 0, 3, 7, -6, -2, 0, 2, -12, 4, -7, 5, -7, 3, -1, 20, -3], [-5, 8, -2, -4, 11, -14, -3, 5, -2, -4, -7, 3, -5, 4, 6, -3, 13, 3, 8, 2], [10, -12, 2, -2, 1, -14, 8, -8, -4, -7, -2, 5, 0, 6, -3, -5, -4, -14, -3, -3], [-1, -6, 6, -1, 3, -11, 2, 1, -2, 18, 4, -7, 7, -4, 4, 14, -1, 3, 2, -6], [45, -11, 16, 7, 3, 14, -3, -8, -3, -9, 8, -6, 0, -10, 8, -10, -8, 6, -1, -2], [13, -6, 2, 3, 0, 5, 1, -15, 0, 5, -2, 2, -6, -3, -9, -6, -15, -7, -6, -6], [23, 7, -1, 5, 6, 6, -8, -3, 0, -1, 9, 0, 8, -5, 9, -10, -1, 2, -3, 1], [6, -11, -1, -2, 8, -21, -6, -1, -2, -6, -13, 2, 8, 1, -6, -21, 2, -5, 6, -13], [21, 5, -2, 12, -10, -11, 3, -7, 8, -3, 3, -2, 0, -6, -7, -7, -9, 3, 4, 1], [5, 9, 14, 15, -11, 12, 9, -4, 5, 0, 23, -9, 12, -7, 12, 7, -6, 17, 12, 11], [-13, -4, 16, -8, -1, -4, 7, 12, -6, 1, 3, -6, 19, 3, 9, 2, 8, 0, 6, 0], [2, -9, 4, 3, -3, -1, 1, -5, -3, -12, -7, -5, -2, -1, 5, -13, -5, -5, 7, 4], [-16, 11, 3, 5, 0, 8, 1, 5, -2, -2, 2, 1, 4, 1, 2, 7, 2, 2, 1, -3], [10, -17, -17, -4, 0, 3, -5, 3, 1, 0, -13, -1, -9, 3, -10, -8, -3, -4, -4, -6], [13, -9, -5, -9, -2, 1, 5, -8, -10, -12, -1, -1, -2, -3, -10, -1, -12, -6, -11, -4], [-7, 0, -10, 12, -10, 0, 4, 1, 0, -7, -3, 1, 4, 2, 0, 2, 4, 0, 8, 5], [24, -13, 0, 0, 6, -14, 6, -5, 3, -10, 0, 1, -6, 3, 2, -4, -5, -8, -9, 5], [-28, 0, -4, -7, 8, 2, -8, 5, -2, 7, -13, 15, -16, 8, -11, 0, 5, -8, -15, -5], [-10, 3, 10, -11, -4, -12, -5, 3, -9, -2, 4, -10, 8, 4, 6, 6, 7, -1, 4, 8], [21, -6, 10, -8, -3, 13, 5, 0, -4, -4, -4, 1, 6, -4, -1, 2, -4, -2, 1, -7], [-11, 12, -10, -13, 0, -5, 2, 11, 2, 14, 4, 5, 1, 10, -1, 10, 9, 2, 2, 3], [-2, 12, 4, 0, -2, -11, -5, 23, -2, 5, 4, -2, 15, 8, 11, 22, 15, 13, 2, -1], [-2, -11, -4, 8, -5, -11, 5, 3, -5, -8, -6, 5, 7, 2, -3, -6, 5, 0, 5, -3], [1, 7, 4, -10, -7, 1, -1, 2, 2, 0, 5, -2, -4, -6, -3, 9, -2, -4, -3, -1], [-14, -10, 5, 3, 4, 9, -9, -3, -5, -8, -6, 1, 2, -2, 2, -15, 3, -4, 5, -1], [-12, 12, -5, -1, 6, 17, -1, 6, 4, 0, 0, 14, -10, 5, -2, 11, 2, -1, -5, 6], [27, -17, -1, 8, 3, 8, 1, 3, 8, 0, 2, 7, -8, 7, 4, -4, -11, -4, -6, 0], [-35, -1, 1, -8, -1, -1, 3, -1, -4, -3, -1, 0, 8, 9, -1, 16, 10, -8, -2, 8], [-6, 4, -2, 3, 2, -5, -5, 6, -6, -4, -5, -7, 12, -1, 7, -2, 11, 11, 16, -9], [-22, 5, -1, 3, 3, 2, -2, -3, -6, -3, -4, 0, 6, 0, 3, 3, 12, -2, 12, 7], [6, 5, 5, -11, -1, 12, -6, 6, -4, 1, 5, -6, 5, -9, 1, -1, 2, 14, 14, -11], [-1, 1, -1, -8, -9, 2, 1, 13, 8, 8, 20, -8, -12, 1, -2, 19, 1, 13, -7, 0], [15, 16, -14, -2, -6, 16, -2, 1, 14, 11, 9, -3, -8, -13, -3, 1, -3, 3, -3, -2], [4, 7, 2, 11, -1, 0, 0, 4, 12, 0, 12, 7, -4, 3, 4, -3, -7, 3, -3, 2], [2, -9, 4, 5, 2, -6, -2, 2, -10, -13, -11, 2, 2, 10, 2, -13, 14, 0, 9, 4], [6, 10, -5, 0, -3, 12, -1, -2, 6, -6, -1, 1, 1, -9, -7, -4, 0, 1, 0, 1], [-12, 2, -4, -4, -4, -10, -6, -1, -6, 4, 4, -4, 2, 4, -2, 9, -2, -1, -9, 9], [-16, -3, 8, -2, -2, -5, -3, 4, -8, -5, -7, -1, 0, 7, -3, 11, 5, 3, -2, -3], [28, -25, 4, -4, 5, -5, 5, -15, 1, -16, -12, 1, -11, -4, -3, -20, -5, -10, 1, -9], [3, -5, -4, 6, 5, -9, -3, 1, 6, -9, -4, 7, -6, 10, 1, 9, 0, -6, -11, 10], [-8, 1, -3, -13, 9, 6, -15, 14, -14, 0, 0, -7, 9, 5, 9, 3, 18, 13, 19, 1], [-5, 14, -20, -1, 2, 14, -2, 3, 14, 5, 1, 17, -12, -1, -9, -3, -7, -5, -12, 2], [15, -24, 10, -13, 3, 1, 6, -1, -4, -7, -6, -3, 0, 5, 1, 1, -2, 0, 3, -7], [-9, -5, -14, -6, 7, -17, -6, -2, 3, 6, -10, 14, -16, 3, -12, -5, 2, 2, -15, -5], [9, 23, 9, -5, -8, 15, -2, 3, 6, 9, 21, -6, 6, -3, 11, 12, 3, 12, -9, 11], [19, 2, -10, 4, -7, -4, -2, 4, 3, 10, 1, 3, -9, 4, -10, 11, -8, 1, -15, 0], [5, -8, 0, -1, 6, -21, 2, 0, 1, 6, -2, -4, 0, -6, -5, -2, -2, -1, -10, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2009_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2009_2_a_u();".
// 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_2009_2_a_u(:prec:=20)
    chi := MakeCharacter_2009_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_2009_2_a_u();".
// 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_2009_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2009_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![49, -42, -1179, 1972, 5690, -9524, -13352, 18446, 17721, -18080, -13797, 9852, 6449, -3090, -1820, 552, 302, -52, -27, 2, 1]>,<3,R![89, -4324, 9466, 27816, -58599, -54676, 133192, 33544, -141680, 7624, 78924, -17616, -23916, 8248, 3770, -1804, -236, 192, -6, -8, 1]>],Snew);
    return Vf;
end function;