
// Make newform 1849.2.a.r in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1849_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_1849_2_a_r();".
// 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_1849_2_a_r();".
// 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 := [-43, 87, 1386, -1004, -11330, 6093, 32885, -21700, -37889, 29618, 20357, -19360, -5068, 6765, 329, -1293, 95, 127, -19, -5, 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], [943797497903, -1145388621680, -15588667852490, 23111814188338, 14545772761564, -99110234446389, 115971133895630, 106811883992698, -222851714467217, -28900607748863, 162485765268892, -14901877981764, -59715411593224, 11484657262735, 11732620801942, -2910628173617, -1171616127168, 330972065901, 46615997016, -14230071983], [1077447576383, -2841030729550, -1917001809428, 22368332065168, -94138817905982, -82784184830195, 440852989621230, 12082802759062, -632498558445865, 115848818962819, 421393038907800, -113171607664568, -148861241315388, 46317478630483, 28657156909476, -9605341367555, -2828666427292, 989932636217, 111749991106, -40192831361], [327793978196, -599774238618, -10227894614659, 7671932147992, 76217290556585, -33672691110928, -202549945339379, 84716752424831, 238528299976955, -100364878110005, -144977562345826, 61123803690812, 48751376649190, -20470715366044, -9117046895340, 3804260095709, 883733858774, -366459861464, -34498144612, 14228637027], [927358146403, -1162405628057, -25874132256391, 14208138715755, 196382266801699, -57782923544782, -539094925429208, 175530105599254, 645124894290285, -230797776420126, -395191025815830, 148499808956902, 133418948270550, -51369446049423, -25003477843045, 9747060758626, 2426492911022, -952275906137, -94788187709, 37344752748], [-775838268705, 4247919237747, 28207382303301, -49064096880753, -219404535918529, 167900179824962, 577328926635576, -320073159163538, -672856286458559, 329108224951442, 406000279232914, -186812864123554, -135847791045306, 60244560176733, 25317574032743, -10953640803422, -2448128554906, 1041101282307, 95400656607, -40076363964], [-1038060570543, 1534054231361, 23096313662912, -20242958750477, -143607640371362, 87259234746636, 343948035834559, -181685495337785, -384942866151136, 190202416504421, 227770708090896, -107767360467754, -75422166461600, 34530559209167, 13972083923691, -6235731993181, -1346077054352, 589133963225, 52329290331, -22565527133], [148383897244, 365413200906, -6725786608347, -7361814401786, 58657760598743, 26408030509110, -179754151581493, -4135372453827, 226414828641029, -35291390286993, -142288509684814, 34481376336088, 48702101793264, -14098932791770, -9199313526368, 2922147432589, 897048356248, -301069644714, -35149906384, 12222313595], [-2574044282979, 6874916674146, 76935016739816, -56463860416428, -587136620124678, 183783042442835, 1619784588572106, -522147572120342, -1943082102381195, 683609133637797, 1191445061787792, -441081865090352, -402330962018060, 153026166000465, 75391201461040, -29101168611029, -7314751041244, 2847790709171, 285669062450, -111811015775], [1778222415337, -1356886460249, -45026666286195, -5290998649529, 337784614049419, 30186985595208, -997254553913076, 140421681280074, 1243989143864211, -314953377096812, -779577845684462, 240735491623326, 266678782011486, -90276060091025, -50381737694825, 17917648163456, 4914739593678, -1799642357535, -192655709901, 71868444078], [1789038068597, -6505454572630, -49961865912316, 71370077753908, 389532750589718, -246006347238045, -1066622998787002, 517697984612838, 1272593937059985, -570751820635207, -777835241874840, 337239578662104, 262139873422644, -111354631835791, -49054829077416, 20544990365311, 4754809536988, -1971310193269, -185555230662, 76368492269], [3615445347609, -6157506806398, -101653872424720, 55146394966836, 783716582169282, -180984174704881, -2212656513798606, 629069854853162, 2688865535032593, -888494600749711, -1660824137675056, 591855997923712, 563309181684564, -208715018002883, -105854505673304, 40072962939383, 10289851728316, -3945463645905, -402386853510, 155548732541], [-1605824368164, 2321635342673, 52922232936785, -11619999502601, -437324187350595, 31159311691865, 1285865117662762, -271286209202392, -1590485442468226, 460951992599979, 991024908111162, -327803387385450, -337763015883766, 119218304069670, 63655869703515, -23283287460119, -6199309247566, 2316290687974, 242725603925, -91935576031], [-1234373979961, 528468761140, 49326510654306, 10161021045118, -433878137753528, -51206133702961, 1320398144934142, -163225402993358, -1655777594146369, 395637183429693, 1038068318046268, -308141908987604, -354861426319336, 116572703951159, 66984208582074, -23257513775093, -6529262514856, 2344007607053, 255780936564, -93833087507], [4807873678325, -7153514089446, -161844984538296, 35323243098284, 1323282669147178, -94127898992397, -3860688296203710, 806514453745394, 4758555148819317, -1368536562842755, -2959745981769472, 973216203616208, 1007740093038052, -354070559699815, -189811990190440, 69180441936035, 18479122112404, -6885247962181, -723381409446, 273386016425], [6688595499263, -9917294383458, -221748050108224, 54517055046444, 1817513534656366, -164041845203031, -5296204244315666, 1162750982341094, 6522655782099111, -1918176825366745, -4055000149086848, 1350599324318704, 1380176974441500, -488977993955301, -259890551008872, 95270005502209, 25295833347988, -9464837304391, -990037727626, 375356809195], [5366073004952, -8139215573933, -179515614702555, 41415463490661, 1470838693878981, -118881150103701, -4295743992084572, 925769465133894, 5297771184472656, -1548195644482233, -3296382641454286, 1095082077575794, 1122664590143306, -397316678434818, -211499274791995, 77503097820145, 20593247638918, -7705358032622, -806215480513, 305725917825], [-11881100268157, 23976035504562, 392460878843472, -183939248550712, -3142535764262346, 633271755666169, 8965145688721578, -2455097015073230, -10930634469065993, 3561178192248795, 6760409747236640, -2392916580413464, -2294382320058436, 846930284261719, 431290860199524, -162900388536371, -41932542212588, 16054290614973, 1639968208882, -633296538777]];
    Rf_basisdens := [1, 1, 1, 210503587696, 210503587696, 52625896924, 105251793848, 105251793848, 52625896924, 26312948462, 210503587696, 105251793848, 105251793848, 210503587696, 105251793848, 105251793848, 210503587696, 210503587696, 105251793848, 210503587696];
    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_1849_a();"
function MakeCharacter_1849_a()
    N := 1849;
    order := 1;
    char_gens := [3];
    v := [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_1849_a_Hecke(Kf)
    return MakeCharacter_1849_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, -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, 0, 0, 0, 1, 0], [1, 1, 0, 0, 0, 1, -1, -1, 1, 0, 0, 1, 0, 0, -1, 1, 0, 0, 0, 0], [-1, 0, -1, 0, -1, 1, -1, 1, 0, -1, -2, -1, -1, 1, 1, 0, 1, 0, -1, -1], [1, 1, 0, 0, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, -1, 0, 0, 1, -1, 0, 0, -1, 0, -1, -1, 0, 1, 0, 1, 1, -1, 0], [0, 1, -1, 0, -1, -1, 0, 1, -1, -1, -1, -1, -1, 0, 0, -1, 0, 0, 0, 0], [-2, -2, -1, 0, -1, 1, -1, 1, -1, 0, -2, -1, -1, 0, 2, 0, 2, 1, -1, -1], [2, 1, 0, 0, 0, -1, 1, -1, 0, 1, 1, 1, 1, 0, 0, 1, 0, -1, 1, 0], [-1, 2, 1, 0, 1, 1, 0, -1, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 0, 1], [3, 1, 1, -1, 1, -1, 1, -1, 0, 0, 3, 1, 0, -2, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, -1, 0, 1, 2, -1, 0, 1, 0, -1, 0, -1, 0, 0, 1, -1, -1, 1, 0], [0, 0, 0, -1, 0, 0, 3, 1, 0, -1, -1, -1, 0, 0, 1, 0, 0, 1, -1, 0], [4, -2, -1, 0, -1, -1, 2, 1, 0, -2, -1, -1, -2, -1, 1, 0, 1, 0, -1, -2], [2, 0, -2, 0, -1, -1, -1, 0, 1, -1, -2, 0, -2, -2, 0, 2, 0, -3, -1, -2], [1, 2, -1, 0, 1, 0, 2, 2, 0, -1, 2, -1, 0, -3, 1, -1, -1, -1, 0, 0], [-3, 1, -1, 0, -2, 2, -2, -1, 2, -1, -1, 1, -1, 1, -2, 3, -1, -1, 0, 0], [-2, 2, 0, 0, 0, 2, -2, -2, 2, 1, -2, 1, 0, 2, -1, 2, -2, -2, 1, 0], [4, 0, -1, -1, -1, 0, 0, -2, 2, -1, 3, 2, 0, -1, -1, 1, -2, -1, 2, -1], [-1, -1, 0, 0, -1, 3, 2, 1, 0, -1, -3, -1, 0, 3, 1, -2, 1, -2, 1, 0], [-2, -2, -2, -1, -2, -1, 1, 4, -2, -3, -1, -3, -1, 0, 2, -1, 1, -1, -1, -1], [-3, 2, -2, 0, 0, 2, -4, 1, -1, 0, -2, -1, -1, 2, -1, 0, 0, 1, 1, 0], [3, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 1, 2, -3, 0, -2, 0, 1, 0, 2], [3, 1, 1, -1, 1, -3, 3, 0, 0, 0, 4, 0, 1, 0, -2, -2, -1, -1, 2, 2], [1, -3, -3, -2, -4, -1, 0, 5, 0, -2, -2, -3, -2, -1, 0, -1, 2, -2, 0, -3], [0, 0, -1, 0, -1, 1, 1, -2, 1, -1, -2, 2, 0, 2, -2, 2, 0, -1, 3, 0], [-2, 0, 0, -3, 1, 2, -3, -2, 1, 0, 0, 2, -1, 1, -2, 2, -1, 0, 1, 0], [2, 0, 0, 1, 1, -1, 0, 0, -2, 2, 1, 0, 1, -4, 1, -1, -2, 1, -2, 0], [0, -1, 0, 0, 2, 1, 3, -1, 0, 0, 1, 1, -1, -1, 2, 3, 0, 2, -3, 0], [-2, -2, -3, -2, -2, 0, 2, 2, 0, -1, 2, -2, -2, 1, 2, 0, 0, 2, 0, 1], [3, 2, 1, 2, 2, -4, 2, -2, -2, 1, 0, 1, 2, -6, 2, 1, -1, 2, -1, 1], [-1, 0, -2, -1, 0, 1, 2, 3, 0, 0, 4, 0, 0, -5, 0, -3, 0, 2, 1, 0], [0, -4, 0, 0, -2, -5, 0, 2, -1, -2, 0, -3, 0, -2, -1, -2, 0, -2, -2, -2], [1, -1, 1, 2, 2, 0, 0, -1, -2, 2, 1, -1, 4, -1, 4, 0, 1, 2, -4, 2], [-3, 0, -2, 2, -2, 0, -4, -2, 0, -1, -4, 0, 0, 1, -1, 0, 1, -3, -1, -1], [5, 2, 1, 2, 3, 1, -1, -1, 0, 2, 1, 0, 2, 2, 0, 1, 0, 3, -2, 3], [4, -4, -1, 0, -3, -5, 4, 3, 0, -1, -1, 1, -1, -7, 1, 0, 1, -2, 2, -2], [-1, -1, 0, -1, 2, -2, -4, -1, -1, -1, -2, 0, -1, 1, 1, -1, -1, 0, -3, 1], [0, 0, -3, 1, -2, 3, -2, 1, 2, -3, -1, -2, -2, -2, -1, 1, 0, 2, -3, -1], [0, -1, -2, 4, -1, -3, 1, 1, 1, -1, 0, -1, 2, 0, -1, 1, -1, -4, 0, 0], [0, 7, 1, 2, 3, 1, 3, -1, 0, -1, 0, -1, -1, -3, -2, 0, -3, 0, 0, 3], [4, 1, 1, -2, 3, 0, 2, 1, 0, 4, 8, 0, 0, -2, -2, -2, 1, 2, 2, 0], [-2, -4, 1, 3, 0, 1, -1, 0, -1, 1, -5, 0, 3, 1, 3, -1, 4, 1, -5, 0], [-4, -2, -1, 1, 0, -2, 2, -4, -2, -1, -3, 1, 0, 1, 3, 2, 2, 1, -3, 0], [4, 1, 0, -2, 1, -1, 1, 0, 2, -2, -2, 0, -1, 2, -1, 1, -5, 1, 2, -1], [4, 0, 4, 1, 3, 0, -3, -7, 2, 3, 3, 5, 4, -1, -2, 1, -2, -1, 4, 1], [5, -5, 3, -1, 1, 2, 1, -1, 2, 3, 2, 2, 1, 0, -2, -1, 1, 0, -2, 1], [13, -1, 3, -1, 2, -3, 3, 0, 1, 3, 5, 1, 3, 2, -4, -2, -3, -2, 5, 2], [5, -3, 1, -3, -1, -3, -4, 1, 1, -2, 2, 0, -1, 5, 0, 2, 1, -1, 0, -1], [-10, 1, -1, 1, -3, 7, -4, -1, 0, -1, -7, 3, 1, 7, -4, 3, -2, -2, 1, 1], [9, -4, 2, 1, 1, -1, -3, -2, -2, 3, -4, 1, 2, 3, 0, -1, 2, 2, -1, -1], [-1, -1, 1, 1, 1, 0, -3, 2, -4, 1, 1, 0, 3, 4, 2, 1, 2, 1, -2, -2], [5, 2, -1, 4, 1, -2, 4, 0, 3, 0, 5, 2, 3, -3, 0, 3, 2, 2, -2, 1], [0, 5, 3, 2, 1, 4, -2, -7, 0, 0, 1, 4, 1, 3, 0, 4, -2, -1, 0, 2], [2, 6, 2, -2, 1, -6, 4, 0, -1, 1, 4, -2, 1, 0, -3, -4, -2, 1, 3, 2], [-3, -1, 1, 0, -1, -3, 0, 1, 0, 4, -1, 0, 1, 0, -2, -3, -1, 2, 1, -1], [3, -4, -3, -1, -4, -7, 3, 4, 2, -1, 2, -2, 0, -2, 3, 2, 4, -3, -2, -4], [-5, 2, 2, -2, 0, -3, -3, -1, 0, -1, -2, -1, -1, 2, 0, 1, 1, 3, 1, 2], [9, 0, 1, 1, 3, -4, 3, -1, 2, 1, 6, 2, 2, -2, 0, 4, -1, 0, -1, 0], [0, -5, -1, 2, 1, -1, 3, -2, -2, 0, -3, -2, 4, 2, 5, -1, 6, 5, -6, -1], [10, -1, -2, 3, -1, -2, 2, 1, 2, 0, 8, 0, 0, -1, 1, -1, 2, 1, 1, 2], [11, 3, -3, -1, -2, -4, 3, 1, 0, 1, 2, 1, 1, -8, -3, 2, -1, -1, 1, -2], [-6, 1, 1, 3, 0, 3, -6, -2, 0, 5, -3, 1, 3, 2, -4, 2, 0, 1, 0, 0], [3, 1, -2, -1, -2, 2, 4, -2, 1, -3, 2, 2, -2, -9, 2, 3, -3, -2, 3, -1], [-4, 1, -4, 4, -4, -9, 5, 6, -2, -1, -5, -3, -1, 2, 2, 0, 0, -3, 1, 0], [-3, 4, 0, 3, 0, 3, -6, -5, -1, 1, -1, 2, 1, -5, 0, 3, -1, -1, -2, 1], [-5, 2, -2, 0, -3, 3, 0, 5, -1, -2, -7, -2, -4, 5, -3, -3, 0, -1, 1, 4], [7, -3, 4, -2, 3, 0, 5, 2, -3, -3, 3, -1, 3, -3, 2, -4, 3, -2, 1, 0], [10, 1, -2, 1, -2, -1, 4, 4, -3, 1, -1, -3, 0, -3, 4, -2, 3, 1, 0, 1], [2, 3, 1, 3, -2, 0, 1, -4, 3, 3, -3, 0, -1, 2, -6, 2, -5, -5, 4, 0], [5, 5, 4, 0, 4, -2, 7, -2, 2, 3, 10, 4, 1, -8, -2, 3, -2, 4, 2, 4], [0, 5, -1, -1, 2, 3, -2, -2, 3, 2, 4, 1, 1, -9, -2, -1, 0, -1, 2, 1], [-2, 0, 3, -2, 0, 2, -3, -1, 2, 2, 7, 5, 1, 5, -3, 1, 1, 1, 0, 0], [-7, 3, -1, 0, 1, 8, 0, 1, 0, -1, -5, -3, -3, 3, 1, 0, -4, -3, -1, -1], [2, -1, -2, 0, -2, -2, 6, 4, 1, 0, 2, 0, 2, -5, -5, -3, -1, -5, -1, -3], [-3, 1, -4, 1, -4, 1, -1, -1, -2, 1, -2, -2, 2, 2, 2, 3, 1, 1, 0, -1], [-1, 4, 2, 0, 3, 4, -1, -1, 4, 3, 11, 4, 3, -4, -9, 0, -4, -4, 6, 4], [-1, 4, -2, -4, -4, 0, 2, 3, 3, -4, 3, 1, -7, -2, -7, 1, -4, -5, 7, -4], [0, 2, -3, 0, -1, -7, 5, 2, 1, -2, -2, -1, -1, 0, 0, 1, 2, 4, -5, -1], [9, 2, 0, -1, 1, -4, 2, 0, 1, 0, 6, -2, -4, -4, -2, -4, 0, -1, 1, -1], [0, 3, 2, 2, 4, -9, 0, 1, -1, 0, 4, -1, 6, 1, 1, -4, 1, 3, 4, 3], [-2, -2, -1, 3, -5, 2, -3, -2, 1, 4, -9, 2, -1, 3, 2, 4, 4, -2, -2, -4], [-2, 1, -1, 4, 2, 7, -6, 0, 5, 1, 2, 3, 2, 3, 2, 3, 3, 5, -1, 1], [-1, -2, 3, -2, 2, 7, -12, -5, 3, 0, -6, 2, 1, 12, 2, 2, 3, -3, 1, 1], [4, -6, 2, 0, -3, -3, 1, -2, -5, -1, -4, 3, 1, 4, 7, 0, 2, -1, 1, 0], [-4, 1, -6, 3, -3, 8, -9, 1, 2, -6, -6, -4, -1, 3, 0, 2, 0, -4, 3, -1], [-3, 2, 3, 1, 2, -4, 3, -3, -5, 2, 1, 0, 6, 1, -1, -3, -5, -2, 0, 0], [-4, 2, 8, -1, 6, 7, -3, -4, 0, 4, 5, 1, 3, 0, -4, 1, -4, 5, 1, 3], [6, 7, 7, -2, 4, 1, -1, 1, 1, -1, 3, 3, 4, 4, -6, -2, -2, 0, 3, 3], [7, -2, 4, 0, 7, -10, -1, 1, -4, 5, 9, 1, 5, -2, -1, -5, -6, 0, 2, 0], [0, 6, 1, 2, 4, -4, 2, 2, -1, -1, -2, -2, 4, 5, -2, -4, -2, 0, -2, 3], [-1, 0, -1, 2, -1, -3, -2, 1, -2, -1, -1, -4, -1, -2, -3, -4, 2, 5, -2, -1], [3, 0, 0, 1, 1, 12, -7, 0, -2, -2, -8, 1, 1, 10, 4, 1, 6, 4, -5, 2], [1, 6, -3, -3, 0, 0, -4, -2, 3, -1, 0, -2, -1, 2, -7, 0, -9, -3, 4, 2], [5, -1, 0, -2, -1, -10, -3, 0, 2, 0, 3, 1, -2, 1, -5, -4, 2, -1, 3, -3], [-5, -1, 2, 2, -2, -5, 1, 0, 0, -2, 1, 1, -2, 0, 0, 0, 1, -1, 4, -1], [6, 6, 1, 3, 3, 1, 2, -2, -1, 1, -1, 1, 2, -1, 0, -4, 0, 1, -3, -1], [-2, -2, 6, 1, 2, -3, 7, 6, -2, 7, -2, -1, 2, 1, 0, -4, 1, 2, -1, 3], [2, 0, -2, 1, 3, -6, -1, 4, 0, -3, 4, -6, 1, -1, -1, -3, -3, -3, 4, -1], [5, -3, -3, -4, -7, -5, 0, 0, -1, -5, -7, -3, -5, -5, 4, 1, -1, -1, -3, -5], [10, -1, 11, 1, 8, 1, 5, -6, -3, 5, 11, 3, 10, 4, -3, 0, -5, 2, 1, 5], [-5, 2, -5, 3, -6, 0, -1, -1, 2, -5, 2, -2, -1, 2, -1, 0, 0, -4, 0, -3], [1, 1, 6, -3, 6, 4, 1, -4, 3, -1, 7, 6, -1, -7, 1, 2, 2, 5, -3, -1], [-1, 0, 6, -1, 4, -2, -1, 2, -4, -1, 8, 2, 2, -7, 0, -4, 4, 7, 0, 3], [-5, 6, 6, -2, 2, -8, 1, -4, -1, -3, 7, 2, -3, 1, 0, 3, -4, -1, 1, 0], [8, -7, 7, -2, 6, -5, 2, -2, -3, 4, 2, 1, 1, 3, 4, 0, 1, 3, -9, -1], [0, -3, 3, 2, 1, -5, 15, 7, -2, -4, 4, -4, 0, -2, 3, -6, 5, -3, 2, 1], [6, 5, 0, 1, 8, 2, -6, -7, 1, 1, 6, 1, 0, -1, -5, 5, -6, 0, 0, 0], [2, -8, 5, -3, 2, 4, 0, -5, 5, -4, -1, 5, -2, 3, 0, 2, 2, -1, -1, -2], [-1, 3, -1, -1, 2, -2, 0, -9, -3, -1, -1, 2, 1, -7, 1, 2, -6, 0, -2, -1], [1, -5, 3, -3, 1, 2, 1, -5, 4, -3, 10, 4, -1, -1, 3, 5, 3, -1, -2, -1], [-13, -4, -5, 0, -6, 4, -5, 6, 0, -12, -13, -5, -6, 11, 7, 1, 4, -3, -7, -7], [-4, 3, 1, 1, 4, -4, 3, -2, -5, 3, 4, -1, 3, -14, 1, -2, 2, 1, -3, 2], [-3, -6, -2, -6, -2, 9, -5, 0, 6, 2, 0, 0, -5, 7, -4, 3, -1, 0, 3, 0], [-2, -7, -4, -2, -10, -8, 14, 13, -3, -10, -7, -6, -6, -3, 2, -4, 6, -8, 3, -6], [4, 1, -3, 5, -3, -1, 2, 6, -3, 0, 2, -5, -2, -1, 1, -3, 4, -3, 2, -4], [11, 5, 5, 0, 8, 10, 1, -5, 2, 0, 12, 1, 2, 4, -1, 0, -2, 4, -2, 8], [0, -3, -4, 3, -4, -7, -1, -3, -1, -6, -4, 2, -1, -1, 6, 4, 4, -5, -4, -7], [-4, -3, 3, -2, 0, 3, 1, -2, 3, -2, -3, 2, -3, 5, 0, 2, 8, 0, -3, -4], [-2, -2, -4, 5, -3, -7, 0, 1, -4, 5, 4, -3, 1, -4, -1, -2, 3, -1, -1, 2], [-7, 1, 1, -5, 2, 11, -11, -1, 1, 1, -9, 0, -2, 13, 0, -1, -4, -1, 3, -2], [8, 2, 5, -7, 7, -1, 5, -1, -2, 2, 11, 3, -4, -5, -2, -3, 2, 6, 4, 0], [0, -1, -6, 2, -2, -3, -6, 2, -5, 5, -13, -1, -2, -4, 4, -2, 0, -1, 1, -3], [1, -2, 2, 0, -3, -6, 4, 4, 3, 1, 6, 0, -3, -6, 4, 5, 4, -3, 0, -5], [12, 0, -8, 1, -3, 1, 6, -2, 5, -5, -7, 2, -7, -7, -3, 8, -6, -3, 0, -3], [-4, 9, -6, 3, -2, 11, -5, 0, 3, -2, -4, 1, 1, 9, -1, 1, 0, -5, 6, 3], [3, -3, 3, -4, 1, 3, -3, 5, -2, 1, -1, -3, -3, 12, 2, -5, 3, 4, -1, -1], [11, -6, 0, -1, -1, 2, -1, -2, -1, 6, 0, 0, 3, 1, 3, 0, -2, -2, 3, -3], [0, 9, -4, 0, 3, -2, 4, 1, 2, 2, 2, 5, 2, -9, 0, 0, -3, -1, 8, 2], [-3, -1, 5, -6, 3, 5, 7, 4, -1, -2, -2, 2, -4, 3, -2, -3, 4, 3, -1, -1], [-11, -2, 8, 1, 4, 2, -5, 3, -5, 6, -7, -1, 6, 13, -3, -4, -1, -2, 2, 2], [-9, 2, 0, 2, -3, -2, -2, 2, 3, 1, 4, -1, 1, 5, 1, -1, 1, -3, 0, -4], [2, 3, 1, -1, 5, -5, -4, -10, 2, 2, 2, 5, -1, -3, -9, 6, -4, -2, 2, 3], [-9, 1, -2, -7, 0, 9, -7, 0, 6, -1, 0, 2, -5, -1, -9, -1, -5, -9, 7, -1], [-6, -3, 5, -3, 3, 4, -5, 7, -8, 0, -8, -4, 0, 8, 0, -7, -2, 4, -1, -4], [1, 5, 1, 5, 3, 1, -5, -5, -3, 11, 2, 2, 2, -4, -3, -2, -1, 3, -4, 5], [1, 1, -3, 1, -3, -2, 11, 0, 3, 5, 2, 2, 2, 5, 3, -3, 5, 4, 6, 4], [-11, -5, 4, -3, -3, 8, -3, 3, -1, -2, -4, -2, -3, 0, 3, -1, -1, -3, -6, -4], [4, 6, 2, 2, 7, -6, 11, 1, -3, 3, 0, -1, 1, -6, -2, -1, -2, -2, -5, -1], [6, -2, 0, -7, -1, -10, 0, 2, 2, 3, 6, 1, -5, -3, 2, 1, 1, 3, -2, -5], [8, 1, -4, 0, 0, -14, 3, 2, 1, -4, 0, -1, -4, -1, -3, -3, 1, -2, -3, 0], [6, -7, -2, 7, -2, -1, 2, 6, -3, -3, -9, -6, 1, 6, 9, -3, 2, 2, -8, -3], [4, -5, 8, -1, 6, -3, 7, 3, -7, 0, -4, -3, 4, 3, 12, -3, 0, 1, -4, 3], [0, 3, 8, -2, 7, -3, -6, -4, 2, 5, 18, 4, 1, -6, -8, -3, 1, 2, 6, 1], [-1, -6, -2, 6, -3, -1, -8, -5, -1, 0, -8, -1, 1, 8, -2, 0, -3, -5, -6, -2], [-8, 4, -8, -2, 1, 4, -5, 2, 0, -3, 1, -1, -3, -6, 5, 2, -3, 0, 5, -3], [-1, -5, 7, 1, 4, 17, -15, -2, -2, 6, -10, -1, 2, 10, 0, 0, -3, 1, 3, 1], [5, 3, 4, 1, 6, -1, 1, -6, 1, 5, 8, 7, 11, 4, -4, -3, 3, 1, 7, 6], [5, 4, 0, -1, 4, 4, -8, -6, 6, 1, 12, 6, 1, 4, -9, 7, -10, -8, 3, -3], [3, -6, -13, 2, -7, -2, -3, 7, 5, -4, -7, -6, -9, -5, 6, 3, -2, 0, -6, -5], [5, 1, 2, 0, 5, -9, 3, 2, -2, 2, 11, 0, 0, -5, 0, -6, -7, 2, 3, 6], [-1, -7, 0, -4, 1, -3, -6, 7, -1, 2, -2, -4, -3, -5, 6, -10, 8, 7, -1, 1], [4, -1, 6, -2, 5, 0, 16, -3, 4, 0, 14, 9, 2, -9, 0, 4, 5, 3, 2, 6], [5, 1, 12, -7, 9, 4, -11, -6, -6, 9, -5, 1, 5, 8, -3, -4, -4, 0, 6, 1], [-2, -3, -6, -5, -5, -4, 2, -3, 2, -7, 2, 0, -6, 2, 1, 3, 2, 2, 3, 2], [9, -7, -3, 0, -3, -1, 3, 3, -1, -2, 0, -1, -4, 3, -4, -1, 0, 3, -3, -1], [11, 1, 1, 4, 1, -5, 6, 1, 1, -4, -2, 0, 5, 3, 6, -6, 7, 3, -5, 5], [-1, -2, -3, 2, 0, -4, 4, 5, -5, 1, 2, 0, 2, 4, 0, 0, 3, 1, -5, -8], [-12, 0, 1, -6, -2, 13, -4, -6, 1, -1, -3, 3, -3, -5, 0, 2, -1, 1, 0, 0], [-3, -3, 0, 4, 4, 9, -12, -1, -4, 0, -6, -1, 6, 2, 8, -1, 6, 11, -8, 3], [-6, 4, -4, 3, -4, 6, -1, 5, -7, -2, -12, -5, 0, 6, 1, -2, 7, 1, -5, 1], [-8, 1, 4, 5, 6, 11, -13, -6, -5, 9, -9, 4, 9, 8, 1, -5, 3, 8, -4, 7], [-3, -2, 0, 8, 2, -2, 5, -1, -3, 7, -6, 3, 5, 0, 2, -4, 5, 3, -1, 4], [12, 5, 0, 3, 5, -6, -7, -6, 5, 9, 10, 5, 2, -8, -2, 3, -6, -1, 3, 1], [-18, 6, -3, -2, 4, 10, -18, -1, 0, -2, -5, -1, -6, 9, 3, 1, -7, -1, -4, 0], [3, -1, -1, 4, -2, -1, -11, -4, -1, -1, -5, -2, 7, 6, -3, -1, 3, 0, 0, 0], [3, -6, -5, 2, -2, -17, 15, 5, -1, -6, 5, -4, 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4, -3, 3], [-14, -9, -9, 6, -8, 4, -13, -1, 4, 2, -5, 3, 0, -11, -4, 5, 2, 0, -5, -7], [4, 1, 7, 3, 4, -12, -2, 3, 1, -4, 2, 0, -2, -17, 4, -1, 8, 8, -2, 0], [10, -4, 7, 6, 2, -9, -1, 7, -4, 5, 8, 0, 6, -11, 0, -7, -4, 2, -4, 1], [5, 8, 5, 2, 8, 5, -3, -5, -3, 11, 14, 3, 1, -11, -16, -2, 1, 1, 3, 3], [-28, -10, 10, -1, 2, 7, -12, 1, -2, 5, -10, -3, 1, 7, 0, 1, 7, 6, 4, 3], [23, -11, -3, -5, -3, 5, 4, -7, 12, -4, 15, 8, -9, -1, 0, 8, 4, 2, -10, -2], [-10, 5, -8, 1, -1, -9, -11, -7, 2, -11, -12, -7, -6, 8, 7, 11, -6, 1, -4, -5], [-4, -5, -9, 6, -15, 3, -7, -8, 3, -5, -20, -2, -4, 13, 3, 12, 4, -7, -8, 2], [24, -5, -6, 1, 2, -14, 17, -3, 2, 5, 25, 6, 6, -9, -3, 4, -5, -7, 4, -1], [-19, -14, 0, -3, -3, 26, -24, 10, -5, 0, -23, -3, 4, 26, 1, -3, 7, 3, -3, -1], [1, -3, -5, -7, -4, -8, -5, 14, -6, -10, -16, -10, -2, 10, 7, -2, 1, -3, 5, -5], [7, 0, 9, 8, 7, -7, -1, -3, -9, 2, -8, 4, 4, -1, 10, -5, 3, 9, -4, 2], [12, -3, 7, -1, -3, 2, 0, -2, -3, 0, 11, 2, 3, -1, 0, -1, -3, -12, -3, -1], [7, 3, 2, 2, 0, -9, 3, 7, -2, -1, 12, 1, 0, -8, -4, 0, 0, 7, -6, 7], [-7, -7, 0, 1, 5, 5, 0, 0, 2, 1, 2, 1, 5, -1, 5, 4, 11, 18, -12, -6], [9, 6, 0, 6, 1, 5, 7, -9, -3, 8, -3, 4, 5, 3, 0, -3, -10, -1, 14, 8], [18, -9, 7, 8, -1, -14, -1, -15, 2, 2, 12, 8, 12, 4, 3, 9, 1, -6, -6, -4], [-15, 15, -8, 1, -3, 3, 14, 8, -5, -5, 1, -6, -2, 5, -3, -10, -12, -8, 10, 8], [-12, 4, 2, -1, 2, 7, -19, -12, 2, 5, -16, 3, 8, 13, 4, 9, -5, 0, 1, -1], [-2, 0, 12, -5, 3, -8, 0, -3, 2, 6, -1, 4, 8, 4, -8, -5, 0, -3, -2, -3], [-4, 2, 2, 3, -3, 1, -15, -7, 4, 7, -15, 3, 7, 24, 2, 3, -4, 3, -9, 2], [3, -9, 3, -7, -5, -11, 9, -1, -4, 7, 7, 5, -1, -15, 0, 1, 4, 12, 10, 0], [4, 2, 8, -10, 14, 10, -7, -14, 4, 11, 11, 2, 7, -7, -1, -2, -3, 5, 6, 2], [9, -6, 3, -2, -6, -9, -13, -7, 4, 7, -7, 10, 7, -4, -9, 4, 2, -5, 11, 2], [9, -14, 1, 5, -1, -12, 1, 0, -6, 3, 9, -3, 1, 12, 1, -5, 5, -2, -3, -2], [2, -9, 3, -11, -6, -13, 2, 4, 5, -6, 5, 1, -5, 2, 0, -2, 0, -6, 4, -1], [-4, -5, 4, 9, -5, -11, -3, 1, 3, -5, -16, -4, 0, 12, 4, -1, 10, -3, -7, -2], [-3, 3, -1, -4, -10, -12, 11, -6, 5, -11, 10, 2, -6, -4, -11, 4, -6, -11, 10, -7], [-24, 0, -6, 3, -9, -10, 8, 21, -11, -6, -17, -15, -4, 9, 11, -10, 8, -2, -12, -2], [-6, -1, -8, 5, -4, -2, -9, -11, 3, -5, 16, 6, -2, 9, -5, 16, 4, -2, -2, 0], [-26, -2, 9, -11, -2, 17, -23, -1, 7, -5, -11, 1, -13, 32, 3, 6, 4, 6, -1, 2], [5, 6, 3, -5, 6, -7, -2, -6, 4, -3, 12, 13, -1, 5, -2, 9, -4, -2, 7, -2], [-3, 2, -14, 4, -20, -20, 16, 12, 2, -11, -8, -8, -6, -24, 4, -1, 3, -2, 3, -3], [-12, 0, -5, -11, -4, 10, -7, 9, -1, -13, -18, -6, -6, -1, 6, 1, -5, 3, -1, -2], [-3, 2, 2, -5, -6, -27, 6, 7, 2, -4, 6, -1, -3, -14, -5, -8, -10, -17, 6, -4], [-6, 10, 13, 4, 9, 12, -7, -13, 3, 16, 5, 5, 7, -2, 0, 2, -9, 3, 8, 12], [15, 6, 7, -7, 8, 8, -1, 2, 6, -9, 0, 1, -4, -6, -7, -3, -6, 5, 4, 1], [-17, 10, -1, 6, 3, 8, -11, -1, -5, -10, -13, 4, -2, 2, -2, 3, 6, 0, -8, -5], [19, 13, -7, -11, -9, 2, 10, 9, 4, -3, 6, -2, -13, 0, -6, -6, -13, -2, 6, 3], [-12, 13, 3, 0, 15, 6, -10, -10, -2, 6, 9, -2, 1, -4, 2, -1, -4, 12, -8, 2], [16, 0, -2, 11, 10, -10, 5, 2, 0, 0, 4, -9, 8, -9, 4, 1, 5, 5, -11, 0], [20, -14, 2, -4, 4, -4, 2, 4, -1, -5, 2, -2, 3, 2, 12, -6, 6, 7, -5, -5], [8, -8, 10, -5, -3, -11, 18, 10, 0, 0, 20, 0, 6, 6, -5, -9, 5, -7, 10, -4], [-4, 6, 7, 0, 4, -3, -5, -3, -2, 1, -11, -4, -10, 11, -6, -4, -7, -2, -4, 1], [2, -17, 0, 2, 2, 3, -16, 6, -6, 3, -18, -9, 2, 12, 10, -5, 2, 5, -9, -8], [-10, 5, -15, 5, -7, 16, -10, -5, 11, -5, -16, 4, -8, 9, -15, 9, -8, -12, 1, -9], [-16, 5, 9, 0, -1, 8, 0, 4, 4, 0, -11, 6, 1, 22, -5, 5, 2, -10, 4, -2], [5, -6, 0, -9, 1, -6, 5, -1, -2, 2, -4, -2, 1, 3, 2, -12, -8, 1, 12, 6], [19, 8, 13, -2, 25, 4, -19, -20, 3, 14, 24, 15, 12, -2, -4, -3, -7, 9, 0, 15], [13, -11, -1, 6, -2, -9, 18, 4, -5, -8, 10, -2, 0, -12, 2, -4, 6, -5, -2, 4], [-14, -2, 5, 2, -8, 24, -7, -3, -1, -3, -16, -1, 1, 22, -7, -2, 0, 2, 4, 7], [44, 5, -4, 11, 3, -13, 21, -5, 2, 1, 20, 8, 15, -14, -2, 2, 4, -5, 5, 1], [16, -6, -12, 6, -4, -9, 1, -4, 5, -5, -6, -2, 3, -15, 1, 13, -12, 5, -6, -2], [-4, 5, 0, 3, -4, -3, -13, 4, 6, -7, -1, -4, -10, -12, -10, 0, 7, -11, 1, -2], [15, 8, 7, -6, 9, 25, -10, -12, 13, -1, 4, 8, 3, 2, -2, 6, 1, 8, 8, 4], [-10, 0, 4, 4, 13, 13, -17, -6, -4, 1, -15, -2, 9, 1, 10, -1, 1, 14, -7, 6], [3, 6, 2, 4, 2, -7, 22, 7, 8, -1, 21, 2, 9, -19, -4, -5, -2, -1, 1, 10], [20, 7, 7, 5, 10, 16, -19, -16, 7, -2, -4, 4, 1, 0, -2, 10, 2, 3, 4, -4], [1, 5, -8, -3, 1, 29, -9, 4, -2, -1, -17, -2, -4, 17, -2, 1, 0, -2, 4, 0], [-13, 19, -4, 6, -1, 6, 8, 5, -3, -8, -3, -2, 2, -8, -2, 7, -1, -3, -1, 0], [11, 2, -12, -2, -7, 0, 20, 13, -3, -20, 20, -5, -16, -5, 6, -1, 10, 2, -4, -7], [-15, 3, -2, 3, -9, 7, 5, -2, 6, -4, 9, 1, 0, -1, -7, 4, 0, -5, 0, -5], [9, 4, -1, 0, 0, -9, 26, 3, -5, 3, 6, -3, -2, -2, -5, -10, -7, -2, 3, -3], [14, -1, -2, 4, -2, 0, 7, 2, -1, -1, 3, 6, 3, -8, -6, 2, 4, -9, 0, 0], [12, 5, -14, 1, -10, -7, -9, 5, -3, -1, 4, -9, -9, -13, 2, 1, 0, -7, -8, -7], [-5, 7, 3, 7, 0, -1, 0, 8, -14, 6, 7, -2, 9, -3, 0, -10, 4, 2, 7, -1], [9, 6, -6, -7, 2, -8, 5, 1, 3, -9, 17, 2, -9, -2, 3, 8, 3, 1, 12, -4], [12, 5, 11, 4, 15, 0, 9, -7, -7, 3, 22, 0, 2, -5, 0, -9, -3, 0, -2, 5], [-4, 9, -4, 5, 6, 5, -4, 3, 4, 0, -9, 1, -8, 11, 5, 8, 9, -4, -8, -2], [14, 11, 9, -2, 25, 5, 11, -10, 2, 8, 23, 7, 6, -16, 2, 0, -5, 3, 12, 13], [-19, 17, 4, 3, 6, -8, -3, -14, -4, -4, -3, 6, 4, 4, 1, 6, -4, -2, 1, 2], [7, 16, 1, 0, 0, 13, -1, -1, 2, 4, 2, 7, -2, 2, -18, 2, -19, 3, 9, 12], [17, -12, 8, 3, 7, -8, -12, -2, -2, 10, 3, -3, 1, -7, 5, 1, 7, 0, -5, -4], [-3, -12, 9, -6, -8, 5, 12, 9, 5, -7, -4, -9, 0, 15, 7, -10, 3, -9, 6, -1], [-10, 11, -15, 8, -13, 12, -11, 10, 1, -17, -16, -9, -13, 23, -1, 5, 5, -2, -8, -3], [5, -22, 6, -5, 2, -16, 11, 7, -13, -2, 13, -8, 5, -11, 19, -4, 21, 16, -7, -2], [0, -2, -5, -2, 3, -4, 7, 4, -3, -13, -10, -9, -10, -8, 5, -3, 6, -3, -9, -8], [3, 8, 12, -9, 17, 0, -8, -1, -10, 6, 11, 4, 3, -9, -7, -7, 1, 7, -11, 2], [14, -14, 0, -10, -6, 3, -7, 1, 8, 2, 3, 3, -4, 15, -11, 8, 5, -4, 3, -13], [14, 3, -19, 1, -12, -2, 6, -2, 6, -3, 14, -3, -9, -18, -6, 2, -4, -4, 14, 3], [0, -2, -14, 0, -8, -19, 8, 6, -3, 2, 4, -1, 1, 3, -3, -1, -8, -1, 6, 2], [-1, 0, -5, 1, -8, 2, 11, 8, 2, -5, 22, -2, 3, 8, 4, 5, 16, -8, 6, -1], [21, -10, 9, -6, 6, -12, -8, -7, 5, 7, 23, 3, 9, -2, -7, 3, 1, -7, 15, -2], [-1, -15, -7, 4, -9, -1, 14, 5, 3, -8, 10, -1, -3, -3, 13, 7, 14, 1, 3, 1], [8, 9, 8, -5, 9, 17, 3, -7, 4, 13, 10, 3, -3, -5, -1, 6, -8, 12, 3, 5], [12, 0, 20, -7, 15, 13, 5, -2, 6, -1, 23, 14, 3, 5, -10, 0, -2, -3, 4, 13], [4, 0, 11, -3, 7, -6, -10, -7, 6, 7, 1, 5, 8, -2, -1, -5, -4, 5, 8, 8], [-23, 3, 0, -7, -3, 15, -32, -6, 5, -5, -22, 8, -10, 23, -7, 9, -3, -3, 8, -3], [-13, -12, 5, 4, -2, -17, 18, 13, -9, 4, -4, -7, 7, -11, 9, -3, 11, -5, -15, -5], [11, 11, -13, 11, -14, -5, 14, -10, 7, 5, -12, 9, 2, 7, -7, 8, 1, -11, 13, 2], [12, 2, -6, -2, 5, -2, -6, -6, -8, 1, 12, -1, -7, -13, 3, 0, 2, 1, -5, -1], [14, -10, 3, -5, 2, 1, 12, 0, -3, 14, 5, -1, 5, 6, -5, -6, 4, 4, -3, 0], [24, 1, 10, -1, 5, -10, 17, 3, 2, -3, 25, 4, 7, -3, 4, -2, 3, -4, 9, 4], [-5, -4, -13, -1, -4, 15, 4, 4, 1, -1, -8, 1, -7, 5, 0, 4, -6, -1, 2, -1], [1, 9, 4, 2, -3, -16, 3, 1, -13, 3, -8, -10, 0, 8, 9, -7, 5, 1, 3, 4], [-3, 5, -11, -5, -1, 16, -14, -9, 17, 3, 4, 4, -10, 0, -8, 16, -11, -1, 4, -2], [3, -4, 3, 3, 12, 10, -11, -17, 2, 4, 0, 9, 8, 4, -4, 0, 6, 11, 0, 10], [-27, 3, -10, 7, -9, 26, -17, 12, -3, 4, -27, -6, -2, 9, 2, -6, 10, 4, -2, 2], [-20, 2, 6, 5, 0, 6, -11, -3, -12, 2, -2, 7, 4, 15, 6, 3, -3, 7, -7, 4], [31, -6, -2, -1, 7, 4, 9, -15, 0, -2, 8, 5, 0, -10, -5, 8, -2, 7, -3, 5], [-14, -1, 0, -8, -5, 11, -15, -4, 0, 5, -22, 0, -11, 10, -3, -3, -5, -11, 4, -3], [43, 6, 1, 6, 10, -20, 13, -8, -2, 3, 11, 8, 4, -24, 0, 0, 0, 7, 1, 1], [3, -9, -4, 6, -8, -11, 10, 14, -12, -9, -14, -22, 1, -4, 8, -14, 2, -4, -12, 0], [20, -2, 0, -2, -6, 14, 4, 6, 3, -7, -4, 2, 0, 7, -4, 2, 6, 2, -10, -5], [4, 3, -5, -8, 0, 0, -14, -10, 0, 1, -5, 7, 2, -18, -8, 5, 2, -10, 7, -5], [-14, 5, 5, 3, -2, 1, -8, -4, -5, 9, -11, -3, 3, 15, -6, -3, -11, -12, 6, 10], [-21, 12, -5, 1, -1, 16, 0, 4, 11, -1, -8, 2, -7, 4, -8, 7, -2, -7, 2, -1], [-31, -5, -7, 3, 1, 22, -3, 13, -6, -5, -34, -11, 4, 21, 6, -11, 5, -1, -4, -3], [-15, 0, 7, -3, 5, 20, 2, -3, -10, 3, -18, -3, 4, -10, 8, -7, 10, 14, -5, 4], [7, 9, 8, 4, 7, -17, -2, 11, -5, 12, 3, -1, 4, -3, -12, -14, -3, -3, -3, -5], [8, 10, -6, -4, 6, 11, -30, -12, 13, -3, 9, 7, -8, 19, -11, 14, -9, -7, 3, 0], [12, -6, -8, -5, -3, 23, -5, -10, 7, -11, 10, 13, -8, 4, 1, 14, -1, -4, 4, 1], [-7, -2, 5, 6, 5, 21, -31, -16, 3, 12, -5, 12, 11, 16, -3, 4, -8, 8, 0, 6], [-4, 7, 3, 10, 6, -4, -5, 0, -3, 2, -18, 0, 11, 5, 6, -1, 11, 9, -17, -4], [4, 14, -15, -6, 3, -4, -5, -6, 9, 1, 6, -4, -12, -24, 0, -1, -8, 5, -3, -4], [17, 19, -2, 8, 14, 9, -3, -6, 8, 3, 17, 7, 4, 5, -10, 7, 3, 2, 0, 15], [23, 5, 13, 1, 22, -15, -2, 7, -1, 11, 13, 3, 11, -4, -2, -16, 2, 9, 3, 9], [13, 4, 2, -4, 7, -17, 7, 12, -6, 0, 15, -11, 1, -7, 2, -19, 12, 6, 2, 9], [15, -14, 9, -5, 1, -9, 3, -4, 5, 15, 22, 1, 0, 1, -14, -2, 0, 1, 5, 0], [10, 8, 11, -2, 4, 2, 16, 3, -7, 0, 19, -1, 5, -11, -6, -14, -3, 5, 0, 12], [26, 1, -6, 6, -6, 4, 1, 2, -1, -8, 8, -1, -1, 8, -1, 4, 11, -10, 5, 2], [-15, -8, -1, -3, -2, 3, 0, 0, 3, 7, -7, 7, 7, 4, -6, 1, -3, -10, 2, -3], [-12, -5, -3, -8, -3, 4, 4, 3, 2, -8, -8, 2, -1, 19, 6, 8, 3, -5, -4, 8], [8, 1, -16, 1, -1, 13, 7, 7, 8, 3, -8, 3, -2, 2, -3, 3, 0, 9, 7, 3], [-3, 3, -6, -2, -2, 2, 15, 10, -1, -5, 0, -4, -7, -4, 9, -7, 12, 1, -2, -11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1849_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1849_2_a_r();".
// 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_1849_2_a_r(:prec:=20)
    chi := MakeCharacter_1849_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_1849_2_a_r();".
// 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_1849_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1849_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-43, 87, 1386, -1004, -11330, 6093, 32885, -21700, -37889, 29618, 20357, -19360, -5068, 6765, 329, -1293, 95, 127, -19, -5, 1]>],Snew);
    return Vf;
end function;