
// Make newform 6040.2.a.p in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6040_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_6040_2_a_p();".
// 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_6040_2_a_p();".
// 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 := [-5628, -40224, -59750, 91643, 206783, -80919, -262172, 41499, 174105, -17345, -67665, 6652, 15895, -1891, -2208, 325, 165, -29, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1011634103784965924, -7576366659179760236, -33212601804754250690, 16339925852923382945, 82631139058765852268, -14636171244761239999, -80173003627501509915, 8934620047045302660, 40067118650304619217, -4500209660384611576, -11294005761518533393, 1588052770850214607, 1792759675120172710, -325977195535059477, -144066698105804193, 33567815192362180, 3843828599512745, -1282570164104816, 55395572792623], [3451667624814453428, 28137440905699534180, 20011205006996581446, -93399395100110890643, -62145344231637892636, 126453980894546361929, 58655996588257041457, -91839653932711203196, -27241571945505753767, 39992548162303710512, 6541382809435982627, -10775530281108362853, -567866092541121870, 1739613051424907459, -67958043505442033, -150533195307394516, 16656356771269549, 5148768030504856, -825471425164009], [1614654699627908572, -6077448197860154164, -9151291982405240270, 28276765452349644391, 27556736624880712228, -43115468783407000837, -47987975251419114109, 39546479291651782084, 37623785569757123579, -23375799369623611568, -14128580590427639255, 8381720526370062953, 2487208350510041006, -1694597100662116159, -143534913928110307, 172205964148202772, -8468181688802937, -6529221237850736, 831149248013573], [1722308039210366348, 29631649698934677180, 34787223627735288026, -91149929208499445773, -64647877824025594908, 119709595019567528583, 23448483493090406239, -72912510979539031644, 13366198949241451367, 18713949331749328624, -11438341612028607971, -100863235120512307, 2836792745171684454, -857864446451730955, -236985751269460111, 143581904507289700, -4374937933861837, -6709911317038048, 893262017742841], [-982608926580490792, -3889761727382701440, -30204773087751264, 8624873138914767615, 4793176495819915334, -7699778770407449598, -7232763185988054269, 4537703098655126976, 4638648378150119924, -2096524180851425584, -1504470703145431741, 676539570406064409, 242072346115647165, -129569857849845676, -14351726530935302, 12662979091017916, -472631481848911, -466859136090342, 56636909905056], [-6696186093159290140, -25973098539530318860, -27312540334604478130, -9135263353863909511, 80227629513941781164, 107127077210264354573, -85870096282012514947, -124009163196399080772, 51035187199415557773, 65042439574070369888, -19088000936660001737, -18426961314159695961, 4616023233902091338, 2894698946932575287, -689632753169362133, -234583820154108356, 56247718798349417, 7487676163763712, -1842573741871917], [-60581224214440872, 2818443801556378476, 7179337289577138652, -13657286033777995806, -19436518983654857783, 25437466164583338201, 20015000123231956768, -22851621137466968613, -10194286201079735993, 11382041130194319182, 2635534383536689278, -3312637470646319049, -271304883452175172, 555872474701520570, -14899268175785925, -48909518613942212, 5145929270516660, 1683217593650907, -265246968233581], [-16633657644450319404, -77031313802425172604, -18400403424873851690, 233698178817215866373, 124888418158671406260, -297154724962151344431, -155286643777086364727, 213623456478839512564, 86611245129153271489, -95642613213777855920, -24123074281514233269, 26864902225484161459, 2875138811829736290, -4507542403554445333, 46336575550144007, 401669051946076876, -38711864397166811, -14007918034232216, 2144199181458703], [11925598053551983396, 89834398748893076436, 116871191622166077710, -225766875158450189399, -363351139346577945276, 267992302397578881109, 399813773255867433917, -206900403791732026524, -217003369085973033403, 104760723853281959600, 62547489597013713767, -32741676166555832273, -9019156893641800262, 5903216243609510583, 419594042994086787, -548476037235782756, 30277054135873193, 19553203998112072, -2543053966709621], [5986425026293859236, 32005685014482075124, 26840551255631058438, -90911545845393234887, -96314212157456324132, 112635352391755816023, 106048900728093493181, -80756436732202726424, -56032071984249745465, 36144995093235280244, 15555986639844284415, -10077000035553477193, -2131813093126046692, 1666477337278914385, 84746839523124309, -145923623282507952, 8667195040728045, 5012064203537208, -648499956243255], [8167086807839149108, 30638708633486468964, -22574056397359268538, -126908161268216007379, 27096772152993291772, 202499225907393933433, -27366939526710095807, -159794259178220365380, 20776420739350952313, 69983881375983107312, -9974466268086971869, -17811058676487844173, 2870798724368680730, 2607313444268021163, -474806648528782897, -201481782286309668, 40860651477333453, 6233607352857088, -1374745276971833], [6670908614249021236, 24582797901195324452, 11835540468138859814, -77599544991083627219, -121490766031181554524, 127604654796804573089, 205522033580513668913, -138163079917435060844, -141094929426519309839, 87392079449956939424, 46387313396762744579, -31024857777623091109, -6978674237471914710, 6020226093986778971, 237007316844629479, -584514148862704228, 43569315816672797, 21385926549656952, -3039888107633601], [-10388116111708522412, -65490892598179360316, -58440175368525168138, 182585464307163511781, 205953568009256349140, -221308435501997123887, -235389831702239684215, 162421707726185401652, 129528865730130573569, -76764217475065407952, -37297691424213731637, 22691217218032930995, 5278789286287108706, -3937975463808284437, -219304735752279001, 357014723638780620, -21564673027731835, -12540283023594488, 1652069990496175], [-17922009821745376204, -97455190035628303772, -72028452155976973882, 316755291677928609789, 296565179915255069556, -457820635907237577503, -349667716662878613727, 373141090695751187588, 191189145105819027985, -181821342301299894016, -52220481059478395469, 53110027235529915035, 6088557687263850570, -8966005037752738805, 86357606179246823, 789591293776478876, -75818776899204467, -27103994474524504, 4125642740291903], [33803425560306121028, 175451898025721634132, 39594315574452729118, -638524446302924062879, -234702087701587701668, 939373349287456002501, 254723974714131028581, -719721317464493735924, -114197716146189228923, 316724397005442684960, 18907168699394828847, -82659828787906528913, 1861805423561134778, 12534591592790226543, -1114941497406725869, -1005499581532864900, 141604639245533105, 32134720242086384, -5734380684972101], [-19426824517150598020, -122429703947055203924, -88184304972892466494, 426181604695001605439, 348640318779203430292, -641770077230588731133, -410391423955509687781, 530761454648421512276, 223294169369261280803, -258328728461468122512, -59542132916546240239, 74989229416502241633, 6177222909748312958, -12588330322487702519, 346768879365177429, 1104923052372439220, -117375998014468801, -37872303245196736, 6000740936115901]];
    Rf_basisdens := [1, 1, 1, 228771461625160096, 457542923250320192, 457542923250320192, 457542923250320192, 28596432703145012, 457542923250320192, 57192865406290024, 457542923250320192, 457542923250320192, 114385730812580048, 228771461625160096, 457542923250320192, 228771461625160096, 457542923250320192, 457542923250320192, 457542923250320192];
    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_6040_a();"
function MakeCharacter_6040_a()
    N := 6040;
    order := 1;
    char_gens := [1511, 3021, 2417, 761];
    v := [1, 1, 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_6040_a_Hecke(Kf)
    return MakeCharacter_6040_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, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [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, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0], [-3, 1, 0, -2, 0, 0, -1, 1, -1, 0, 0, 0, -1, 1, 1, 0, 0, 1, 1], [-2, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0, -1, 1, 1], [-1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1], [0, 0, -1, 1, 0, 0, 1, -2, 1, 0, 0, -2, 1, 0, -1, -1, 1, -1, -1], [-1, 0, 0, -1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 1], [0, -1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0], [0, 1, 0, 0, -1, 0, -2, -2, 0, 1, 2, -1, 0, 0, -1, -1, 0, 0, -1], [-2, 1, 1, -2, 0, -1, 0, 2, 0, -2, 0, 1, 0, 0, 1, 2, -2, 1, 2], [-3, -1, -1, -2, -1, 0, 0, 0, 0, 0, -1, 1, -2, -1, 1, 0, 0, 1, 0], [-5, 0, 0, 0, 1, 1, 1, 1, -1, -1, -1, 0, -1, 0, 0, 0, 1, 0, 0], [0, -1, 0, 2, 0, 0, 1, -1, 1, 0, 0, 0, 1, -1, -1, -1, 0, -2, -2], [-5, 2, 0, -2, 1, 1, -2, 0, -2, 0, 0, 1, -2, 0, 0, 0, 0, 2, 1], [1, 1, 0, -1, 1, 1, -1, 0, -1, 1, 0, 1, -1, 0, 0, 0, 0, 0, 0], [-5, 2, 1, 0, 1, -2, 1, 2, 0, -3, -1, 0, 4, 2, 0, 1, 1, 0, 0], [-6, 3, 0, -4, 1, -2, 0, 3, -1, -4, -1, 1, 1, 1, 1, 1, -1, 1, 0], [1, 0, 1, 1, -1, 0, 0, -2, 1, 3, -1, -1, 0, 0, 0, -1, 2, -1, -1], [1, 1, 0, -1, 1, 0, 0, 1, 0, 1, -1, 1, 0, 1, 2, 1, 0, 0, 1], [0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 2, 0, 3, 1, 0, 0, 1, -1, -1], [2, 0, 0, 3, -1, 0, 1, -3, 2, 2, 1, -1, 0, 0, -1, -1, 2, -2, -1], [-3, 1, 0, 0, 1, 0, 2, 1, 1, 0, -1, 1, 1, 1, 1, 0, 1, 0, -1], [-6, 1, -2, -3, 1, -1, 1, 1, -2, -1, -2, 0, -1, 1, 1, 1, 1, 1, 0], [-2, -1, 2, -1, 0, 0, 1, 1, -1, -1, 0, 0, -1, 0, 1, 0, 0, 0, -1], [3, 0, 1, 1, -1, 0, 1, 0, 1, 1, 0, 1, 1, -1, 1, 0, 0, -1, 0], [-6, 1, 1, 0, 1, -1, 2, 2, 0, -3, -1, -1, 1, 1, 2, 2, 0, 0, 0], [-5, 0, 0, -3, 2, -2, 1, 4, -3, -3, -2, 1, -2, 1, 4, 1, -2, 1, 2], [-2, -2, -2, 0, 1, 1, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, -2, 2, 2], [4, 0, -1, 4, 0, 2, -1, -4, 2, 5, 2, 1, 1, -1, -3, -1, 1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 4, 1, -2, 0, 0, -1, 1, -1, -1, -1, 0, 1, 2, 1, 2, 1, 1, 0], [-9, 1, 2, -3, 1, -2, -1, 4, -3, -4, -2, 0, 0, 1, 1, 1, -3, 0, 2], [0, 2, 3, 4, 0, 1, -1, -2, 0, 3, 1, 1, 0, -1, -2, -1, 1, 0, -1], [5, -2, -2, 2, -1, 2, 0, -6, 2, 7, 2, -1, -1, 1, -1, -4, 4, -1, -2], [-5, -1, -1, -3, 1, 0, 0, 2, -3, -3, -2, 2, -3, -2, 0, 2, -2, 2, 2], [5, -3, -1, 3, -3, 1, 0, -5, 2, 3, 2, -2, -1, -1, -3, -3, 1, -3, -3], [-6, 3, 1, -2, -1, -1, 0, 1, -2, -2, 0, 0, 1, 3, 1, 1, 1, 1, 1], [-1, -2, 2, -2, -2, -2, 0, 1, -1, 0, 0, 0, -1, 1, 2, 0, 0, 0, -1], [7, -4, -1, 3, -3, 3, 0, -4, 2, 6, 0, 1, -3, -1, 1, -2, 1, -1, 0], [2, -4, -1, 3, -3, 1, 1, -5, 2, 5, 1, -1, -2, -1, -1, -1, 3, -2, -1], [-6, 1, 0, 0, 2, 0, -1, 2, -2, -1, 1, 0, -1, 0, 0, 0, 0, 0, 0], [-5, 1, -1, -3, 0, 0, 0, 0, -1, -1, -4, 2, -3, -1, 2, 1, -1, 3, 3], [-8, 0, 0, -2, 0, -1, 1, 2, 0, -4, 0, -1, 0, 1, 0, 0, -1, 1, -3], [1, -1, -1, 4, 1, 0, 1, 1, 3, -2, 1, 1, 2, -2, -2, 0, -1, -2, -3], [0, 0, 1, 2, 0, 1, 0, 1, 1, 0, -1, 1, -1, -2, 0, 1, 1, -2, -2], [-6, -1, 0, -4, 2, -1, -1, 4, -2, -4, -2, 2, 1, 0, 3, 3, -3, 1, 3], [1, -2, 0, 2, 0, 1, -2, -1, 0, 2, 1, 2, -2, -2, -2, 2, -2, 0, 1], [-1, -2, 1, -1, -3, 1, 0, -2, 2, 3, 0, -2, -1, 0, 2, 0, 1, 0, 0], [8, -1, -1, 4, -3, 2, -2, -6, 2, 4, 2, -1, -1, -1, -3, -4, 1, -1, -3], [1, -1, -1, 0, -1, 3, 0, -2, 2, 2, 1, -1, 1, -1, -2, 0, 0, 1, 0], [-8, 0, 2, -3, 0, -3, 1, 8, -1, -9, 1, 0, 4, 2, 3, 4, -3, 1, 1], [0, 1, 0, 4, -1, 1, 1, 0, 0, -2, -1, -2, 0, -1, -1, -1, -1, 0, 2], [2, 2, -1, 1, 0, -2, 0, 1, 0, -1, 1, 0, 2, 0, -1, 2, 1, 0, 1], [6, -3, 2, 4, -2, 0, -5, -2, 0, 1, 5, -2, 1, -2, -4, -2, -2, -2, -2], [-1, -2, -3, -2, 1, 3, -1, 0, 0, 1, 1, 3, -3, -1, 0, -1, -2, 2, 3], [6, -4, 1, 3, 1, 1, 2, 1, 1, -3, -1, 0, 3, -2, -2, 0, -1, 0, -1], [-2, -1, -2, -4, -1, -1, 0, 0, 0, 2, -4, 0, -3, -1, 2, 2, 1, 0, 3], [-2, 2, 3, 5, 1, 1, 1, -1, 1, 0, 0, -2, 2, 1, -2, 0, 2, -2, -1], [-2, -1, 0, 0, 2, 0, 1, 3, -1, -3, 0, 0, 1, 2, 1, 0, 0, 2, 3], [-7, 0, -3, -5, 3, -3, 1, 5, -4, -7, -3, 0, -1, 1, 3, 3, -4, 2, 3], [-1, -1, 1, 1, 0, 0, 0, -3, 0, 3, 1, -3, 0, 0, -2, -5, 3, -1, -2], [4, -1, 1, 0, 0, -1, -2, 0, -2, -1, 2, -1, 1, 1, -1, -2, 1, -2, -4], [-2, -1, 0, -2, 1, -1, -1, 4, -2, -3, 2, 0, -1, 0, -2, 0, -4, 2, 1], [-5, -3, 2, 4, -2, 0, 0, 2, -2, 1, -2, -1, -3, -2, -1, 0, 1, -2, -1], [-2, -3, -3, -2, 1, 2, -1, -1, -1, 1, 0, 0, -3, -2, -1, -3, 0, 2, -1], [7, -1, -1, 1, -2, 1, 0, -3, 4, 0, 3, 0, 4, -1, -2, 1, -1, 0, 1], [2, 2, 0, 4, -1, 1, -1, -2, 2, 2, 4, -1, 2, 0, -2, 1, 0, -2, -2], [0, -7, -1, -2, 0, 0, 0, -2, 0, 5, 0, -1, -4, 0, 3, -2, 2, 0, 1], [2, -9, 2, 5, -1, 0, 1, 0, 3, 1, 2, 0, -1, -2, 0, -4, -2, -4, -1], [11, -2, 1, 5, -3, -1, -2, -8, 5, 7, 4, -2, 1, -1, -3, -4, 1, -4, -3], [7, -5, -1, 5, -2, 1, 0, -4, 5, 2, 2, 0, -1, -4, -2, 0, -2, -3, -1], [3, 3, -2, 0, 1, 2, -1, 0, 0, 2, -1, 0, -1, 0, -2, -2, 0, 1, 2], [-10, 4, -1, -4, 0, -1, 0, 1, -4, -1, -4, 1, -1, 2, 0, 2, 1, 3, 2], [-6, 0, -2, -4, 3, -5, 0, 5, -1, -5, -1, 0, 1, 2, 2, 1, -1, 0, -1], [9, -4, 3, 3, 0, 0, -1, 0, 1, -3, 4, 0, 3, -1, -3, -2, -2, 1, -2], [-10, -3, -2, -3, 2, 2, 0, 2, -3, -3, -1, 3, -3, 1, 0, 0, -3, 5, 3], [-5, -4, -3, 2, -1, 0, 2, -4, 2, 0, -3, -3, -2, -1, -2, -2, 1, -3, -1], [6, -1, 2, 0, 0, -3, -1, 0, -1, 0, 0, 0, 1, 1, 0, 0, -2, -1, 0], [-1, 1, 1, 1, 1, 1, 0, -1, -1, -1, 2, 0, 2, 3, -1, -2, 0, 0, -3], [-5, 1, -2, -3, -1, 1, 0, -1, -1, 2, 0, 2, 1, 0, -1, 3, 0, 3, 3], [13, 2, 2, 1, -2, 2, -4, -4, 3, 6, 2, 1, 0, -1, 0, -1, -1, 1, 2], [11, -3, 2, 6, -2, 1, 0, 0, 2, 1, 2, -3, 2, -2, -3, -2, -1, -3, -3], [-11, -2, -1, -2, 0, -1, 3, 1, 2, -2, -3, 2, -4, -1, 3, 3, -3, 2, 1], [-9, 1, 0, 0, 1, 2, 0, 1, -3, 1, 0, 1, 0, 2, 2, 0, 5, 3, -1], [-4, 1, -2, -6, 1, -2, 0, 1, -1, -5, -1, -1, 1, 2, 1, 2, -3, 1, 3], [-8, -4, -1, -7, 0, 0, 3, 6, -3, -4, -2, 1, -2, -1, 4, 3, -3, 1, 1], [-1, 2, 1, -2, -3, 2, -1, -5, -1, 3, -2, -2, 0, -2, -3, -1, 1, 2, -1], [-1, -1, -1, 3, 1, 2, -2, -1, 1, 5, 2, -2, -1, 0, -1, -2, 0, 1, 0], [9, -1, 3, 6, -3, 2, -2, -3, 3, 2, 1, -2, 3, 1, -1, -1, 3, -4, -5], [-12, 0, -2, -9, 2, -1, 0, 4, -5, -1, -1, 3, -7, 2, 5, 2, -2, 5, 2], [-7, 4, 1, -4, 4, -2, -1, 3, -5, -6, -1, 2, 2, 3, 1, 2, -2, 2, 2], [9, 0, 1, 7, 1, 2, -3, -3, 3, 8, 4, -1, 0, -1, -6, -5, 3, -3, -3], [4, 2, 0, -2, 1, -3, -3, 2, 0, -5, 0, -1, 3, -1, -1, 1, -3, -1, 0], [-10, 3, 1, -3, 2, -2, 3, 2, -1, -1, -1, 0, 3, 2, 1, 0, 3, 0, -3], [2, -5, -1, 6, 1, 2, 3, -3, 3, 5, 0, -2, 2, -1, 0, -1, 4, -4, -1], [11, 5, 1, 8, -2, -2, 1, -6, 6, 2, 2, -4, 6, 0, -5, 0, 4, -4, -1], [-9, 0, 4, -2, 1, -2, 1, 4, 0, -6, 2, 1, 3, 0, 1, 0, -3, 3, 0], [-4, 6, 3, -3, -1, -2, -3, 1, -3, -4, 2, -2, 1, 3, 0, -2, 2, -1, -2], [-12, 4, -3, -3, 0, -1, 4, -2, 3, 2, -5, -2, -1, 1, 3, -1, 6, 0, 1], [0, 2, -2, -9, 1, 1, -2, 3, -4, 0, -1, 4, -4, 1, 1, 1, -1, 3, 0], [-11, 4, 1, 0, 1, 0, -2, 5, -2, -3, -2, 3, -4, -2, -1, 5, -4, 3, 0], [2, 1, 0, -1, -1, 1, -3, 3, -2, 0, 0, 4, -2, -1, 1, 3, -2, 4, 6], [-8, 0, 1, -4, 1, 2, 1, 1, -3, 0, -4, -1, -5, -3, 0, 2, 0, 0, 2], [-17, -2, 2, -8, 3, 0, 0, 5, -7, -4, -3, 3, -3, 4, 3, 2, 0, 3, 2], [9, 0, 2, 8, -1, 0, 3, 1, 4, -4, 0, -2, 7, -3, -2, 2, -1, -5, -2], [8, -4, 1, 5, 0, 1, 0, -1, 2, 4, 1, 0, -1, 1, 0, -2, 1, -1, 4], [-1, -8, -3, -1, -2, 2, 3, 2, -1, 0, -3, 2, -4, -2, 4, 2, 0, 0, 2], [1, -1, -3, 1, 0, -1, 1, 1, -2, -1, -1, -3, -2, 2, 0, -2, 3, 1, -2], [6, 2, 2, 4, -4, 1, -2, -8, 2, 8, 3, -2, 3, 0, -5, -4, 4, -2, -3], [-4, 0, -1, 2, -1, 2, -1, -4, 2, 4, 2, 2, 0, 0, -4, -1, 3, 1, -3], [-6, 5, 4, 0, 5, -2, 0, 5, -1, -7, 1, 0, 4, 4, 2, 2, -1, 3, 3], [-7, 7, 1, 0, 2, -2, 3, 2, -2, -6, -1, -2, 5, 3, 0, 2, 1, 0, 1], [3, 5, 5, 8, 0, 1, -5, 0, 2, -1, 3, 3, 5, 0, -4, 2, -3, -1, -2], [0, 0, -7, 2, 1, 1, -3, -7, 1, 6, 0, 0, -1, 0, -5, -2, 4, 1, -2], [1, -5, -1, 2, -1, 0, 0, 0, 2, -2, 0, -4, 1, -4, -2, 1, -3, 0, 0], [10, -7, -4, 3, -2, 2, 2, -6, 5, 4, 2, 1, 1, -2, -2, -2, 1, 0, 1], [1, 1, 5, -4, 4, -3, 2, 9, -6, -8, -1, 2, 2, 5, 4, 1, -3, 2, 1], [-8, 0, 1, 0, 1, -5, -1, 3, -3, -2, -2, -4, 2, 3, -2, 0, 1, -2, 1], [-7, 7, 2, -7, -3, -2, -5, 3, -4, 0, -1, 3, -3, -1, 3, 4, -2, 3, 4], [7, -6, -2, 2, -4, 5, -1, -6, -1, 7, 1, -1, -2, 0, 0, -1, 2, 0, -1], [0, -3, 1, 0, 0, 2, 0, 2, -1, -2, -1, 2, 0, -3, 2, 4, -4, 1, 4], [4, 0, -1, -4, 2, -1, -3, 3, -3, 1, 2, 3, 1, 5, 4, -2, 0, 3, 3], [0, 4, 0, 3, -4, -2, 0, -7, 2, 2, 0, -5, 0, -1, -1, 0, 2, -2, -2], [18, -6, -3, 9, -1, 3, 2, -7, 6, 5, 1, 2, 2, -3, -2, -1, 3, -5, -1], [-4, -1, 0, 4, 3, 3, 1, -1, -1, -2, -2, 0, 4, 0, -6, 1, 1, 0, -1], [-13, -1, 0, -2, -2, -3, 4, 2, 0, -5, -6, -3, 1, 0, 2, 1, -2, -2, -2], [-9, 4, -1, -1, -1, 0, 0, 0, -1, 5, -1, 3, -4, 3, 6, -1, 2, 1, 3], [4, -5, -3, -2, 1, 0, -1, 0, -2, 1, -1, 1, -2, -1, 1, 2, -1, -3, 1], [3, -3, 2, 0, -1, -2, -2, 3, -1, -2, 1, -6, -1, 0, 1, 0, -1, -2, 1], [0, 4, 2, -3, -1, 0, -4, 0, -1, 0, 1, 0, -1, 1, 3, -1, -1, 1, 2], [2, -4, 0, -5, -4, 0, 1, 2, -1, -3, -5, 0, -4, 1, 2, 1, 0, 3, 0], [8, 1, 1, 4, -3, 2, 1, -6, 2, 5, 1, -3, 0, 2, -3, -4, 7, -3, -7], [4, 1, -4, 1, -1, -2, 1, 0, -1, 2, 3, -3, 0, 3, 1, -2, 4, 1, -1], [-6, 1, -1, -5, 3, -2, -3, 7, -2, -5, -1, 4, -1, 2, 1, 3, -6, 2, 8], [0, 6, 4, 2, -2, 1, -3, -2, -2, 3, 5, -1, 0, 1, -4, -2, 1, 2, -1], [-8, -4, 1, 3, 0, 2, -1, -5, 0, 4, -1, 1, -5, -4, -2, -4, -2, -2, -3], [4, 1, 1, 1, -1, 1, -2, 2, -1, -1, 2, 0, -1, 1, 0, 2, -4, 4, 1], [8, -5, -2, 4, -1, 0, -2, -3, 1, 5, 3, 2, -4, -1, 0, -6, -3, 1, 3], [-8, 3, -5, 1, -1, 2, 2, -5, 1, 5, 0, 3, -7, -1, -2, -4, 2, 4, 0], [4, 7, -1, 2, 0, -1, -2, -5, -2, 3, 1, -6, 3, 3, -9, -4, 6, -3, -4], [-3, -6, -1, -3, -2, 2, 0, 2, -3, 3, -5, 3, -9, -1, 5, -1, 2, 1, 1], [2, -5, -10, -3, -2, 3, 1, -8, 1, 7, -2, 0, -4, 0, 0, 0, 3, 4, 0], [-1, -9, -1, 2, 0, -2, 5, 3, 5, -4, 0, -3, 3, 0, 3, 1, 0, -1, 1], [1, 4, 1, 2, 0, 1, 0, 1, 0, 2, 2, -3, 1, 1, 0, 0, 1, -1, 2], [2, -1, -4, -5, -2, -1, 4, -3, 6, 4, 0, -3, 0, 0, 4, -3, 4, -3, -4], [-10, 0, -2, -4, 0, -2, 2, 2, -2, 0, -6, -4, -3, 1, 3, -3, 3, -1, -1], [2, 3, -6, -2, 0, -2, 2, 0, 6, -2, 3, -1, 7, 2, 2, 1, 2, -1, 1], [-8, 9, 8, -1, 4, -6, 0, 10, -1, -11, 0, 2, 4, 1, 2, 5, -5, -2, 0], [8, -1, -1, 6, 4, -1, 2, 2, 2, -4, -2, 2, 4, -1, -3, 1, -1, -4, -3], [-5, 0, -2, 0, -1, 2, 2, 1, 3, -3, -2, 3, 2, -3, 1, 4, 0, 5, -2], [-8, 8, -1, -12, 3, -3, -4, 8, -4, -3, 2, 4, -5, 2, 5, 2, -7, 6, 4], [7, 3, 0, 2, 4, -1, 0, 0, -2, -1, 4, -4, 7, 0, -6, 0, 1, -1, -1], [-14, -2, 1, -15, 5, -6, 4, 15, -10, -14, -6, 5, -2, 4, 7, 6, -5, 5, 5], [-16, 11, 3, -2, 3, -3, 2, 7, -1, -7, -6, -1, 4, 6, 4, 3, 3, 1, 4], [-3, -3, -1, 2, 1, 3, -3, -1, -3, 4, 0, -1, -4, -3, -4, -1, 1, -1, 0], [-3, 1, -2, 1, 3, 0, 1, 7, 4, -8, -3, 5, 6, -1, 0, 7, -3, 0, 1], [6, -7, -5, 6, 0, 5, -2, -8, 0, 9, 2, -1, -2, -1, -1, -5, 2, -1, -1], [5, 3, 0, 0, -2, 1, 1, -10, 2, 8, 0, -9, 2, 2, 1, -4, 6, -3, -3], [15, 4, 0, 5, -4, -2, -1, -4, 6, 4, 2, -4, 2, 0, -2, 1, 6, 0, -3], [-3, 7, -2, 3, 3, 2, 2, -3, -1, -2, -1, 4, 3, -1, -3, 2, -3, 2, 3], [-5, -1, -2, -6, 2, -1, -1, 8, -5, 0, -2, 1, -7, 1, 1, 1, -1, 4, 5], [14, -5, 1, 3, -1, -3, 1, 0, 7, 0, 6, 0, 3, 0, -2, -2, -3, -2, 3], [7, 2, 0, -1, 0, 2, -1, -5, 0, 5, -1, 0, -1, 2, 2, -1, 4, -1, -1], [-12, 3, 0, -5, 3, -2, -1, 4, -1, -1, -3, 5, 0, 4, -1, 3, 0, 3, 2], [15, -4, -2, 10, -3, 2, -2, -10, 6, 8, 5, -1, -4, -3, -6, -7, -1, -1, -5], [5, 1, -1, 0, 0, 0, 0, -8, 0, 5, 1, -2, -4, -3, 1, -3, 1, -1, -2], [-22, 1, -3, -14, 3, -1, -1, 7, -10, -6, -2, 3, -9, 0, 4, 2, -3, 6, 6], [12, -7, -3, 4, -4, 2, -6, -10, 4, 9, 8, -5, 2, -2, -1, -5, 1, -1, 0], [14, -1, -4, 5, 0, 4, -1, -6, 3, 2, 1, 4, -2, -5, -7, -2, 0, -3, -7], [6, -3, -1, -6, 1, 2, 0, 1, -3, -1, 1, 3, -1, -1, 3, -1, -1, 5, -1], [-5, 4, 0, -3, -2, -3, 0, -4, 1, 5, -3, -1, -1, -1, 1, 2, 2, -3, 1], [-27, 2, 2, -7, 3, -5, 1, 12, -7, -12, -5, -4, 1, 5, 1, 2, -1, 3, 5], [3, -2, 2, 5, -1, 5, 2, -6, 3, 7, 1, 0, 2, -2, -1, -1, 4, -4, -5], [-3, -1, 0, 0, 2, 3, 0, 1, -5, 3, -5, 4, -11, -4, 1, 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1, 3, 0, -10, -2, 1, 0, 1], [-24, 7, 9, -2, 3, -2, 1, 8, -4, -3, -5, 3, -5, 0, 8, 1, -4, 0, -2], [13, -5, 1, 12, 0, 3, 6, -4, 7, 1, -3, -4, 10, -1, 1, 0, 8, -7, -5], [-14, 6, -3, -9, 1, 0, 5, 9, -4, -11, -4, 7, -6, -3, 1, 4, -8, 5, 5], [-8, 2, -5, -2, 3, -6, 0, -2, -1, -4, 1, -4, 7, 3, -1, -2, 1, -1, -8], [-2, 4, 3, -4, 4, 5, -1, 5, -3, -3, -1, 4, 2, 3, 0, -1, 2, 2, 0], [-29, 4, -5, -6, 2, -4, 0, 11, 3, -9, -3, 2, 4, 0, 3, 12, -9, 8, 8], [1, 3, 2, 11, 0, 5, 0, -9, 2, 10, 7, -4, 3, 0, -5, -1, 2, 1, -6], [0, 9, 6, 0, 3, 2, -5, 3, -3, 0, 7, 2, 2, 3, -4, -2, 2, 1, 1], [19, -2, -3, 5, -5, -1, 0, -6, 7, 1, 7, -7, 6, 0, -8, -3, -2, -3, -5], [-9, 10, 3, -8, -4, -6, -8, 9, -3, -11, 6, 2, 2, 5, 6, 3, -5, 3, 3], [1, 14, 0, 2, 4, -6, 5, 8, 7, -12, 1, 2, 12, 3, 2, 3, -6, -1, -2], [1, -7, 2, 5, 2, -3, 7, 9, 6, -12, -4, 4, 10, -3, 0, 10, -6, -7, -1], [-21, 9, 2, -18, 6, -4, -6, 17, -9, -11, 5, 7, -10, 4, 6, 4, -9, 13, 10], [-5, 15, 10, 6, -1, 2, -4, -3, -1, 1, 2, 3, 10, 0, -4, 4, 1, 1, -1], [5, 3, 6, -14, 8, -9, 0, 27, -5, -22, 1, 4, 3, 8, 11, 5, -14, 7, 6], [-6, 7, -1, 0, 3, 0, 3, 0, 2, 3, 5, 5, 2, -1, -1, -2, 1, -1, 3], [-2, 3, 6, 4, -3, -8, -1, -1, 1, -9, 4, -12, 14, 4, -4, 1, 5, -5, -6], [-2, -11, 5, 2, 3, 2, -1, 5, -11, -1, -2, 4, -14, -1, 0, -1, -4, 0, -1], [-19, 1, 5, -3, 3, -4, 2, 8, 1, -7, -1, 2, 4, -1, 5, 0, -2, 2, 1], [4, -1, 4, 5, 0, 5, -8, 3, 4, -1, 7, 7, 1, -2, 1, 1, -6, 2, 6], [-20, -2, -2, -11, -1, 2, 2, 0, -1, 3, -10, -4, -4, 1, 6, 6, 6, 1, 3], [-6, 5, -2, 1, -1, -1, 6, -7, 0, 8, 4, -5, -3, 0, 1, -2, 11, 4, -1], [-6, -1, 6, 10, -4, -1, 6, -3, 7, 6, -4, -5, 3, -3, -3, -1, 4, -9, -10], [2, -1, 4, 0, -4, -4, -4, -3, -3, -3, 4, -11, 2, -3, -10, -5, -3, 2, 0], [23, -16, -6, -2, -1, 5, 1, -4, 1, 14, 4, 7, -12, -3, 6, 0, -7, 8, 10], [15, -4, 4, 1, -3, 0, -2, 1, 2, -3, 3, 0, 4, -6, -5, 0, -7, -1, -4], [0, 18, 6, -3, -3, -5, -8, 2, -1, -2, 6, -2, 8, 8, -4, 3, 3, -1, 1], [9, 7, -1, -7, 5, -3, -1, 10, -7, -12, -2, 0, 4, 6, 1, -1, -1, 1, 6], [-23, 4, 3, 4, 0, -1, 4, -1, 2, 5, -2, -2, 0, 1, 0, -3, 0, 1, 3], [13, 7, 7, 2, -1, -3, 0, 6, -2, -7, -4, -7, 2, 9, 3, -2, 4, -4, -2], [-36, 3, 5, -7, 2, 2, -5, -2, -8, 5, -4, 10, -9, -5, -1, 2, -3, 5, 1], [-5, -7, -1, -3, -6, 0, 1, 1, 8, 2, -3, 9, -8, -8, 6, 3, -8, -1, -2], [23, 0, -4, 7, -1, 4, 1, -7, 6, 6, 6, 6, 0, -1, 1, -1, -6, -2, 5], [-7, 1, -3, -14, 1, 3, 2, 8, -8, -6, -5, 8, -8, -6, 9, 9, -9, 10, 5], [-7, 14, 2, 9, 1, -1, -2, -5, 0, 8, -2, -5, -2, 1, -9, -5, 9, -7, -2], [-8, 0, 0, -8, -5, -1, -3, -3, 3, 6, -1, 0, -7, -3, 0, 2, -3, -3, 0], [20, -10, -3, 2, 0, 0, -1, -12, 0, 12, -3, 2, -2, 0, -4, -3, 3, -3, -4], [21, -3, 0, 4, -5, -3, -2, -7, 8, -1, 8, -9, 14, -2, -6, -7, 4, -5, -10], [-20, -2, -3, -12, 5, -3, 3, 7, -5, -16, -5, 0, 3, 7, 4, -2, 0, 2, 3], [-3, 5, 2, 14, -6, 1, -3, -9, 4, -2, 11, -4, 9, 0, -10, -2, 0, -1, -7], [12, -1, -2, 9, -3, 6, -3, -17, 7, 10, 3, -8, -7, -5, -7, -5, 5, -4, -2], [-11, 3, 0, 5, 3, 0, 6, -2, 1, -2, -4, -6, 5, 2, -3, -2, 10, -4, -1], [14, -10, 4, 9, -1, 5, 6, 1, 4, -1, 0, 2, 4, -10, -5, 6, -4, -3, -3], [-22, 5, 2, -14, -2, -4, -3, 1, -5, -5, -7, -5, 1, 4, 4, 0, 5, 4, 1], [5, 2, 2, -5, -1, -2, -4, 3, -6, -6, 0, -9, -1, 6, -3, 1, 1, 8, 4], [15, 3, 8, 9, 3, -2, -2, 11, -2, -9, -3, 4, 0, -1, 0, 8, -7, 3, 0], [-13, 5, 2, -19, 2, -4, 0, 13, -13, -8, -4, 5, -7, 0, 8, 9, -8, 7, 8], [-23, -7, 4, -6, 6, -3, -4, 17, -11, -9, 3, 0, -1, 2, 1, 4, -7, -1, 8], [10, -3, 3, 3, -1, 0, -2, -4, 3, 2, 4, 3, 13, 3, -4, -6, -2, -3, -4], [-4, 7, 1, -5, -1, -8, -5, 14, -5, -13, 2, 0, 5, -1, 4, 7, -8, 0, 9], [-1, 6, 6, 2, -1, 3, -4, 8, -4, -6, 5, 4, 0, -2, -6, 6, -3, 6, 9], [-15, 20, 0, -16, 4, -13, 0, 12, -4, -13, -7, -3, 6, 8, 6, 5, -3, 0, 6], [-18, 1, 10, -12, 8, -6, 2, 16, -6, -14, -1, 3, 0, 7, 3, 2, -1, 6, 2], [-11, 8, 3, -14, 3, -1, -1, 0, -9, -2, 2, 6, -3, 5, -3, 2, 2, 6, -7], [-5, 4, -7, -11, 5, 0, 0, 1, -6, 3, -3, 8, -4, 4, -3, 1, -4, 3, 5], [-1, -4, -3, 6, 0, -7, 8, -4, 6, -2, -2, -8, 9, 4, 1, -2, 9, -9, -8], [-2, -4, -4, 6, 1, -1, 4, 6, 4, -13, 0, -6, 10, -7, -4, 2, -8, -3, 3], [-31, 8, 5, -14, 4, -2, -2, 20, -6, -14, -2, 5, -6, 6, 11, 6, -10, 9, 13], [1, 2, -6, -6, 5, 5, -3, -3, 1, -1, 1, 1, -1, 1, 2, -1, -4, 3, 1], [32, -18, 5, 27, -8, 4, 2, -21, 16, 21, 7, -7, 12, -4, -9, -9, 10, -15, -16], [18, -3, 6, 4, 0, -5, -2, 0, 6, -6, 4, -5, 9, -7, -6, 2, -5, -6, -3], [-13, -6, -1, 9, -1, -4, -1, 3, 3, -3, 4, 0, 9, -2, 0, 2, -2, -5, 0], [-12, -5, 0, 1, 5, -2, -1, 9, -4, -5, -5, 6, 0, 2, 0, 5, -5, -1, -1], [-1, 5, 2, 1, -1, 5, -6, 0, -13, 0, 6, -5, -5, 8, -2, -1, 2, 5, -3], [22, 7, 2, 10, 1, 1, 3, -3, 7, 5, 3, 1, 8, -7, 0, -2, 4, -8, -4], [6, 10, 6, 11, 3, -3, -1, -6, 5, 5, 1, -6, 15, 10, -6, -5, 9, -5, -7], [-10, 10, 6, -7, 4, 1, -2, 7, -4, 7, 2, 2, -3, 8, 6, 3, 5, 2, 8], [6, -2, -6, 5, -3, 1, 7, -18, 7, 6, -3, -9, 10, 2, -6, -1, 10, -4, -11], [10, 5, -3, -9, -1, -6, 0, 0, -1, -4, -8, 3, 0, 5, 7, 1, 1, 0, 1], [17, -4, 1, 17, -2, 5, 4, -10, 7, 14, 5, -3, -1, -6, -9, -2, 5, -5, 3], [0, 4, 0, 7, 2, 0, 1, -5, -1, 1, -6, 1, 1, -4, -7, -1, 6, -5, -10], [21, 1, -5, 11, -8, 4, 5, -9, 9, 6, -8, 3, -4, -5, 2, 4, -1, -4, -2], [-9, 8, 8, -10, 8, -6, 4, 19, -5, -14, -5, 8, -2, 0, 12, 9, -13, 2, 9], [5, 2, 7, -5, -4, -1, -1, -1, 2, 2, 6, -1, 10, -1, -1, -2, 0, -8, -5], [-15, 7, -9, -17, 2, -3, -2, 4, -9, 0, -2, 2, -8, -1, 7, -1, -1, 4, 8], [-1, -4, -6, 8, -9, 4, 6, -7, 13, 8, 0, 1, 1, -7, -6, 0, 0, -8, -3], [-30, 1, 2, 3, 3, -2, 7, 5, -2, -5, -12, -2, -3, -1, 3, 11, -3, 1, 4], [19, 11, 7, 15, 3, 4, -8, -6, 5, 11, 9, 5, 2, -4, -5, -1, -6, -1, 3], [28, 9, -1, 13, -10, 4, -13, -19, 14, 10, 10, -6, 8, -7, -16, -7, 0, -5, -2], [-4, 0, -9, 3, -4, 2, 1, -9, 0, 6, -9, 1, -14, -6, -5, 1, 3, 5, 0], [-29, 6, -3, -21, 6, -2, -1, 13, -8, -9, -7, 2, -3, 5, 12, 2, -4, 7, 3], [9, 4, 2, 10, -4, -4, 7, 1, 9, -8, 5, -10, 11, -2, -2, -4, 3, -7, -11], [8, -7, -4, -6, 0, 2, 5, 2, 6, 5, -7, 4, -1, 0, 14, 2, -2, -2, 6], [-42, 9, -6, -13, 4, -3, 3, 12, -9, -14, -10, 0, 0, 8, 7, 11, 1, 4, 11], [28, -1, -6, 13, -2, 7, -2, -8, 1, 5, 0, -2, -2, -12, -11, 1, -6, 0, 3], [1, 2, 0, 8, 1, 0, 7, -4, 6, 3, -1, -6, 0, 0, 2, 0, 5, -4, -2], [4, 9, -2, 9, 3, -5, 1, -1, -4, 0, 4, 0, 4, 5, -3, -5, 2, 2, 0], [9, -3, -5, 7, -3, -4, 3, -4, 15, -1, 7, 1, 12, -2, -3, 1, -2, -4, -5], [20, -4, -5, 4, -1, 0, 7, -2, 6, -3, 2, -2, 3, -4, -7, -3, -3, -2, -2], [-10, 9, 5, 11, 0, -1, 1, -2, 9, -6, 8, -6, 9, -2, -10, -3, 4, -5, -10], [4, 4, 4, -16, -2, 1, -11, 4, -12, 2, -2, -1, -12, 4, 4, -2, -1, 13, 5], [-14, 1, 6, 3, 5, 1, 0, 8, -1, 0, 3, 3, -1, 1, 2, -4, -3, -4, -4], [6, 4, 2, 9, 1, 1, -2, -6, -3, 4, 1, 6, -4, -4, 2, -4, -2, 0, 1], [-19, 4, 1, -7, -1, -10, -1, 9, -2, -13, -1, 0, -3, -2, 2, 5, -15, 6, 4], [1, -2, -4, 18, -3, 7, 10, -7, 11, 11, -2, 5, 1, -2, 0, -2, 8, -9, -4], [2, -6, 2, -2, -2, 1, 1, 7, 3, -6, 1, 6, -5, -2, 6, 4, -11, 1, 8], [-4, -1, 6, -7, -6, -1, -4, 0, 5, 4, 4, 6, -7, -1, 8, -3, 0, 3, 0], [-17, 1, 9, 0, -3, 2, -1, 4, -10, -2, 1, 3, -2, 2, -3, 0, 3, 0, -7], [2, 3, 3, -8, -2, -3, -9, 8, -8, -3, 6, 6, 1, 3, 4, 2, -7, 5, 4], [25, 0, 2, 13, -1, 8, -4, -9, 10, 13, 2, 1, 6, 0, -5, -2, 6, -7, 2], [10, 5, -4, 2, -4, 3, -2, -7, 3, 11, 4, 5, -5, 2, -7, -2, -1, -1, 2], [-17, 12, -1, 2, 3, 0, 3, 4, -2, -1, -5, -1, -2, -3, 0, 0, 4, -1, -5], [-41, 8, -12, -23, 2, -3, 7, 9, -6, -6, -15, 9, -12, 1, 9, 10, -7, 13, 13], [-19, 9, 10, 3, -1, -5, -8, 11, -6, -13, -1, -5, 2, 4, -5, 2, -3, 2, 0], [-12, 15, 6, -13, 5, -6, 2, 23, -4, -18, -3, 3, 2, -1, 10, 8, -13, 0, 2], [-24, -1, 3, -9, -1, 2, 4, 6, -7, 4, -4, 1, -13, 4, 6, -2, 0, 3, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6040_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6040_2_a_p();".
// 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_6040_2_a_p(:prec:=19)
    chi := MakeCharacter_6040_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_6040_2_a_p();".
// 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_6040_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6040_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![5628, -40224, 59750, 91643, -206783, -80919, 262172, 41499, -174105, -17345, 67665, 6652, -15895, -1891, 2208, 325, -165, -29, 5, 1]>,<7,R![-1728, 35712, 241472, -294336, -1635684, 664688, 2923099, -718804, -2052760, 290672, 680951, -37587, -116773, -1504, 10488, 660, -465, -46, 8, 1]>,<11,R![-71561024, -609210640, -476487589, 671346706, 732532046, -179591575, -376254655, -34467342, 82896806, 22378192, -7442617, -3458077, 134882, 231329, 18635, -6616, -1110, 38, 18, 1]>],Snew);
    return Vf;
end function;