
// Make newform 8004.2.a.k in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8004_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_8004_2_a_k();".
// 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_8004_2_a_k();".
// 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 := [-115488, 1509552, -5659296, 5001480, 4260960, -6184176, -855636, 2789438, -114554, -655793, 76571, 88434, -13745, -6888, 1222, 288, -55, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2924380632926426855889005136, 9713974267365832529985440160, 4381380361918556505105405528, -36435941553545754661166169408, 1490503315719243326926821168, 26795700891125903440393467120, -2437178316183588229476738912, -8751517417440225088359606592, 877757215823034564311919639, 1539624796713181155670596745, -148490565710478946852143558, -154879314725306781562364529, 13256749203470338601883184, 8810425247375992739248360, -605877232513046416552126, -258457330225268973228733, 11207570645722098877963, 2927284725800794255337], [-5257001432083204378379490912, 38674162220774257482841017720, -96757902891444698299605604224, 33691959097630028455317049464, 91328367074137931967426591756, -51815390154058981684358411460, -34093655015930802739541252754, 22070720640860118940025133811, 6674535572415086879154761238, -4600025462646453193930691635, -744048668514131700022046961, 529806407574936006061495707, 47118126192082374503085128, -34245685489149462402114880, -1558114069513149064931567, 1158425410995214634885614, 20481619523436301822646, -15856898046747700680671], [10695232811231753481967711104, -90209877165370623418178176440, 131671159993904411990283618408, 46698395583386654989372541112, -115592059513900897373079693552, -6839508136673965432165924380, 37182686942633677528041243018, 1410447997834672133763623113, -5389983465140058970299406446, -733187440012258688665178305, 283654137960019281643287837, 171388383393455234023881081, 10935750146712076138961624, -18690205504873808659084540, -1738114125253927213119461, 975993069880071227811562, 50260217613457649535518, -19834097645988095110793], [8636480002243850177143274688, -61660011299197642888375646280, 62697508676133717281912410776, 67246741158110098928910237264, -106229023753839406893910752744, -24356837027641604097167219460, 63119926944263630390491559796, 1733250830043557095125535111, -18559515190192405343680079712, 1059451342617790598718407015, 2995662900716414698105237689, -309995813495353279322649693, -269664845965959593069360572, 35986054589389871064974270, 12677064418113551600564233, -1962686869754752843704686, -242158390012619827991104, 41412917201663495765029], [-3999378399434756613970855632, 36895192385313984461759761920, -83071322267623402723018709064, 6400046343306190805161804704, 102362362568865716687445020016, -32780948216448336579167996760, -51386911406851359650899423494, 18455128601695451156405460296, 13740225505480615539555664893, -4822124902841509064269946885, -2102329179432804940444823796, 692826517348627412394938427, 183420481011514812288520708, -56323942595387279210912330, -8452255223194401946523362, 2431529415250997934497279, 159157061769265235549581, -43333738854790069367731], [6040669928408817177236365008, -27283903064457130077361534080, -37524938006783953794801845184, 85756363701189933343131301224, 52502575807393264684246688796, -72956988236190067499062374360, -31224180682863786864906001914, 29161315269817013716219877426, 9523526946947109432701706333, -6449064309911489713591369835, -1599623366103212950263091626, 835895921149036982929059537, 148845418283002898662051348, -63334249081299279545880230, -7171121978121236543096572, 2600708965267858817240399, 139307793084177255893761, -44718954707623369104061], [3008097760086639408616881408, -1947118300677229386765953880, -85374333119104489519707116184, 115083049679871086523463233024, 79631386952190925576037748096, -115261731745435518607127291160, -33261156855348442216504401864, 47040810137737578862956680351, 8073008091972462186683944458, -10105200954165235091298156935, -1196491429946642089001785251, 1238799475775927751864882537, 104492819933826308740565248, -87281697845317926776793230, -4873239849396061715171647, 3297035103635045498715224, 92927380737163194056686, -51853525011373513017361], [1128076375034555595788640588, -11820928042756235246108674680, 36578590925784190753196736876, -17846465682043634395873270686, -38051816390374258619272801644, 25717986162282620789330129565, 16353771186501568785007642521, -11594313592548550448496538039, -3822807420872789710614397737, 2604920670161152338084814165, 524448178573778345675274639, -327118924112844966934060443, -41944197979170167366928197, 23367301898416757955738445, 1801536096414810102638408, -889402633050528900276736, -31995528324989864271104, 14033228130829226246429], [-30698113361234511387302736, -1120246991235604368346854960, 6088158990970631638713408288, -4728364354650308547608786748, -6097135374116470568948991732, 5181563055234374441047816730, 2546930838129243530860438458, -2108208898351544702370197467, -575770363835084394423021311, 443986295934300244350812190, 76002317937248245684942997, -53144243972831898803072104, -5839633611313119245563386, 3649111112421406449063920, 241274852048796945543259, -134035659386486971516363, -4132887294948038749297, 2043657375292274692832], [256355102982801246997433712, -3318033069803232178388716680, 12421021416576272215336647024, -7801609743285289466141240424, -13516888294155117008705914116, 9594786857378271634723498540, 6096668891211704391560605624, -4102827281786056450482443111, -1480746492115686220737314128, 899747708956589888775125945, 208604337202095783110453411, -112169083731814423593511687, -16989568684753651016724188, 8049631490531154088016450, 739666407122397339495607, -310484410604219777652934, -13287830637189807515156, 4995941761396610786931], [-7052986200113986695186856608, 74031494629762249738102621080, -195042416927436587160124849416, 55058426415419523236831591376, 223571380953424962748150067604, -101794241554796089134885020040, -104351645621829317437821724536, 48375436972321660440373605049, 25997863081338479397499348917, -11305719829974373633144705240, -3734779360628703213576040299, 1483458306376376908666109838, 308918686637788241218752752, -111548002620698527873025470, -13621216980384200787892703, 4499212025046647057467651, 247359119346637422602489, -75583454060712120333914], [1694910894136937542040729664, -320043913744284501580902840, -53002926748150652601934426272, 56203037642536860065732144592, 53267848295130048169108835868, -56795024945993943716391326080, -22875642795226859604150349662, 23238852539553548864635576433, 5438953100444389328700375114, -4996130029700500791392041605, -766444611082574756244411983, 612247323123168837291118721, 63124567150850560089419284, -43062987187257089314087440, -2787059132695762560041201, 1621391099742668687407242, 50706308357888305174438, -25375359856559521468813], [-11905134249553848674588421504, 114258078842512541583720498840, -241090682282959142754718657008, 27593374043982656861939793288, 285458046825356976426144078252, -111926710864861937694075735420, -139996025850649010318350558818, 61819842828166455730611430487, 37064898126902626404185670246, -15845377455265245439711524695, -5662871533265845603953326637, 2224551856745135776014595119, 495073566837989110647323776, -176391017767978015129757960, -22876400833337976918678139, 7430842493517809979493238, 431773041270738106994782, -129493678954510444776307], [8960410328410972391900913984, -79273804513783476304358061240, 173857065163518210547271807568, -47142890042521339455822020448, -177373672886268339449404300692, 93384499139860437324993568320, 73708886232166849341296927778, -43601062938602665583605596777, -16553666775885714098903130716, 9839305921704552663533784895, 2179566689407526344392923527, -1231346651820545843191010399, -167826455047710272981218396, 87406529355025077003262660, 6974997754173978279065069, -3301094205914111580435448, -120497952335555890666272, 51630065955629125660147], [-1409401304765454177605790576, 9444633811713278028244127160, -36322685239692333567154626252, 23299424567320790024088093072, 48029100724302248127293500938, -32697004031765348651644703630, -25833094437610044877549984242, 15667914927487977019971647903, 7327142908386201768821339124, -3809427495290295870543010255, -1174501743296248588497210103, 522015234781145833499003461, 106083697589212727903688994, -40912295647699343410768990, -5012332624848445488632241, 1714342125406004953414072, 96084227747052021743608, -29816274986940605729983]];
    Rf_basisdens := [1, 1, 1, 340787117604243100821037200, 340787117604243100821037200, 340787117604243100821037200, 340787117604243100821037200, 340787117604243100821037200, 340787117604243100821037200, 340787117604243100821037200, 85196779401060775205259300, 11359570586808103360701240, 22719141173616206721402480, 340787117604243100821037200, 113595705868081033607012400, 340787117604243100821037200, 113595705868081033607012400, 56797852934040516803506200];
    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_8004_a();"
function MakeCharacter_8004_a()
    N := 8004;
    order := 1;
    char_gens := [4003, 2669, 3133, 553];
    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_8004_a_Hecke(Kf)
    return MakeCharacter_8004_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], [1, 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, 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, 1, 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], [1, 0, 0, 0, 1, 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], [1, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 1, 0, 1, -1, 0, 0, 0, 0, 0, 0, -1, 1, 1, -1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0, -1, 0, -1, 0, -1, -1, 0, -1, 0, 0, 0, 0], [2, 0, 0, -1, 0, 0, 1, -1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0], [0, 1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0], [1, -1, -1, 0, 0, 0, 1, 0, 0, 0, -1, -1, 1, 1, -1, 0, 0, 1], [0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 1, 0, 1, 0, 0, -1, 1, 1, -1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1], [2, 0, 1, -1, -1, 0, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, -1, 0], [1, 0, 0, 1, -1, 0, -1, 0, -1, 0, -1, -1, 0, -1, 0, 0, 0, 0], [4, 2, 1, 0, 0, 1, -2, 1, -1, 2, 2, 2, -1, -1, 2, -1, -1, -1], [1, 0, 1, 0, 1, 0, -1, 0, -1, 2, -1, -1, 0, -3, 0, -1, 0, 0], [3, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 0], [0, 0, -1, 0, -1, -1, 1, -2, 0, -1, -2, -2, 2, 2, -2, 1, 1, 1], [4, 0, 1, 1, -1, 0, -1, 0, 0, 2, 0, 0, 0, 0, 1, 0, -1, -1], [2, 0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, -1], [-1, 0, 0, 1, 0, 1, -1, 2, 0, 1, 0, 0, 0, 0, 0, 0, -1, -1], [1, 0, 0, 0, 0, -1, 1, 0, -1, -1, 1, -1, 0, 0, -1, 0, 0, 0], [3, 1, 0, -1, 0, 1, -1, 0, 0, 1, 2, 3, -1, -1, 2, -1, -1, -1], [3, -1, 2, 1, -1, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, -1], [5, 2, 1, 0, 1, 1, -1, 0, -1, 1, 0, 1, 0, 1, 1, -1, -1, 0], [1, -1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, -1, -1, 1, -1, 0, -1], [2, -1, 1, -1, -2, -1, 0, -1, 0, -1, 1, -1, 0, 0, 0, 1, 0, 0], [6, -1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0, -1, 0], [1, -1, -1, -2, 0, -1, 2, -2, 0, -1, 2, 1, 1, 2, 0, 1, 0, 1], [-4, -1, -1, 0, 0, 0, 1, -2, 1, 0, -3, -3, 2, 0, -1, 1, 1, 1], [1, -2, -1, -1, 1, -1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 1], [3, 1, 0, 0, 0, 0, 0, 2, -2, 2, 0, 0, 0, -1, 0, 0, -1, 0], [0, 0, 0, 2, 1, 1, -2, 0, 1, 1, -1, 1, -1, -1, 1, -1, 0, -1], [2, 1, 1, 0, 1, 0, 0, 0, 1, 2, 0, 1, 1, 0, 0, 0, 0, -1], [0, -1, 0, -1, 1, -1, 0, 0, 0, -1, 0, 1, -1, 0, 0, 0, 0, 1], [-2, -2, -2, 0, 1, 0, 1, -2, 2, 0, -2, -2, 1, 1, -2, 0, 0, 1], [1, -2, -2, -1, 1, 0, 1, -2, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1], [6, -1, 1, 0, -1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0], [5, -1, 2, 0, 1, 1, -1, 2, -2, 3, 1, 1, 0, -2, 1, -2, -1, -1], [0, 2, 0, 1, -1, 0, 0, 0, 0, 2, 1, -1, 1, -1, 0, 0, 0, 0], [1, 1, 0, 1, 1, 0, 0, -2, 1, 0, -1, -1, 0, 1, 0, 0, 0, 0], [1, 2, 0, 0, 1, 1, 1, 0, 0, 3, -1, -1, 0, 0, 0, 0, 0, 0], [3, 2, 1, 2, 1, 2, -3, 2, -2, 2, -1, 0, 1, -1, 0, -1, -1, -1], [2, 0, 1, -1, 0, 0, -1, -2, 1, -2, 2, 2, 0, 1, 1, 0, 0, 0], [-1, 2, 1, 1, 0, 2, -3, 0, 1, 2, 0, 1, -1, -1, 2, -1, 0, -1], [-3, -1, 0, 2, 1, 2, -1, 1, 0, 1, -2, -2, 1, -2, -2, 0, 1, -1], [0, 1, 0, 0, 0, -1, -1, -2, -1, -1, 0, -2, 0, 0, -1, 0, 0, 0], [1, 0, 1, 0, -2, -1, -1, 2, -2, -1, 2, 1, -1, -1, 2, 1, 0, -1], [-4, 0, -2, -1, -1, -1, 2, -4, 0, -3, -1, 0, 1, 1, -2, 1, 2, 1], [5, 0, 0, -2, -1, -1, 2, -3, 0, -1, 1, 0, 1, 1, 0, 1, 0, 0], [5, -2, 0, 1, 2, 1, -2, 0, -2, 1, 0, 0, -2, -1, 1, -1, -1, 0], [-4, 1, 0, 0, 0, 1, -1, 0, 1, 1, -1, -1, 0, 0, -1, 0, 1, 0], [1, -3, 2, 1, -3, 0, 0, 0, 1, 0, 0, -3, 1, 0, -1, 1, 0, -1], [-6, 1, 0, 1, -1, -1, 0, 2, 2, -1, 1, 0, 1, 1, -2, 0, -1, 0], [1, -3, -1, -1, 0, -1, 0, -2, 0, -1, -1, -1, -2, -2, 0, 0, 2, 0], [1, 0, 1, 1, -1, 1, 0, 2, -1, -1, 1, 0, -1, 0, 2, 0, 0, -1], [1, -1, 0, -1, -1, -1, 1, -2, 3, -3, 1, 1, 0, 1, -1, 1, 1, 0], [2, 2, 0, 0, 0, 2, -1, -2, 0, 0, 0, 0, 1, 0, 2, 1, 0, -1], [-5, 0, -1, 1, -1, 0, -1, 2, 1, 0, 0, 1, -1, -1, 0, 0, 0, -1], [0, 1, 0, 1, 0, 1, -1, 2, 0, 3, -1, -1, 0, 0, 0, -2, -1, 0], [0, -3, -2, -1, 3, -2, 2, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 1], [0, -3, -1, -2, 1, -1, 0, -1, -3, -1, 0, -1, -1, 0, 1, 0, 0, 1], [3, -1, -1, -2, -1, -2, 3, -4, 2, -5, -1, -1, 1, 3, -1, 1, 2, 1], [2, -1, -1, 0, 0, -1, 2, -2, 1, 1, 0, -1, 1, 1, 1, -1, 0, 1], [7, -3, 0, -1, -1, -1, 2, -4, -1, -3, 0, -2, 2, 2, -1, 2, 1, 1], [-1, -4, -1, -2, 0, -2, 2, -2, 0, 0, -3, -3, 0, -2, -1, 0, 1, 2], [1, -4, -1, -1, 1, -1, 1, -4, 0, 1, -1, -2, 1, 0, -1, 0, 1, 1], [9, 1, 1, 2, 1, 1, -4, 2, -2, 1, -1, 1, -2, -3, 2, -2, 0, 0], [4, -4, 0, -1, 2, -1, -1, -2, 0, -1, 0, 1, -1, -1, 0, -2, -1, 1], [1, 4, 0, 1, 0, 1, -1, 2, 1, 1, 3, 3, 0, 1, 1, -1, -2, -2], [3, 3, 0, 2, -1, 2, -1, 0, 4, 2, 0, 3, -1, 2, 1, -1, -1, -1], [6, -2, -1, -4, 0, -2, 3, -2, -3, -4, 1, 1, 1, 3, 1, 2, 1, 2], [-3, -5, -1, -2, 0, -2, 5, -4, 3, -2, 1, -2, 1, 3, -2, 3, 2, 2], [5, 0, 3, 3, 0, 0, -1, 3, 0, 1, 2, 1, 0, -1, 0, -1, -1, -1], [4, 1, -1, -1, 3, 1, -2, 0, -2, -1, -1, 2, -1, -1, 1, -1, -1, -1], [1, -5, 0, -2, 0, 0, 1, -2, -1, 0, 0, -1, 1, -2, -1, 2, 1, 0], [3, -1, 1, -1, 0, -1, 0, -2, 1, -3, 0, 0, 0, 1, -2, 0, 1, 1], [7, -1, 1, 0, 2, 1, -1, 0, -2, -1, -2, 0, 0, -1, 0, 0, 0, -1], [-2, -2, -2, 0, 1, 1, -1, 2, -2, 1, 0, 2, 0, 0, 1, 0, -1, 1], [-1, 0, -1, -3, 0, -2, 1, -2, 0, -4, 2, 1, 1, 3, 0, 1, 0, 2], [-4, 1, 2, 2, -1, 0, -1, 2, 0, 4, 2, -2, 1, 0, 0, -1, -1, -1], [2, 0, 1, 1, 1, 1, -3, 4, 0, 2, 1, 1, -1, -2, 4, -1, -2, -1], [-5, 0, -1, 0, -3, -1, 2, -2, 0, -1, -2, -3, 3, 2, -1, 2, 1, 2], [-1, 3, 2, 1, 1, 1, 0, 4, -2, 3, 1, 0, 3, 0, 0, -1, -1, 0], [6, -2, 2, 0, 0, 1, 0, 0, 2, -3, 1, 1, -2, 0, -1, 0, 1, -1], [1, -3, -1, -1, 1, -1, 2, -3, 2, 1, -1, -1, 0, 3, -2, 1, 0, 2], [-5, 3, 0, -3, -1, -1, 2, -4, -1, -1, 2, 1, 1, 1, 0, 3, 0, 0], [2, 3, 2, 1, 0, 2, -3, 1, -1, 2, 1, 1, 0, -1, 2, -1, -1, -2], [4, -2, -1, -2, -2, -1, 3, 0, 1, 1, 4, 3, 0, 0, 1, 1, -1, 0], [-3, 1, 1, 0, 0, 0, 0, 2, 4, -4, 0, 1, -1, 1, -2, 0, 0, 0], [-2, -4, 0, 2, 2, 0, -2, 2, 0, 2, -2, -1, -1, -2, -1, -1, -1, -1], [4, 0, 1, 2, -1, 1, 1, 2, 0, -1, 0, 0, 2, 2, -2, 0, 0, -1], [2, 0, -1, -2, 4, 0, -2, 2, -3, 2, -1, 3, -4, -2, 1, -2, 0, 1], [0, -3, 1, 1, 0, 0, -3, 0, 1, -1, 0, 1, -2, -2, 0, -1, 0, 0], [3, -3, -1, -1, 0, -1, 0, -2, 1, 3, -1, -4, 2, 0, -1, 1, 0, 1], [-1, 3, -1, 1, -1, 0, 0, 4, -5, -2, -1, -3, 0, -1, -1, 1, 0, 1], [-6, 1, -1, -1, -3, -1, 1, 2, 0, -1, 0, -1, -1, 1, 1, 0, 1, 0], [-5, 3, 0, 3, -2, 1, 0, 2, 3, 1, -2, -1, 3, 3, -2, 0, 0, -2], [-7, 3, -1, 0, 3, 2, -1, -2, -2, 2, -5, -2, 1, -1, 0, -2, 0, 1], [4, 2, -2, 1, 2, 0, 0, 0, -1, -2, -2, 0, -2, -1, 0, -1, 0, 1], [-4, 2, -1, 0, 2, 1, 0, 2, 1, -1, 1, 1, -2, -2, 0, -1, 1, -1], [1, 2, 1, 1, 3, 1, -2, 4, 0, 3, 1, 3, -4, -4, 2, -2, -1, -2], [-1, 0, -1, 0, -2, 0, 0, 2, -3, 1, -1, -2, 2, -2, 0, -1, -1, 1], [6, 1, -3, -2, 2, -1, 1, 0, -5, -1, 0, 2, 0, 3, 0, 1, 0, 2], [4, 1, 1, 1, 0, 0, -1, 2, -4, 0, 2, 0, 0, 0, 3, -1, -1, -1], [-7, -1, -2, -1, 0, -1, 2, 0, -2, 1, 0, -2, 2, 0, -1, 1, 3, 2], [3, -2, 1, 0, -1, 1, -1, 3, 1, 3, -2, -2, 2, -1, 0, 0, 1, 0], [-12, 1, -4, -4, 2, -1, 3, -4, 1, -3, 0, 3, -1, 2, 1, 0, 3, 1], [-1, 3, 0, 0, 1, 0, -2, 4, 1, 2, 3, 3, 0, 0, 2, 1, -2, -1], [2, -2, -2, -1, -2, -2, 3, -5, 1, -2, -1, -3, 2, 2, -4, 2, 2, 2], [-4, 7, 0, 5, -2, 3, -4, 2, 0, 3, -2, -2, 2, 2, 1, 0, -1, -2], [11, 0, 2, 0, -2, 0, -2, 0, 1, 0, 2, 1, 1, -1, 3, 0, -1, -2], [4, -4, 0, 0, 1, -1, 1, 0, -1, 0, -1, 0, 0, -1, 0, -1, 0, 2], [-8, 2, 1, -1, 1, 1, -2, 2, -3, 3, -1, 0, 1, -3, 2, -1, 0, -1], [7, -2, 0, -1, 1, 2, 2, -4, 1, 2, -1, 0, 1, 1, -1, 0, 0, 1], [5, 1, 0, -1, 0, 1, -2, 4, 2, 3, 2, 5, -2, 1, 4, 0, -3, -2], [5, 0, -1, 1, 3, 1, 0, -2, -1, -1, -5, -1, 0, -2, 0, 0, 3, 1], [-3, -1, 2, 2, 1, 2, -3, 0, -2, 4, 0, 1, -1, 0, 1, -1, -1, -2], [3, 1, -2, -2, 3, 0, 3, -2, 1, 1, 0, 0, 1, 0, -2, 0, -1, 0], [0, 2, -1, 1, 1, 0, -2, 2, 3, 1, 2, 7, -3, -1, 2, 0, -1, -3], [-10, 2, -3, 1, -1, 2, 0, 2, 3, 0, 0, 0, 0, 3, 0, 0, 0, -1], [-4, 3, 2, 2, -1, 2, 0, 0, 0, 4, 2, -1, 0, -1, -1, -1, -1, -2], [0, 4, 3, 0, 1, 2, -2, 0, 0, 2, 3, 5, 0, -2, 3, -2, -1, -2], [1, 0, 0, -3, 0, 1, 1, 4, 0, 1, 2, 3, -3, -3, 2, -1, -1, -1], [4, 7, -1, 2, 1, 0, -1, 2, -3, 0, 1, 1, 0, 0, -1, -2, -1, 1], [-1, -1, 3, 3, -3, 0, -2, 2, 2, -2, 0, -1, 1, 1, -2, 1, 1, -2], [-3, 2, 0, 1, 1, 0, 1, 2, -2, -1, -3, -4, 0, 0, -1, 1, 0, 1], [-6, 0, 0, 0, -3, -3, 1, 0, 1, -1, -1, -4, 2, -1, -2, 1, 0, 0], [7, -3, -2, -5, 3, -1, 2, -4, -1, 0, -2, 1, -2, -3, 0, -1, 1, 1], [1, -3, 3, 2, -1, 0, -3, 4, -1, 4, 3, 1, -2, -3, 2, -2, -3, -2], [3, 2, 1, 1, -2, 3, -2, 0, 2, 1, 1, 3, -2, 0, 2, -1, -1, -4], [-2, 2, 0, -2, -1, 1, -2, 2, -3, -1, 1, 1, 0, 2, 1, 1, -1, 0], [-6, -2, -2, 0, -4, -2, 4, -6, 1, -2, 1, -3, 2, 1, -2, 4, 1, 1], [6, 2, 0, 2, 1, 2, -1, 2, 2, -2, 0, -1, 1, 2, -1, -2, -1, 1], [1, -3, -1, -4, 2, -2, 4, -8, 0, 0, -2, -1, 1, 3, 0, 0, 2, 2], [-7, 2, 0, -2, -2, -2, 0, -2, -3, -3, 0, 0, 2, 1, 0, 1, 0, 1], [4, 4, -1, 1, -1, 1, 1, -2, 1, 1, 1, -2, 1, 2, -1, -1, -1, 0], [2, -4, 0, 2, 2, 1, -4, 0, 0, 3, -1, 0, -3, -6, 2, -1, 0, 0], [2, -1, 1, -1, 2, -1, -1, -2, -3, -3, -3, -2, -1, -2, -1, -1, 2, 2], [-15, 1, 2, 1, -4, 0, 0, 0, -1, -4, -3, -3, 4, 5, -2, 1, 2, 1], [-4, 6, 3, 5, 3, 4, -3, 6, 0, 4, -3, -3, 0, -3, -1, -2, 0, -2], [-8, -3, 0, -4, 1, -2, 3, -2, -1, -4, 0, -1, -1, -2, -1, 0, 1, 2], [-1, 3, -4, -2, 1, -2, 2, -4, 0, 2, -2, -1, 1, 1, -1, 0, 2, 3], [0, -3, 1, -1, 0, 1, -1, 0, 2, 1, -2, 0, -2, 1, 0, 0, 0, 1], [8, -1, 1, 3, -1, 2, 0, 0, 3, 2, 0, -1, -1, 0, -2, -1, -2, -1], [-6, -2, 5, 2, -2, 0, -2, 0, -1, -4, -1, -3, 0, -2, -1, 2, 0, -1], [6, 3, 0, -3, 1, -1, 1, -2, 3, -3, 2, 5, -1, 3, 0, 0, 0, 0], [2, 1, 0, 0, 2, 1, -3, 2, -6, 5, -1, 0, 1, -1, 2, -1, -2, 0], [-5, -1, -2, 2, -2, -1, 2, -2, 2, -1, -7, -6, 5, 3, -4, 1, 3, 2], [-2, -4, 1, -1, 1, -3, 1, -4, 1, -3, -3, -4, 1, -2, -1, 0, 2, 2], [-6, 4, 4, 4, -3, 1, -2, 4, -1, -1, 0, -2, 0, 1, -1, -1, -2, -3], [3, -3, 3, 0, -2, 1, 0, 1, 1, 3, 0, -1, -1, -2, 1, 0, 0, -1], [-2, 3, 1, 3, -2, 2, -1, -2, -3, 2, -4, -4, 0, -2, -1, -2, 2, 0], [-2, 1, -2, -2, 2, 0, 3, 0, -4, -3, 2, 2, -1, 1, 0, -1, 0, 1], [-1, 1, -3, 1, -1, -1, 2, 2, 1, -3, -3, -4, -2, 0, -3, 2, 4, 1], [-9, 0, 0, 0, -1, 2, 0, -4, 4, 4, 2, 1, 2, 2, -1, -1, 0, -2], [2, 0, 0, -2, -1, -4, 1, 2, 0, -4, 2, 1, -3, -1, -1, 1, 2, 1], [1, -2, 3, 0, 0, 1, -1, 2, 3, 1, 1, 0, -1, 0, 1, -2, -3, -2], [-5, 2, -1, -1, -1, -1, -2, 2, 0, -1, 3, 3, -2, -2, 2, 0, -1, 0], [-12, -1, 2, 2, -3, 1, -2, 4, 1, -3, 4, 2, 0, 1, -1, 0, -3, -3], [-3, 3, 1, 0, 2, 2, -2, -4, -4, 0, -3, -3, 2, 2, 1, -1, 0, 2], [12, 0, -1, -2, -1, 0, -1, 4, 2, 0, 4, 3, -1, 1, 2, 0, -2, 0], [-6, -2, 1, 1, 3, 1, 2, 4, 2, 3, 1, 3, -4, -5, 0, -3, -1, -2], [2, -5, 0, -2, 0, -1, -1, 3, -3, -1, 1, 3, 0, 0, 1, 0, -1, 0], [-1, 3, 1, 0, -3, -1, 1, -6, 1, -1, 4, 0, 0, 3, -2, 1, -2, 0], [4, -3, 1, 1, 2, 3, -1, 0, -1, 3, -1, 1, -1, -2, 2, 0, 2, -2], [1, 1, -1, -2, -1, -1, 1, -4, 1, -1, -2, -3, 1, 1, -3, 2, 1, -1], [-8, 1, 2, 2, 1, 2, -4, 0, -1, -2, -1, 0, -1, -3, 1, -1, -1, -1], [-8, 5, 0, 4, 1, 2, -5, 4, -4, 1, -4, -4, -1, -1, 2, -1, 1, 1], [-3, -1, -3, -3, 1, 0, -1, -2, 1, 0, 2, 4, -2, -1, 1, 0, 2, -2], [-6, -2, 0, -1, 0, 0, -2, 2, -2, -2, 1, -1, -2, -4, 3, 0, -1, 0], [1, -4, 2, -3, 1, -2, 2, 0, 0, -4, 5, 3, 3, 2, 0, 1, -2, 0], [7, 0, 2, -1, 1, 2, -3, 2, 1, 0, 3, 4, -3, -2, 4, -2, -1, -2], [1, 1, 0, 4, -2, 1, -1, -2, 1, 3, 1, -2, 3, 4, 0, 0, -2, -1], [-8, 2, 0, -1, -2, -4, 2, -4, 1, -8, 1, 0, -1, 4, -1, 4, 0, 1], [-2, 3, -2, 0, -1, 0, 0, -2, -4, 0, -6, -5, 4, -2, -1, 1, 3, 0], [-6, -5, 1, -1, 2, 2, -3, 4, 1, 2, 4, 4, -2, -4, 3, -2, -2, 0], [-1, 1, -2, 0, 2, -1, 5, 0, 1, -2, -1, -2, 0, 2, -4, 2, 3, 0], [-7, 2, -4, 1, -1, 2, -1, 0, 1, -2, -1, 1, 0, 0, 2, 2, 0, -2], [-4, -2, -1, -1, 7, 1, 0, 4, 0, 1, -2, 3, -5, -4, 3, -2, -1, 0], [0, 4, -2, 3, 3, 2, -4, 4, -4, 2, -1, 2, -3, -4, 4, -4, 0, -1], [-15, -4, 1, 2, -3, -2, 2, 2, 1, -2, -5, -5, 2, 2, -5, 3, 1, 1], [0, -3, 3, 0, 0, -1, -1, 0, -1, 0, 0, -3, 4, 2, -1, 0, -2, 2], [6, -3, 3, 3, 0, 0, -2, 4, -3, 0, 1, 1, 0, -4, -1, -1, 0, -1], [-10, -6, -4, 1, -1, 0, 3, 2, 0, -4, -5, -5, 2, 6, -4, 2, 1, 3], [-6, -2, -3, -5, 3, -1, 6, -5, 3, -5, 1, -1, 0, 1, -4, 3, 5, 3], [-3, -3, -4, 1, 2, 2, 2, 0, 3, -1, -3, -1, -1, 1, -4, 0, 2, 1], [6, -4, 1, 0, -2, -2, -3, -4, -1, 0, 3, 0, -1, 0, 2, 1, -3, 0], [9, -5, 0, 2, -1, 1, 3, -3, 3, -1, 3, 1, 0, 4, 1, 0, -1, 0], [-1, -6, 1, -1, 3, 3, -2, 2, -1, 5, 1, 1, 2, 0, 1, -3, -3, 0], [4, -6, -2, 0, 2, 3, 3, -2, 2, 1, -1, 1, 0, 2, -1, 1, 2, 1], [-4, 2, 0, 0, 2, 2, -6, 2, -4, 4, -2, 1, 1, -6, 3, -1, -1, -1], [0, -4, 1, 3, 0, 0, -2, 4, -2, 0, 0, -1, -1, -9, 0, -1, -1, 0], [-4, 7, -1, 0, -1, 1, -1, -5, -6, 3, -4, -2, 2, 1, 0, 0, 0, 1], [-3, -2, 2, 3, -3, 1, -1, -3, 2, 1, 2, -1, 1, -1, -2, 1, -2, -3], [0, -2, -1, 0, 1, -1, 0, -2, 2, -3, 4, 4, 0, 5, 2, 0, -2, -1], [-8, 2, 1, 2, -1, 1, 0, 2, 2, -1, 0, -5, 1, 2, -3, 1, -1, 0], [6, 5, 0, 4, -1, 1, -1, 2, -1, 5, 0, -3, 1, 0, 3, -1, -1, -1], [0, -6, -3, -2, 1, -1, -2, -6, 0, -1, 0, 0, -2, -1, 2, 0, -2, 1], [-6, -3, -3, 2, -1, -2, 3, -4, 5, -4, -2, -1, -1, 1, -4, -1, 1, 2], [8, 6, 2, -2, 1, 2, -1, 2, -5, 2, 2, 1, 1, -1, 3, -1, -2, 0], [-10, 1, -3, 1, -3, -2, 0, 2, 1, 6, 0, 0, 0, 1, 1, 0, -1, 0], [0, 0, 1, 0, -2, -1, 0, 0, 2, -1, 0, -1, 1, 0, -1, 0, -1, 0], [3, -1, 3, -1, 0, 0, -4, 3, -2, -2, 2, 1, 1, -1, 2, 1, 0, 0], [-13, 8, 0, 1, -2, 1, -1, 0, 0, 1, 2, -1, -1, 0, 0, 0, 0, -1], [8, -1, -3, -1, -1, 0, 2, -8, 1, -2, -2, -1, 1, 3, 0, 3, 2, 1], [7, -6, 0, -2, 2, -1, 0, 2, -3, 1, 1, 1, -2, -1, -1, -1, -3, 0], [-9, -5, 3, 3, 2, 2, -1, 2, 0, 4, 1, -1, 0, 0, 1, -3, -1, -1], [-11, 1, 0, 0, -3, -3, 0, 2, -3, -7, 3, 0, -3, -1, -2, 2, 1, 1], [1, -2, 3, 1, -1, -1, -2, -2, 3, -1, 1, 0, -3, 0, 0, 0, 2, -3], [1, -7, -1, -4, 4, -4, 4, -4, 4, -6, 4, 3, -3, 1, -2, 2, 0, 0], [14, 3, 2, -2, 1, 1, 0, 2, 3, 5, 3, 6, 1, 2, 2, 0, -2, -1], [9, -4, 4, 1, -1, 0, -1, -2, 1, 2, 0, -2, 2, 1, -2, -2, -1, -2], [-8, 1, -3, 0, 4, -1, 3, 0, -5, -1, -2, -2, 0, -1, -2, 1, 2, 0], [-8, -2, 1, -1, -6, -2, 4, 0, 3, -2, 3, -2, 1, 2, 0, 5, 2, 0], [5, -1, 1, 0, -1, -2, 3, -2, 1, -8, 2, 0, -2, 0, -1, 5, 2, -2], [-8, -5, 0, -1, -7, -4, 5, -6, 3, -2, 3, -3, 2, 4, -2, 4, 1, 2], [-10, -3, -3, -3, -3, -1, 2, -8, 2, -1, -2, -1, 3, 2, 1, 1, 2, 2], [10, -3, 5, 1, -2, -3, 2, 0, 1, 1, -1, -3, 2, 0, -2, -1, 0, 1], [-8, 1, -1, -3, 4, 1, 3, -2, -6, -3, 3, -1, 0, -2, -1, -1, 1, 3], [-3, 5, -1, 0, 5, 3, -1, 0, -1, 7, 1, 4, -3, -2, 3, -2, -2, 0], [7, 6, 5, 2, 0, 3, -2, 5, 3, 3, 4, 5, -3, 4, 3, -1, -3, -3], [-3, 10, 3, 0, 0, 1, -2, 2, 1, -1, 4, 7, -1, -1, 3, -1, -1, -3], [1, -3, -1, 0, -4, 0, -1, 0, 1, 2, 0, 0, 0, 2, -1, 0, -2, 0], [1, -5, 0, -3, -3, -1, 2, 0, 1, -3, 2, 0, 0, -1, -1, 1, 0, 3], [6, 4, 5, 2, 0, 1, -5, 1, -1, 6, 2, -1, -2, -6, 1, -5, -1, -2], [1, 0, -3, -1, -4, -4, 5, -4, 3, -6, 2, -2, 0, 4, -3, 3, 0, 2], [-6, -3, 1, 0, -2, -1, 0, 0, 2, -1, 0, -1, 3, 0, -1, 1, 0, 2], [-7, 3, -2, -1, -2, 2, 3, 4, -1, 0, 1, -1, 4, 4, -1, 1, -1, 1], [-23, -1, -4, 0, 2, -2, 3, -2, 7, -6, -1, 1, -4, 1, -2, 1, 3, 2], [-7, 3, 1, 1, 3, 1, -6, 7, -2, -1, -1, 1, 0, -5, 2, -1, 0, 0], [-10, -5, -5, -5, 0, -3, 7, -6, 2, -9, -1, 1, 0, 6, -2, 4, 5, 4], [11, 6, 1, 1, 0, 0, 3, 0, 6, 2, -2, -1, 1, 5, -4, 1, 2, 0], [-2, -4, 2, 8, -1, 3, -2, 4, 8, 3, 0, 2, -2, -5, 1, -3, 0, -4], [-3, 2, 1, -5, 3, 0, -1, 2, 0, -2, 4, 8, -4, -3, 3, -1, -1, 1], [2, -5, -1, -5, 1, -2, 0, -8, -1, 2, -2, 0, 0, -1, 1, 0, 1, 4], [-5, 0, 2, 1, -4, -1, 3, 2, -1, 3, 0, -5, 3, 2, 0, 0, 0, 0], [-15, 0, -6, 2, -3, -1, 1, 0, 4, -1, -5, -3, -2, 4, -3, 3, 2, 1], [2, -1, 3, -2, -4, -1, 4, -8, 0, -1, 0, -6, 5, 3, -2, 2, 2, 2], [-10, -2, 4, 4, -2, 3, -1, 2, 1, 3, 1, -2, 4, 0, 0, -2, 0, -2], [-4, -2, -4, 3, -1, 1, -1, -3, -2, 6, -2, -4, 3, -1, 3, -1, 0, 1], [17, -6, 1, 2, 2, 0, -1, -2, -1, -2, 1, 2, -3, 1, -2, -3, 0, 0], [1, 2, 3, 7, -1, 2, -5, 3, -3, 8, 0, -2, 2, 0, 3, -1, -3, -2], [13, -2, 5, -1, 1, 0, -1, 6, -1, 5, 4, 5, -2, -5, 0, -3, -4, -2], [0, 1, 4, 0, 0, 2, -1, 4, -1, 2, 1, 1, 4, 4, 0, 0, -1, 2], [-12, -2, -7, -1, 0, 0, 2, -6, 1, -4, -7, -4, -1, 6, -2, 1, 6, 3], [11, 5, 5, 4, -1, 3, -3, 3, -3, 6, 2, -1, 6, 5, 1, -2, -6, -2], [14, -5, 5, -3, -2, 0, 0, -2, 1, 1, 6, 4, -1, -5, 3, 1, -2, -3], [-4, 3, 0, 0, -1, 0, 1, 0, 2, -4, -2, -2, 2, 1, -2, 1, 0, 2], [-8, 3, 4, 4, -2, 0, -3, 2, 0, 4, 1, -1, 0, 0, -1, 1, -1, -4], [-23, 0, -4, -3, -7, -2, 7, -2, 4, -3, 3, 0, 2, 8, -3, 3, 0, 1], [-5, 8, 1, 6, 0, 4, -4, 7, -1, 0, -1, 1, -2, -5, 1, 0, 1, -3], [-3, -1, 2, 2, -1, 4, 0, 2, -2, 6, -2, 0, 2, 5, -1, -4, -3, 1], [-8, -5, 4, 0, -4, 0, -1, 2, 0, 1, -2, -6, 5, 3, 0, 1, 0, 1], [6, 1, -2, 0, -2, -2, 0, 6, 2, -2, 0, 2, -4, -4, 2, -1, -1, -2], [8, -3, 1, -4, -2, -3, 1, -2, 1, -7, 2, 2, -2, 5, -2, 1, 0, 4], [-3, -2, 2, -4, -3, -1, 2, -4, 1, -1, 3, 4, -1, 2, 0, -1, -2, 3], [-4, -2, 1, 1, 0, 3, -2, 0, 1, -1, 3, 1, 0, 6, 2, 0, 1, -1], [-5, -7, -8, -3, 2, -1, 6, -2, -3, -6, -4, -5, 0, 3, -4, 2, 3, 5], [-2, -3, -5, -1, 1, -2, 6, -6, 2, 0, 0, -5, 1, -1, -2, 3, 2, 4], [-1, -4, 3, -4, -1, 2, 3, 2, 0, 2, 4, 0, 4, 0, 1, 0, 1, 0], [7, -2, -2, -3, -2, -5, 4, 9, -2, -7, 0, 0, 0, 1, -1, 2, 2, 2], [3, 6, 1, 3, -2, 3, -1, 0, 0, 5, 1, -2, 1, 3, 3, -1, -1, -3], [-17, -3, 0, 2, -2, 0, -1, -4, -5, -4, -8, -11, 7, -1, -7, 0, 1, 2], [14, -1, 0, -1, 0, 2, 2, 2, 3, -2, -1, 2, 1, 2, -3, 1, 1, 1], [-17, -1, 2, -2, -1, -2, 2, 4, -5, -4, -1, -4, 1, 2, -1, 4, 3, 3], [-5, 6, -2, 1, 3, 1, 2, 4, 2, -1, 1, 2, -1, 4, 0, -2, -2, 0], [15, -2, 1, 3, -2, 4, -3, 2, 1, 6, 1, 0, 3, 0, 4, -1, -3, -2], [-33, 6, 0, 3, -4, -2, 0, 2, -1, -9, -3, -6, 1, -1, -3, 3, 4, -1], [-9, 2, -3, 1, 0, -1, -2, -2, 5, 1, 1, 3, 2, 4, 0, -1, -3, -1], [-9, -1, 4, 2, -2, 3, -5, 4, -1, 5, 1, 0, 3, 0, 2, -4, 0, 0], [-16, 3, -1, -2, -1, -2, 1, -3, 4, 2, -3, 0, 1, 2, 2, 1, 0, 1], [-6, 3, -1, 3, 3, 1, 2, 4, 2, 7, 1, -1, -2, 1, 2, -4, -2, 0], [3, 1, 1, -2, 2, 1, -3, 4, 0, 1, 2, 6, 2, 1, 4, 0, 0, -1], [-8, -4, 0, -1, 4, 2, 1, -6, 4, 2, -1, 1, -2, -3, -4, 0, 4, -3], [-20, 5, 0, -2, -1, -3, 2, 0, -1, -7, 0, 2, 0, -3, -3, 4, 1, 1], [-8, -1, 1, 0, 7, 2, -5, 5, -6, 6, -1, 0, -1, -1, 4, -3, 0, 1], [7, 2, -2, -5, 2, -2, 1, -4, -5, -4, -1, -1, 0, -3, -2, 1, 0, 0], [-7, 2, -1, -1, 4, -2, -4, 2, -6, -4, -2, 1, -5, -6, 2, -2, 3, 2], [-15, 5, -2, -1, -1, -2, 2, -4, 3, -4, 1, 1, -2, -1, -2, 4, 4, 0], [13, -2, 7, 1, -2, 2, -2, 1, 2, 6, 4, -1, 1, -4, 2, 0, -1, -4], [-12, -3, -4, -1, 0, -2, 1, 0, -2, 2, 1, -1, -2, -1, 4, 1, 1, -1], [-10, 3, 0, 0, 3, -1, -1, -4, -2, 1, 1, 0, 1, -1, 4, 0, -2, 0], [4, 2, 3, 4, -2, -1, -1, -5, -2, -5, -2, -4, 2, 1, -2, 2, -1, -1], [-6, 1, 5, 6, 3, 6, -4, 6, 1, 10, -4, -1, -1, -2, 2, -4, -2, -5], [8, 1, -3, 1, -1, -2, 4, -6, 0, -8, -4, -3, 1, -1, -2, 2, 3, 2], [16, 1, -3, 1, -2, 0, 2, 2, 3, 8, 2, -1, 3, 10, 0, -2, -4, 1], [-13, -8, 0, 5, 2, 4, -5, 2, 2, 4, -5, -6, 1, -6, -3, -2, -1, 0], [5, -2, 0, 2, 2, 3, -3, 4, -1, 9, 0, 4, 0, -3, 1, 0, -1, -2], [0, 3, -2, -5, -2, -1, 3, 0, -2, -1, 3, 1, -2, 0, -1, 3, 3, -1], [16, 3, 4, 0, -4, -3, 1, -2, 0, -1, 1, -3, 2, 4, -7, 1, -2, 1], [6, 1, -5, -4, -2, -2, 3, -8, -6, 0, -7, -7, 5, 8, -3, 3, 2, 5], [-25, 0, 1, 1, 2, 1, -1, 8, -1, -2, 1, -1, -1, 1, 0, 0, -1, -2], [5, 5, -3, 1, 2, 3, 1, 0, 4, 3, 2, 2, -2, 0, 0, 1, 0, 1], [0, 6, 6, 10, -5, 2, -8, 6, -5, 5, 0, -3, 2, -2, 1, -4, -4, -5], [8, -3, 0, 1, -1, -2, 2, -10, 0, 0, -6, -5, 3, 1, -7, 0, 0, 1], [-10, 0, 2, 6, -3, 3, -4, -2, 1, 3, 1, 0, 3, 4, 2, 2, 0, -3], [-2, -2, -2, -6, 1, -3, 5, -2, -3, -7, 1, 3, 0, -1, -1, 2, 2, 4], [-4, 4, -2, -5, 2, -1, 3, -8, -4, -5, 2, 3, 1, 3, 0, 1, 0, 5], [-8, 3, 2, 2, -2, 0, -1, -4, 3, -2, 0, 0, 4, 3, 0, 2, 0, -1], [-23, 8, -2, 1, 3, 2, 1, 4, -2, -2, -4, 0, -2, 3, 1, 1, 3, 3], [7, 2, -7, -6, 2, -2, 3, -8, -1, -3, -1, 3, -1, 3, -1, 3, -1, 2], [14, 6, 0, 2, 2, 5, -4, 6, -2, 7, 1, 4, -5, -2, 6, -4, -3, -2], [-3, -2, 3, 1, -2, 1, -1, 4, -4, 1, -1, -3, 0, -1, 2, -2, -4, 0], [-1, -2, -1, 0, 0, 2, 3, -4, 1, 4, -1, -4, 5, 3, -4, 2, 3, 4], [-4, -7, 2, -2, 5, -1, 2, 4, -1, -1, 5, 2, -3, -4, 2, -1, 1, -1], [9, -9, 4, 3, -3, 4, -3, 0, -1, 8, -1, -3, 4, 2, 0, -3, -1, 0], [-7, 2, 0, -2, 1, -4, 2, -4, 0, -2, 2, 3, 1, 1, -3, -1, 1, 2], [-7, 2, 2, -4, 0, -4, 1, 0, -1, -8, 2, 3, -3, 0, -1, 3, 2, 0], [-2, -2, -1, -4, 6, 1, -1, 2, -2, 3, 4, 9, -5, -6, 2, -4, -2, 1], [-1, -3, -3, -1, 3, -4, 4, -4, 2, 1, -6, -2, 5, 4, -5, -1, 4, 6], [15, -4, 0, 2, -4, -2, 1, 4, 1, -4, 4, 3, -1, 6, -1, -1, 0, -2], [4, 2, -1, 0, 0, 2, 0, 0, -2, 0, 5, 2, -1, -1, 4, -2, -2, 1], [-13, -4, -2, 2, 2, 0, -2, 2, -1, -2, -7, 0, -1, -2, 1, -2, 2, 1], [14, 1, 6, 1, 2, 1, -3, -1, -1, 1, 4, 1, 1, -1, 0, -4, -3, 0], [11, 0, -3, 4, -3, 0, -4, 0, -7, 3, -7, -4, -2, -6, 2, -3, 0, 1], [1, -4, 4, -1, 1, 2, 3, 2, 3, 0, 1, -1, 2, -4, 0, 1, -1, 0], [0, 1, -1, 0, 1, 1, 2, 4, 0, 7, -2, 0, 2, -3, -2, -1, -1, -1], [7, 0, 3, 0, 3, 3, -5, 6, 1, 0, 0, 7, -6, -3, 3, -6, -2, -2], [1, 3, -1, 2, 5, 4, -2, 8, -4, -4, -3, 2, -4, -8, 2, -5, 0, 0], [-14, 0, 4, 3, -2, 2, -3, 6, -10, 2, 3, -1, 4, 1, 4, 0, -4, -3], [-1, 0, -5, 3, 2, -1, -1, -2, 1, 1, -3, 1, -4, 1, 0, 0, 0, -1], [-2, 1, -3, 1, 2, -2, 1, 2, -2, 6, -2, 1, -1, -8, 2, -1, 1, -2], [3, 9, 6, 1, 2, 6, -2, 0, 3, 4, 7, 8, 1, 5, 5, -2, -3, -3], [-3, -2, -6, 0, 0, -1, 2, -4, 4, -3, -7, -2, 1, 8, -4, 0, 2, 0], [4, -6, 0, -2, 3, 0, -1, 0, -9, -2, -1, -3, 0, 0, 2, -2, 2, 4], [12, 1, 1, 0, 1, 2, -1, 6, -3, -4, -1, 0, -1, 0, 0, -1, 0, -2], [-5, -6, 5, 0, -3, -1, -3, 2, -5, -2, -2, -7, 4, -3, -5, -3, -3, 0], [-12, 3, 0, 1, 1, 0, -2, 0, 1, 0, 4, 4, -4, 3, 4, 1, -1, -1], [10, -10, -2, -3, 4, -3, 5, -10, -3, -5, -4, -7, 1, 5, -2, 2, 3, 8], [-7, 1, -2, 2, 0, 2, -1, -8, 6, 0, 1, 5, -2, -3, 0, -3, 1, -3], [21, -1, -3, -1, -2, -1, -2, -2, 1, -3, 5, 5, -2, 2, 2, 1, -1, -1], [15, 0, 0, 0, 0, 0, 2, 4, 1, 4, 0, -1, -1, -3, 0, -1, 3, -1], [9, -1, 4, 3, -2, -2, -3, 0, 0, -4, -1, -7, 0, -2, -4, 1, 0, -1], [-17, 1, -8, 0, -3, -1, -2, 3, -4, -3, -3, -2, -3, -3, 0, 2, 1, 2], [-7, -1, 3, 3, -2, 1, 2, 2, 1, -1, -5, -3, 2, -4, 0, 0, 2, -1], [16, 3, 9, -3, 1, 3, -1, 0, 3, 7, 8, 6, 0, -3, 2, -1, -2, -2], [-6, -2, -3, -4, -4, -4, 4, -8, 2, -10, 3, 2, -1, 11, 1, 5, 2, 2], [-14, 1, -4, -2, -1, -1, 2, 0, 0, 2, -5, -5, -1, 0, 1, 3, 0, 0], [8, 9, -3, 1, -3, -1, 4, 2, 2, -3, -3, -5, 2, 2, -6, 6, 4, 0], [7, 2, 0, 1, 0, 2, 2, 2, 1, 2, -3, -3, 2, -1, 0, 0, 3, 2], [-2, 2, -7, -3, 2, -2, 3, -6, -2, -8, -4, -2, 4, 8, -1, 3, -1, 6], [-1, 3, 3, -1, -4, -2, 1, -2, 1, -2, 4, 3, 1, -3, 1, 1, -1, -4], [6, 1, 1, -4, 0, -3, 0, 2, 1, -7, 8, 7, -1, 3, 3, 0, 1, 1], [-4, 1, -3, 6, 0, -1, -3, 3, -6, 0, -2, -1, -4, -2, 3, -4, -2, -1], [11, -6, -2, -4, 2, -4, 0, 0, -4, -4, -3, -1, 0, 0, 0, -3, -1, 4], [-15, 3, -1, 6, 0, 3, 1, 2, 1, 5, -4, -6, -2, -2, 0, -1, 1, 0], [-20, 6, 3, -1, -1, -3, -1, -2, -1, -1, -1, -2, 3, -4, -3, 1, -1, 0], [-1, 1, -2, -4, 1, 0, 0, 0, -1, -4, -3, 3, -4, 0, 2, 1, 4, -1], [-13, 11, -5, 1, -1, -2, 0, 0, -1, -4, -6, 1, -1, 4, 0, 0, 2, 0], [-5, 0, -6, -2, 1, -2, 8, -6, 0, 2, -6, -4, 4, 7, -4, 2, 2, 2], [2, -2, 1, 3, 0, -2, 1, -4, -5, 6, -4, -4, 2, 0, -3, -3, 3, 2], [-1, 4, 7, 2, 1, 6, -6, 4, -1, 6, 7, 5, 0, -4, 5, -3, -7, -3], [-29, 5, -1, 6, -2, 1, 0, 0, -2, 1, -6, -9, 5, 2, -5, 2, 2, 2], [12, 3, 4, 6, -1, 4, -4, 8, -7, 0, -3, -3, -2, -3, 0, -2, 0, -1], [8, 0, 1, 4, -1, 2, -8, 4, -5, 2, -5, -3, 0, -7, -1, 0, 1, -1], [-3, -2, -3, 0, 5, 2, 3, 0, 0, -2, -1, 4, 1, 1, 0, -1, 3, 1], [-17, 7, 0, 2, 3, 3, -1, 0, 5, 3, -1, 1, -2, 2, 1, 1, 0, 0], [0, 3, 1, 2, 0, 1, 3, -3, 2, 5, 2, -6, 6, 3, -4, -1, 0, 5], [15, 0, 4, -5, 1, -1, 3, -2, 7, 3, 9, 7, -2, 1, 1, 0, 0, -2], [-4, 3, 1, 4, -3, 2, -1, -2, 0, -2, -5, -5, 6, 6, -5, 0, 1, -2], [-26, 0, -1, 1, -1, -2, 3, 2, 0, -6, -4, -5, 1, 2, -4, 1, 4, 4], [6, -6, 1, 2, -3, -3, 4, -8, 1, -1, -2, -7, 5, 4, -5, 5, 3, 3], [3, 6, 4, 2, -3, -1, -2, -6, 4, 1, 1, 1, -1, -1, 1, 0, 0, -2], [-2, -1, 3, -2, -3, -2, 5, -8, 2, -4, 2, 2, 0, 3, -1, 0, -2, 0], [-2, 6, -3, 1, 0, 3, 5, 2, 0, 9, 1, 4, 1, 8, -1, 2, -1, -1], [-2, -2, 3, -3, 2, 4, -4, 0, -2, -1, 4, 10, -7, -9, 7, -3, -2, -4], [-1, 1, -2, -5, 2, 0, 1, -4, -2, 0, 3, 6, 0, -1, 1, 1, 1, 3], [4, 4, 0, 4, -6, 3, 0, -2, 4, 5, 1, -2, 5, 2, -2, -2, -1, -2], [32, -6, -3, 3, 9, 4, -2, -2, 3, 4, -6, 0, -4, -3, 1, -3, 4, -2], [4, -2, 0, -4, 0, -3, -3, 0, -6, -3, 1, 1, 0, 2, -1, -1, -2, 1], [-7, 4, -4, -1, 0, 1, 5, 4, 7, -3, 0, 4, -7, -2, -1, -1, 4, -2], [13, 0, -8, -5, 1, -2, 1, 0, -4, 0, 0, 2, 2, 3, 0, 0, 1, 3], [20, -2, 1, 2, -2, -2, 5, 0, 2, 0, -1, -3, 0, 6, -2, 1, -2, 1], [9, -5, -3, -1, 5, 1, 5, -5, 3, 1, -1, -4, -3, -1, -1, 1, 4, 2], [3, 2, 2, -1, -2, 1, 0, -2, 0, 5, 1, -2, 3, -2, -2, -1, 2, -3], [-9, -5, 3, 5, -5, 5, -3, 4, 4, 7, -1, -6, 7, 5, -3, -1, 2, -1], [-1, 5, -3, 4, -2, 2, 0, 2, 5, -2, 3, 3, -2, 2, -1, -3, 0, -1], [-5, 6, 2, 0, 0, 1, 0, 2, 0, 3, 1, 0, 7, 4, 0, 4, -4, 0], [-4, 6, -7, -1, -3, -3, 2, -4, -4, -3, -1, -4, 4, 8, -3, 2, -1, 6], [15, -2, -2, -6, 0, -2, 6, -2, 2, 0, 4, 6, 0, 2, -4, 0, -1, 2], [-11, 5, -2, -1, -2, -1, 2, 4, 6, 5, 0, 0, -2, 1, 1, -1, -1, -2], [10, 1, 2, -3, 1, -3, 4, -4, -2, -1, 1, -4, 3, -3, -6, -2, 3, 4], [21, -4, -1, -6, 3, -3, 4, -12, -3, 3, -4, -2, 0, -4, -1, 2, 4, 1], [-3, 11, -1, 2, 0, 2, -3, 2, -7, 0, 3, 2, -1, 6, 6, -2, -5, -2], [-2, -2, -1, -3, 2, -3, -2, 0, -3, 1, 5, 5, -2, -5, 2, 2, -1, 1], [1, 4, 6, 7, 1, 4, -7, 13, 0, 14, 3, 7, -4, -4, 6, -4, -4, -5], [4, -1, -3, 1, -1, 4, 3, 6, 4, 2, -2, 2, 0, 3, -1, 1, 0, -1], [21, 5, -1, 4, 0, 4, -5, 2, 0, 0, -1, 2, -1, 5, 2, 0, 1, -3], [-7, 0, -5, 2, -7, -3, 5, -2, 6, -5, -7, -8, 5, 8, -7, 7, 3, 3], [2, -1, -3, 4, -4, 2, 0, 2, 2, 2, -2, -4, -2, -2, 1, -2, -1, 0], [-22, -7, -4, -4, 1, -3, -1, -4, -4, 3, 0, -4, -2, -2, 1, -1, 2, 3], [-3, 3, 4, 1, 2, 3, -8, 4, -5, 1, 2, 4, -4, 0, 5, -5, -1, -1], [21, 3, 2, 1, 2, 3, 0, 2, 2, 1, -2, 3, 2, -1, -2, 0, -3, -2], [16, 3, 2, 1, -1, 3, -4, 2, 1, 3, 4, 6, -2, -2, 5, 3, -1, -1], [11, 4, 3, -4, -6, -3, 1, 0, -4, -7, 6, 3, -1, 3, 3, 4, 0, 2], [-1, 10, 0, 5, 1, -2, -3, 2, 0, 4, 1, 1, 2, 0, 0, -2, -4, -1], [1, -1, -2, -1, -6, -5, 6, 4, 10, 1, 8, 3, 1, 5, -1, 3, -2, 0], [2, 11, 2, -1, -1, -1, -2, -4, -2, 1, 2, 0, 2, -3, 3, 3, 1, -2], [-7, 9, -1, 9, -2, 2, -6, 6, 1, 2, -2, -1, -1, 2, -2, -3, -2, -4], [-16, 5, 2, 2, -1, 4, -5, -6, 2, 0, -3, -1, 2, 0, 1, 0, 1, -4], [23, -2, 2, 4, 4, 5, -8, 6, -7, 3, -1, -2, -1, -3, 4, -2, -2, -3], [10, -4, 3, -1, 3, 1, 3, -2, 1, 1, 1, -4, 1, 2, 1, -1, 3, -2], [19, 2, 0, 0, 1, 1, -4, -5, 0, 7, -5, -6, -1, -5, 2, -1, 2, 2], [18, -12, -1, 0, -3, -3, 1, -8, -3, 3, -6, -7, 6, 3, -1, 2, 3, 4], [11, -5, -2, -4, 2, 3, 3, 2, 1, 1, 4, 5, -3, 1, 2, 0, 1, 2], [1, 0, -3, -2, 1, 3, 0, 8, 1, -3, 1, 5, 0, 3, -2, -3, 1, 1], [11, -2, 1, -2, 1, -2, 2, 0, 0, 0, 1, -4, -1, 2, -4, 0, 0, 1], [-1, 1, 1, -3, -2, 1, -2, 8, -3, 1, 3, 1, 0, -5, 0, -2, 2, -1], [-7, -2, 0, -1, 0, 0, 4, -6, -7, -1, -7, -8, 5, 1, -3, 1, 4, 1], [19, -4, 2, 0, -5, 0, -2, 2, 3, -2, 6, 4, 0, 1, 6, 1, -5, -3], [-10, -1, 7, 4, -1, 4, 1, 1, -2, 0, 5, 0, 1, 3, -2, 1, -2, -1], [12, 0, 3, 2, 3, 4, -6, 8, -3, 6, -1, 1, -2, 3, 5, -2, 1, 1], [11, -2, -1, 5, 4, 0, -6, 2, -3, -2, -7, -5, -2, 1, 1, -3, 3, -1], [22, -6, 2, -1, 6, 1, 2, 4, 6, 7, 8, 10, -4, 0, 4, -2, 0, -3], [-10, 3, -1, 1, 1, -1, 8, 0, 0, -1, -1, -3, 4, 7, -6, 2, 0, 4], [5, -2, -7, -1, 2, -2, 1, -10, -2, -12, -4, -2, -2, 1, -1, 2, 3, 3], [4, -6, 0, 1, -1, -4, -4, -2, -6, -4, -5, -10, 5, -2, -3, 1, 2, 4], [1, 2, -1, -5, 2, -2, 4, -6, -2, -2, -2, 1, -3, -6, 0, 0, 3, 2], [9, -5, 2, 2, -3, 1, -2, 0, -3, 1, -2, -6, 6, 4, -1, 1, -1, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8004_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8004_2_a_k();".
// 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_8004_2_a_k(:prec:=18)
    chi := MakeCharacter_8004_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_8004_2_a_k();".
// 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_8004_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-115488, 1509552, -5659296, 5001480, 4260960, -6184176, -855636, 2789438, -114554, -655793, 76571, 88434, -13745, -6888, 1222, 288, -55, -5, 1]>,<7,R![2162624, -10553984, 2287648, 34280000, -4020656, -31927888, 2685216, 12561120, -1269308, -2471372, 299782, 264892, -36453, -15706, 2356, 483, -77, -6, 1]>],Snew);
    return Vf;
end function;