
// Make newform 4010.2.a.m in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4010_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_4010_2_a_m();".
// 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_4010_2_a_m();".
// 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 := [104, 1972, 7920, -8984, -68876, 37365, 211360, -189098, -150697, 190617, 34362, -83577, 2776, 19157, -2591, -2392, 460, 154, -35, -4, 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], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-192689250572281045604052, -4243524215589906115557284, -15302329716260491274790272, 30317438886744979785281938, 100582635779098028690027499, -154547417955549848542242656, -75240595290174938068450304, 160504605873151474966194043, 14421889160306297367063167, -73746841119611087873283396, 3393190587424715294223925, 18123369656041119825105522, -1859977858221430827617547, -2523901525854569513964825, 307904447861665559694004, 196621751166453406494308, -22774025091904881148580, -7785496031020130083777, 640111674902866325002, 116014587172107068127], [849481945617866039571448, 13944746847373487468891484, 30692107501390467607144180, -111351583060334801827646554, -175380389696309148203994441, 441432831515493023022840972, 27830621819326022712984952, -409808669434183097900157517, 91242894083313732643391463, 165865851676723905907971088, -58676991135624483297555223, -33787734343004053799434818, 15512929663601058950255437, 3353365447882748541181147, -2080239065107749043928048, -103781999398782250499212, 139473444254698661552340, -6105654787245623330365, -3713157145230569889126, 381890322228470127811], [-97930463161670898984784, -1845600069658615177331918, -5912694544531963846562872, 11232664459172077565024340, 37348188371556988720349338, -45795645395669983298002057, -42812984088725050790439808, 51173281785468624732206197, 22585670582814874853777716, -26717309844529810196229795, -6679433655930631630668038, 7734560140578400203662932, 1176238060510791095985110, -1316500543950236316611309, -122430844130049582041055, 131042000829305332961906, 6947979307521970282602, -7052680498361920376066, -165510479941526953732, 158358215677478871834], [308889410401021481347968, 3538350924370565587810236, 10241607847011599675503092, -9142653565077423376943002, -68619860824489316187794735, -7317961243969399947208616, 176608360040356555042609360, -27583030355109342007533899, -153750683910395021484611183, 39549692162843203014824044, 63680802429345329998519419, -19257689277831527744719262, -14199801235848615666407733, 4617869516112734577765049, 1747121076757043457794012, -589292053025298126485872, -111572241785130709379400, 38333738780464460561397, 2885395428797447040394, -999626326932643253811], [93583911115562606944987, 1244849369564183836427184, 2035241542476548031276896, -14032845569575371093974027, -10652934185109545792409197, 67677154972691604550812769, -36975255093475330305977433, -54931703343995084463191829, 49857169321253674143592880, 15509572235364018390928239, -23857680923758730001730199, -491328919934575895320239, 5694219644280705889090321, -596980868306248709499010, -727150517520315475409018, 125023760355703914577064, 47466929437924614832456, -10071449640775877619681, -1244183378690323911445, 296221760166791911206], [-218947124132096447481041, -3225994613560182407576658, -5656087873568691359199317, 31635282983204416494482189, 30085096393798773969719460, -139293281261527691706815030, 52681976363325771101082567, 117185728932173939708309293, -81409465321846185964993384, -37519361493098511832066873, 40154851697151314424725489, 3735812431564036078098954, -9703424217813463228470465, 573279854859614309256400, 1246937604319414623253837, -173438948868376514272501, -81685678338935034438956, 15259569747340112833082, 2145576029452580751067, -466793766632834090905], [768531102416619265207312, 12710119782824726532399896, 27708583065443687293291992, -125681247228007480688403482, -154815115746377582396262601, 591889518781073264019445620, -187674378859922962869132764, -508326460010982070729117693, 324059764027864699964818695, 167764158738860977503275800, -163388966210012147429801151, -18909251345769732840700930, 39837967846741924857698677, -1830244294205914165632057, -5143190390781541573654824, 668813633126194984703628, 337841240953512241566252, -60790240391370671408381, -8888942055680230084478, 1884753628400079777355], [-224710702256973187588725, -3401240547623802135591091, -6701212743343895922814451, 33496443289456748286697156, 36907829246212195311101221, -151689817123212191455930115, 50366895624856063009545583, 129606374627203981236496536, -83625563617360858132525289, -42739873769995057284338591, 41844531593334313944906169, 4839230598661196559956655, -10170243128634954925798368, 456367020084764491730201, 1310730952554843700891871, -168702378818310521319570, -86004232101683687400687, 15358368994970655094252, 2261150940496645137064, -476462524614547076662], [-436036784036438682904452, -8868469585737039803341956, -30774867882955314066751758, 48538361927323472489944564, 195466303198729993721916239, -184388049715590943196801386, -276670047108631459193912154, 227872220297451973399592059, 178427262859425888212445389, -133118622709114169808015760, -62509875650115713349464691, 42848505233762839661010308, 12603138058775366850027407, -8027508521515750685593553, -1455508542425966469741592, 869790392702661565032060, 89232430953300415098274, -50414302737871564246701, -2247181342857484764768, 1207434850304812365853], [-604059387067058925497476, -9609884254476182545264624, -13211705517797761836745184, 114546479174603106008396202, 53641870346786670161614791, -536002055774682948870807056, 354916157103821177416797104, 420010142839881237390997231, -443335161158479647005455289, -107367204487314058684431104, 207921164446610883661717853, -2749870479204769967432866, -49189944811880306907760159, 6216476889098677147808871, 6252075359393404533340596, -1172351233751431179109504, -406992850767994113620008, 91288581180531005559555, 10649506491248448238214, -2640649899689473379501], [140031249837851789305768, 2631491299065404623481847, 4483077563939623557009137, -34766170116891192798108945, -20381918000793002939944745, 177362450703633048197133085, -118400990854284243160166845, -139857317203626593295263391, 149861361592849943117898630, 35339158994767751030520578, -70558132248270910576003880, 1249533456063875953741397, 16723705361063596849228437, -2169161541444692884268581, -2127899778284846061265956, 404589169952395976706645, 138618825668018039168042, -31390865586963716408176, -3628977296486894225234, 906457072838216964082], [1592074908429308896508416, 27270917625149273833516268, 59277512190399136368455356, -264382287138187932221509550, -330977420585354991276338559, 1218727562507744425200435480, -346966847568919206376515488, -1053311329079702402249544959, 630419774302201966751461757, 354889209213996383882036604, -320693015892015725084839045, -43613024108359836061648198, 78462784018346905024421023, -2634367907510142021186183, -10147240693829979483090548, 1231316305717842238789600, 667188840556892290729636, -115636723222934464488163, -17564545293762038295530, 3631258242295806765453], [-611989356765519066222632, -8895736270030353418450328, -6832345975076365549363456, 130813845334220221247135166, 9139608244597743615157309, -647285058589129120834044824, 560935202377639861594338668, 481313367239573210807895753, -647802148768359369079590651, -98015177947681939839931512, 297161588029706448826147683, -18108209528281622156357818, -69620842627828531311759201, 10989958517098690230175977, 8803688782975893353907588, -1855859584849898459966444, -571419830443223297042128, 138695996226101670224337, 14925956319362018390438, -3927785207346788767135], [434790140985479929088453, 7120300586889498935989802, 13193929518238796506121678, -74014035244757325495950890, -69210761944566981329902851, 341092711768412943785063238, -146866351638555918297252989, -284185995548277354367416141, 216735518039565914836472611, 87543514595202460841969015, -105875503238112161972589831, -6940209720507964758129297, 25485667188491878171747223, -1946728139186136232579558, -3268651599263988205081725, 495060092922914141164018, 213893582685869905990658, -41989787193908721570809, -5614620561958718180829, 1264472658655511227494], [1112816409198791104770760, 16294502047601695803672512, 14691710720940172080593128, -209919950187223911336578014, -36563629108135351662405247, 975470236198143672379935044, -768988415983968580414341056, -737414111761402929820796123, 904241211612978215609104833, 165875733340926319740473736, -416911015802940054516558769, 18120464240992583538283722, 97889157905570962471010043, -14315694482819121711656027, -12391881937746586803605320, 2506202389660641475350108, 804798273735627353154448, -189937094677161838202411, -21028882424189033473846, 5418404036695537037889], [788415911390967734805626, 11773889608624996970981568, 16388938799762510418203468, -132983362883294444683458648, -74734627074977725735521141, 609995251439267990222801910, -364231038870957598427223512, -486641734722183014335399537, 468939029513702380312779877, 132771934441679099734661132, -221689602467094470043371235, -1813842703208030192357358, 52626949058217068420362635, -5911219384637382066178013, -6700748010287355896296002, 1187274336422850598138538, 436634595061272592855124, -94397143893885790930495, -11431752151240422468314, 2758753361192571885413], [3071590555772710668815584, 49110444353489173637753464, 75869399959125235523461804, -558302259460451814779557586, -357276643406505762646451513, 2605018497515346382262540644, -1501016739616879718371822192, -2091395692889556420271389561, 1967670775033323362634671215, 578082175331391735988689832, -934616652952177256655492303, -11918526908097317367907938, 222332963227085352961603569, -24389190301742277875629049, -28340846153083168985306712, 4972099334320587284612548, 1848029398972936980518076, -397193027005254290163733, -48406217720396724392326, 11634694858096714461999]];
    Rf_basisdens := [1, 1, 1, 12660924903950932044476, 12660924903950932044476, 3165231225987733011119, 12660924903950932044476, 3165231225987733011119, 3165231225987733011119, 12660924903950932044476, 3165231225987733011119, 6330462451975466022238, 12660924903950932044476, 3165231225987733011119, 12660924903950932044476, 12660924903950932044476, 3165231225987733011119, 12660924903950932044476, 6330462451975466022238, 12660924903950932044476];
    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_4010_a();"
function MakeCharacter_4010_a()
    N := 4010;
    order := 1;
    char_gens := [2407, 3211];
    v := [1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_4010_a_Hecke(Kf)
    return MakeCharacter_4010_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 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, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 1, -1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1], [1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 1], [2, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [1, 0, -1, 0, 0, -1, -1, 0, -1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0], [0, 0, 1, 1, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, -1, 0, 0, 0, -1], [-1, 0, 3, 1, 0, 1, 1, 1, 2, 0, 0, -1, 1, 1, 1, -2, 1, 1, 2, -2], [1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1], [3, 0, -1, -2, 0, -1, 1, 0, 0, 0, -1, 1, -1, 0, -1, 2, -1, 0, -1, 2], [4, -1, -3, 1, 1, 0, -2, 0, -2, -1, 1, 1, 0, 1, 1, 2, -1, -1, -1, 1], [1, 0, 0, -1, 0, 0, -1, 0, -1, 1, -1, 0, 1, 0, -1, 0, 1, 0, 1, 0], [3, -1, 1, 0, 0, -1, 0, 1, 0, -1, -1, -1, -1, 0, -1, -1, 0, 0, 0, 0], [3, -1, 1, -2, 0, 1, 2, -1, 0, 0, -1, 0, 1, 1, -1, 1, 0, 1, 1, 1], [2, 0, -3, -2, 0, 0, 1, -1, -1, 0, 0, 2, 0, 1, 0, 2, 0, -1, -1, 1], [4, 0, -1, 0, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, -2, 1], [-1, 1, -1, 2, 1, 0, -1, 1, 0, -2, 1, 0, 1, 0, 2, -1, 0, 0, 1, -1], [0, 1, 1, 0, 1, 1, 1, 1, 2, 0, -2, 0, 1, 0, 1, 0, 0, 0, 1, -1], [1, 1, -3, -1, 2, -1, -1, 0, -1, -1, 2, 1, 0, 1, 1, 1, 0, 0, 1, 1], [-2, 2, -1, 0, 1, 0, -1, -1, -1, 2, 1, 1, 0, 1, 0, 0, -1, 0, 0, 0], [-1, 2, 0, 2, -1, -1, -2, 0, 1, 1, 0, -1, 0, -1, 0, -2, 0, 0, 0, -2], [0, 0, 1, 0, 0, -1, 0, 1, 1, -1, 0, -2, 0, -1, 0, -2, 0, 2, 0, -1], [2, 0, -1, -1, 0, 0, 0, -1, 0, 1, 1, -1, 0, 1, -1, -1, -1, 0, 0, 0], [-5, 2, 5, -1, -1, 1, 3, 0, 2, 2, -1, -2, 0, -1, -1, -3, -1, 1, 3, -1], [3, -2, 1, -3, -1, 0, 2, 0, 1, 1, -2, 1, -1, 0, -1, 1, 0, 0, -1, 1], [-2, 0, -1, 3, 1, 0, -4, 2, 0, -1, 2, 0, 1, 1, 3, -2, -1, 0, 1, -3], [-3, 0, 1, 1, -1, -1, -1, 1, 0, 1, -1, 0, 0, -1, 2, 0, -1, 0, 1, -2], [1, -1, 2, -1, 0, 1, 3, 0, 0, -1, 0, 0, 1, 1, 0, 0, 1, 1, 2, -1], [-2, 1, 2, -2, -1, -1, 2, 0, 1, 2, -1, -1, 0, -2, -3, -2, 0, 1, 0, 1], [4, -1, -1, -3, 0, 0, 3, -2, 2, 1, 0, 2, 0, 2, 1, 2, -1, 0, 2, 0], [4, -3, 1, -1, -1, 1, 5, -1, 2, -3, -2, 0, 1, 1, 0, 3, 1, 0, 2, 2], [-2, 4, 2, 1, 1, 1, -1, -1, 0, 2, 0, -1, 0, -1, -1, -2, 0, 1, 0, -1], [-1, -2, 0, -1, 0, 1, 0, 0, 2, -1, 0, 0, 2, 2, 2, 0, 2, 1, 3, -2], [2, -1, -2, -1, 1, 0, -1, 1, 0, -2, -1, 2, 0, 3, 2, 3, 0, 0, 1, -2], [0, 0, 0, 0, 0, 0, -2, -1, -2, 3, 0, 0, -1, -3, -1, 0, -1, 0, 0, 0], [-1, 1, 2, 1, 1, 1, -1, 1, 0, 1, 0, -1, 2, 0, 0, -1, 2, 0, 3, -1], [-5, 2, -1, 1, -1, -1, -5, 0, -2, 2, 1, -2, 2, -3, -1, -3, 1, 1, 1, -1], [4, 2, -1, -2, 1, 2, 3, -1, -1, 0, -1, 0, -2, 0, -2, 4, 0, 0, -3, 4], [-3, -3, 2, -1, -1, 0, 2, 0, 1, 1, -1, 1, -1, 1, 1, 2, -1, -1, 0, 0], [9, -1, -7, -2, 0, -3, -2, -2, -3, -1, 0, 3, -1, 2, 1, 6, -1, -1, -2, 2], [10, -3, -6, -4, 0, -4, -4, -1, -3, -1, 0, 3, -1, 2, 1, 5, -2, 0, -1, 2], [4, 2, 1, 1, -1, 0, 2, -1, -1, 0, 0, -1, -1, -2, -2, 0, -2, -1, -2, 2], [-3, -4, 4, -1, 0, 2, 1, 1, 1, 1, -2, -2, 2, 0, -1, -1, 1, 0, 1, 1], [2, 1, 0, 2, 0, 0, -2, 2, 0, 0, -1, -1, -1, 0, 0, 0, 0, 1, -2, 0], [0, 1, -1, -3, 2, -1, 0, 1, 1, -2, 1, -1, 0, 0, 0, -1, -1, 3, 1, -1], [1, -2, 1, -1, 0, 0, 2, 2, 1, 1, -2, 1, -1, 0, 1, 1, -1, -2, -1, 1], [0, 2, -1, 2, 0, 0, -2, -2, -2, 3, 2, 1, -1, -1, -1, 1, 0, -2, -2, 1], [7, -1, -4, -4, 2, 0, 3, -1, -2, -1, -1, 3, -1, 2, -1, 4, -2, -1, -4, 5], [-1, -6, 5, -4, 2, 3, 6, 2, 3, -2, -2, -1, 0, 3, -1, 0, 0, 2, 1, 0], [6, -5, 5, -4, -2, 2, 8, 0, 4, -3, -4, 0, 0, 4, 1, 2, 0, 2, 4, 0], [5, -1, -4, 2, 0, -1, -2, -2, -1, -1, 4, 0, 0, -1, 2, -1, -1, -1, -2, 0], [-1, -2, 3, 2, -4, 2, 1, -1, 2, 3, -3, -3, 0, -5, -1, -3, 3, 0, 0, -2], [9, -6, -7, -2, -1, -1, -2, -3, -5, -3, 2, 2, -1, 2, -2, 7, 0, -1, -3, 6], [-7, 0, -1, 3, -2, -1, -4, -1, 0, 3, 2, 0, -1, -3, -2, -4, 2, -1, -2, -2], [4, 1, 1, -4, 3, 1, 4, -1, 0, 1, -2, 2, 2, 3, 0, 2, 0, 1, 1, 1], [-1, -1, 0, 2, 1, 1, -2, 1, -1, -1, 1, -2, 0, 2, 0, 0, 2, -1, 1, -2], [3, -1, -5, 1, 1, -1, -4, -1, -2, 0, 0, -1, 0, 0, 1, 0, -3, -1, -1, 1], [2, -3, 0, -4, -1, 3, 3, -2, 0, 3, -3, 3, 1, 2, 0, 4, 1, 0, 1, 0], [2, -3, 1, -4, 0, -2, -1, 2, 0, -1, -1, -1, -1, 0, -1, -1, -1, 2, 2, -2], [1, -5, 5, -1, -2, 0, 1, 2, 1, -3, -1, -3, 0, 0, -1, -1, 3, 1, 2, 1], [10, -3, 1, 2, -3, 1, 1, 0, 0, -2, -4, 0, 0, 0, 1, 3, 0, -1, -1, 0], [-4, -1, -1, 2, 0, 1, 0, -3, 0, 2, 2, 2, 0, 0, 1, 0, 0, -1, 2, -1], [0, -3, -2, 1, -1, 0, 1, -1, -2, -1, 1, 1, 0, 1, 1, 2, 0, -1, 1, 0], [4, 2, 1, -1, -3, 0, 5, -3, 2, 2, 0, 1, -3, -1, -2, -3, -1, -1, -2, 0], [-3, -2, 1, 1, -1, -3, -2, 4, 0, -2, -1, -3, -2, -3, 1, -5, -1, -1, 0, -1], [1, -1, -3, -1, -2, -1, 1, -1, 0, 1, 0, -1, -2, -3, -2, -2, -2, -1, -2, 3], [3, -1, -2, -2, 1, 2, 0, 0, 0, -1, 1, 1, -2, 1, -3, 1, -1, 0, -2, 3], [-3, 3, 4, 0, 1, 3, 4, -2, 3, 3, -1, 0, -1, -1, 0, 0, 2, 1, 2, 0], [7, -1, 3, 0, 1, 1, 3, 0, 1, 3, 0, 1, 4, 1, 2, -1, 1, 1, 4, -2], [1, -2, -3, 4, 0, -2, -7, 2, -1, 0, 2, 0, 4, 1, 2, -3, 1, 2, 1, -4], [-1, 3, 1, 2, -1, -1, -3, 2, 3, 3, -1, -1, -3, -6, -2, -4, 0, -1, -3, 0], [2, -4, -5, -3, 1, -2, -3, -1, -2, -1, 0, 3, 2, 4, 1, 2, -1, 0, 1, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, -1, 1, 1, -2, -1, -2, 1, -2, 0, 0, -2, -2, -1, -3, 1, -2, -1, -4, 2], [19, -5, -4, -2, -1, 0, 0, 0, -2, -5, 0, 1, -1, 3, -1, 5, 1, -2, -1, 3], [10, -4, 1, 0, -1, 0, 0, 0, 1, -1, -2, 2, 1, 4, 1, 3, 0, 0, 0, 1], [-2, 4, -4, -1, 3, 1, 1, -1, -4, 1, 1, 3, 0, 3, -1, 4, -4, 1, -4, 3], [7, 1, -6, 2, -1, 1, -4, -1, -4, 0, 0, 3, 1, 1, 1, 4, -2, -1, -5, 2], [11, -4, -3, -3, 1, 1, 1, 0, 1, -3, 1, 1, 3, 2, 0, 3, 4, 2, 3, 0], [-2, 2, -3, 3, 0, -4, -5, -1, -1, 0, 4, 0, -3, -5, 0, -3, -2, -2, -2, 0], [6, -5, 5, -3, 0, 0, 5, 3, 5, -4, -3, -2, 0, 0, 2, -3, 1, 1, 5, -2], [-5, 1, 6, 3, -2, -1, -1, 4, 3, 1, -3, -1, -2, -4, 0, -2, -2, 1, -3, -3], [-2, 2, 2, -2, -2, 0, 4, -2, 0, 3, -1, 0, -2, 0, -3, -2, -2, -1, -3, 5], [0, 3, -3, 0, 2, -3, -4, -2, -1, 2, 4, 0, -2, -1, -3, -1, -2, -1, 0, 0], [6, -3, -1, -4, 0, 1, 4, -1, -1, -2, -3, 2, 0, 2, -1, 6, 0, -1, -1, 4], [2, 1, 1, -3, 0, 0, -1, 1, 3, 0, -1, 0, 2, 2, 2, -1, 1, 3, 3, -2], [6, 0, -3, -2, 2, -1, 3, -2, 1, -2, 2, 3, 0, 2, 1, 1, -2, 0, 1, 1], [-1, -2, 5, -3, 2, -3, 0, 3, 3, -1, -2, -1, 0, 1, 0, -2, -4, 2, 5, -1], [-4, 4, 0, 3, 1, 0, -3, 2, 1, 0, 2, 0, 1, -3, 2, -5, 1, -1, 2, -4], [-7, -1, 8, 5, -2, 3, 2, 4, 4, 5, -3, -5, 2, -5, -2, -8, 3, -1, 0, -3], [5, 2, 2, -2, 2, 2, 3, -2, 1, 1, -4, 0, 2, 0, -1, 1, 0, 2, 0, 0], [-8, 2, -4, 0, 3, -2, -3, -2, -4, 0, 5, 3, 0, 3, 0, 0, -2, -1, -1, -1], [6, -1, -1, -4, 1, 1, 3, -1, 0, 2, -1, 2, 0, 3, 2, 2, -3, 1, -1, 3], [1, 2, 0, 2, 2, 1, -4, 0, -2, 2, 2, 1, -1, -1, 1, 1, 0, 0, -2, 1], [-1, -1, 2, 1, 2, 4, 0, -2, 0, 3, 0, 2, 3, 3, 1, 2, 0, 0, 2, -3], [-11, 2, -8, -2, 4, -2, -4, -4, -2, 3, 6, 5, 1, 3, 3, 2, 0, 0, 4, -3], [1, -3, 5, -4, -2, 1, 4, -1, 3, 1, -3, -1, -4, -2, -4, 1, 1, 0, -1, 2], [4, -2, -4, -4, -1, -2, 0, -2, -3, -1, 1, 1, -3, 1, -1, 2, -1, -4, 2, 2], [10, -1, -8, -3, -1, -2, -1, -4, -1, 0, 3, 2, -1, 1, 0, 5, -2, 1, -2, 2], [-3, -4, 0, -3, 3, -1, 1, 2, 0, -6, 1, 1, 0, 5, 1, 2, -3, 0, 5, -1], [-1, 3, -3, -6, 2, 1, 4, -2, -1, 2, -1, 4, 1, 3, -3, 4, 0, 1, 1, 4], [-13, 3, 6, 0, 2, 4, 7, 1, 5, -1, -1, -1, 0, 0, 0, -6, 3, 1, 5, -3], [-1, 0, 2, 2, -3, 2, 2, -3, -2, 3, 1, -3, -2, -5, -1, -2, 2, -3, 1, -2], [1, -1, 6, -4, -1, 2, 6, 2, 4, 1, -5, 1, 1, 2, -1, 1, 0, -1, 1, 0], [6, -5, -3, 0, 1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1, -1, 0, 0, -1, 1], [-1, 4, 0, 1, -1, 1, 1, 0, -4, 3, -2, -3, 0, -4, -2, -2, -1, -1, -6, 4], [-2, 2, 3, 3, -2, -1, -4, 0, 1, -2, -1, -4, 2, -4, -1, -2, 5, 3, 2, -3], [14, 2, 0, -3, 0, 0, 5, -3, 2, -1, 0, 1, -1, -1, -3, 1, -2, 3, -2, 4], [-3, 3, -5, 2, 4, -1, -4, 0, 0, -3, 4, 2, -2, 1, 2, -2, -1, -1, 0, -2], [-1, 1, -2, 0, 1, 1, -2, 0, -1, -3, 2, -1, -1, 2, 4, 5, 1, -1, 2, -3], [7, -1, -5, 1, -2, -2, -6, -1, -6, -2, -2, 2, 2, 0, 2, 6, 3, 0, -2, -1], [-12, 1, 3, 6, -4, 0, -6, 1, 0, 2, 1, -3, -2, -7, 0, -6, 4, -1, -3, -4], [-7, -1, 4, 3, 0, 2, 0, 1, 4, 2, -1, -2, 2, 1, 0, -6, 4, 1, 2, -4], [2, -6, -1, 1, 0, 4, 0, -2, -1, 2, -1, 1, 4, 2, 0, 0, 4, -2, 3, -1], [-4, 3, 1, 1, 3, -3, -4, 4, 1, -5, 1, -2, -1, 0, 2, -2, 1, 1, 4, -4], [5, -3, 2, -1, 0, 3, 3, -1, -1, 0, -2, 1, 3, 1, -1, 1, -1, 3, -2, 2], [-11, 0, -8, 4, 0, -2, -9, -1, -4, 2, 2, 4, 1, -1, 4, 1, -1, -3, -4, 0], [1, 2, 5, 1, 0, 0, 4, 2, 3, -3, -3, -2, 1, -2, 1, -1, 1, 1, 0, 2], [1, 2, 1, -1, 0, 3, 3, -2, -5, 3, -3, 0, -1, -1, -4, 2, 1, -1, -5, 4], [-5, 0, -5, -6, 2, -3, 3, -2, 2, -2, -1, 6, -2, 1, 3, 4, -2, 0, 1, 3], [4, -2, 1, -1, 1, 1, 0, 3, 3, -1, -4, 1, 5, 2, 2, 3, 0, 1, 6, -2], [5, 3, -4, 4, 1, -4, -5, 3, -1, -4, 0, -2, 0, -1, 2, 0, -1, 1, -4, 1], [7, -2, -1, -1, -2, -2, 1, -1, -2, 4, -1, 2, 1, 2, -2, 2, -1, 1, -1, 3], [0, 1, -9, -1, -1, -6, -2, -3, -2, -3, 4, 2, -4, -4, -3, 3, -1, -2, -5, 5], [2, 0, 8, 1, -2, 0, 2, 0, 5, 0, 0, -3, 0, 1, 0, -6, 1, 1, 3, -2], [-2, -3, -4, 1, 2, 1, 0, 0, -2, -2, 2, 1, -2, 2, 3, 5, 1, -4, -1, 4], [-1, 1, -6, -2, 3, 1, -2, -4, -3, 1, 4, 1, 1, 0, -2, 1, -1, 1, 0, 1], [-15, 3, 3, -1, -3, 0, 0, -2, -2, 9, -2, -3, -4, -4, -7, -6, -7, -4, -5, 2], [-12, 3, 3, 4, -1, 3, 2, -3, 3, -1, 2, -1, 1, 0, 4, 0, -2, -2, 4, -2], [13, 2, -9, -2, 1, -3, -3, 0, -4, -2, 1, 7, 1, 4, 1, 8, 0, -2, -5, 6], [-13, 3, 9, 4, 2, 1, -3, 2, 2, 2, -1, -6, 1, -4, 1, -8, 3, 5, 2, -4], [-6, -2, 3, 0, 2, -2, -1, 2, -1, 1, 1, 2, 0, 2, -1, -1, -1, 1, 1, -3], [-14, 0, 5, 5, -3, 2, 4, -3, -1, 4, 0, -1, -2, -6, -4, -5, -1, -3, -4, 2], [-1, 3, 6, 4, 1, 3, 4, 0, 1, -2, 2, -3, 2, 5, 1, -1, 1, 2, 2, 0], [1, 4, -14, 2, 1, -4, -8, -5, -6, 1, 5, 6, -1, -5, -1, 2, 1, -4, -7, 4], [-10, 2, -7, 3, -3, 1, -4, -3, -4, -2, 1, 0, 4, 0, 3, -1, 2, -1, 0, -2], [-9, 0, 2, 1, 0, -1, -9, 3, -2, 5, 0, -4, 1, -3, -2, -7, 2, 0, 3, -6], [2, -6, 1, -1, 3, 0, 2, 2, 2, -5, 3, 0, 1, 4, -1, 2, 2, 3, 3, 2], [-1, 0, 2, 0, 0, 0, 1, 3, 0, -2, -1, -1, 0, -3, -3, -4, 1, 2, -5, 1], [-3, 1, -1, 2, -2, 1, -3, -3, 1, 1, 5, -1, -3, -1, 2, -3, 2, 0, 4, -3], [1, -2, -3, -3, 1, -1, 0, 1, 1, 0, 0, 6, 5, 7, 3, 4, -1, 1, 0, 1], [-24, 3, 0, 3, -2, 2, 0, -1, -1, 5, 1, 0, 0, -4, 1, -2, -2, -2, -1, 0], [17, -7, -3, -5, 0, -4, 1, 0, -5, -4, -2, 2, -2, 5, -2, 6, -3, -1, -3, 4], [-3, 2, -9, -1, 0, 1, -6, -5, -8, 4, 0, 4, 1, 1, -3, 6, 1, -3, -5, 8], [-6, -6, 3, 1, 2, 7, 0, 1, 0, 1, 0, -2, 4, 6, 0, -1, 1, 0, 5, -1], [-13, 1, 3, -2, -1, -2, 0, 0, 1, 2, -1, 1, 1, -1, 0, -6, -2, 0, 2, 0], [6, -3, 8, -2, 0, 3, 6, -2, 0, -1, -4, -2, -3, 3, -4, 0, 1, 1, 0, 1], [-2, -3, 2, 3, -1, -3, -2, 3, 1, 0, -1, -1, 4, 4, 3, -2, 5, 1, 3, -2], [13, 0, -8, -4, 0, -4, -2, 3, -1, -6, 1, -1, -3, -2, 0, -2, -3, 2, -2, 1], [-9, -4, 6, 8, -6, 2, -1, 4, 1, 0, -3, -4, -2, -2, 0, -3, -2, -3, -5, 1], [-2, -2, -8, -4, -2, 0, 2, -5, -4, 2, 2, 5, -1, 0, 0, 5, 2, -1, 1, 0], [2, 4, -5, -8, -2, 0, 4, -8, 0, 8, 0, 5, -2, -2, -4, 5, 0, 1, -2, 2], [-10, -4, 0, 2, -1, -1, -4, 4, -3, -3, 0, -6, -1, -6, -4, -6, -1, 0, -1, -1], [-10, 0, 3, 4, 1, -1, -4, 4, 2, -6, 3, -2, -2, -3, 3, -5, -1, -1, 1, -3], [-4, -4, -5, 2, -3, -3, -5, 1, -2, 0, 0, 0, -3, -3, -1, -2, -2, -3, -5, 0], [0, -11, -2, -3, -3, 2, 3, -2, 0, -4, 0, -1, 0, 4, 2, 5, 1, -1, 1, 2], [1, -1, 3, -6, 1, -1, 4, 0, 1, -2, 0, 0, -1, 2, 1, 2, -4, 4, 3, 1], [-7, -1, 2, 3, 2, 1, 0, 0, 1, 4, 2, -3, 1, 1, 1, -6, -7, -1, -1, 0], [8, -5, -6, -7, 2, -1, 1, -4, -1, -2, 4, 1, -1, 6, 1, 7, -5, 1, 5, 1], [-13, 3, 8, -2, 1, 3, 1, 3, 4, 6, -4, 2, 6, 4, 3, -4, 3, 4, 7, -7], [-7, 5, 1, 0, 4, -5, 0, 3, 0, 2, 1, 3, 0, -3, -3, -5, -1, -1, -3, 0], [-5, 3, 4, 2, -2, 2, 5, 3, 6, 1, -3, -2, 2, -4, 3, -4, 1, 0, -2, -3], [-1, -2, -10, 3, 1, -1, -4, -1, -3, -2, 3, 0, -2, -2, 1, 3, -3, -2, -4, 2], [1, -5, -4, -7, 4, 2, 9, -2, 0, -1, 1, 7, -3, 2, 3, 6, 1, -1, 1, 2], [4, -3, -1, 1, 1, 0, 2, -2, 0, -3, 1, 3, 2, 5, 5, 2, -3, 3, 3, -3], [2, -6, 10, 1, 0, 5, 6, 3, 9, -3, -4, -3, 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5, -6, -3, 3, 3], [8, -2, 7, 1, -3, 1, 1, 3, 2, -2, -5, -4, 2, -1, -1, -2, 4, 2, -3, -2], [-5, 6, 4, 4, 1, 3, -1, -1, 5, 3, 1, 0, -1, -3, 2, 0, 1, -1, -2, -3], [7, 4, -3, 1, -1, 0, 2, -3, -1, 1, 0, 0, 0, -1, -3, 8, -2, -1, 0, 5], [-2, 8, -3, -9, 5, 1, 4, -4, -3, 1, -2, 5, 3, 2, 0, 6, -3, 2, 1, 1], [5, -9, -5, -3, -2, 5, 2, -8, 0, 0, 1, 4, 5, 8, 3, 7, 3, -1, 8, -2], [-4, -3, -7, 5, 0, -2, -6, 1, -4, -5, 9, 0, -1, 0, 1, -3, 2, 0, -6, -3], [2, 1, 10, 4, -2, 5, 3, -1, 1, 3, -4, 0, 4, 2, 2, -1, 6, 0, 1, -2], [4, -6, 2, 2, -3, 0, 5, 2, 0, -4, -2, -1, 0, 0, 2, -2, -1, -4, 3, 0], [-11, -3, 10, 1, -1, 7, 10, 3, 8, 2, -2, 0, 4, 4, -3, -5, 1, 2, 5, -3], [-1, 2, -2, 1, -1, -2, 1, -3, 1, 4, 0, 2, -2, -7, 0, 1, -3, 0, -4, 1], [-1, -2, 5, 1, 2, 1, 1, 6, 6, -8, 0, 0, -3, -1, 5, -4, 3, 3, 2, -7], [-3, -7, 8, -5, 4, 4, 12, 3, 7, -3, -6, 0, 2, 5, 3, 1, -5, -1, 5, -1], [-12, 3, 3, 5, 0, 5, -5, -2, -3, 6, 0, -4, 1, 0, -3, -4, -3, 0, -1, 1], [-8, -5, 17, -1, -4, 4, 8, 2, 7, 2, -6, -7, 2, -2, -4, -10, -1, 0, 3, -1], [-6, 1, 3, 1, 7, 4, -7, 1, -1, 6, 3, 1, 3, -1, -4, -7, 3, 1, 1, -1], [-5, 11, -6, 4, -2, -2, -5, -3, -4, 3, 5, -1, 1, -4, 1, 0, -2, -1, 0, 3], [13, -6, -8, -3, 3, -1, 3, -1, -5, -4, 0, 3, -2, -1, -1, 9, -4, 1, -7, 9], [-5, -1, -1, -1, -1, -1, -2, 1, -1, -2, 0, 0, -1, 2, -1, -2, -1, -2, 6, -6], [0, -5, 2, 0, 0, 0, -3, 0, -1, -2, 0, -3, 3, 4, 4, 2, -4, 0, 3, -1], [-18, 1, 3, 6, 1, 1, -2, 3, 7, 4, 2, 1, -1, -4, 1, -4, 1, 0, -3, -4], [-3, 4, -6, 0, -2, -2, 6, -1, 2, -5, -1, 3, -4, 2, 1, 6, -7, -3, -1, 2], [-8, 1, 12, -5, 4, 1, 4, 6, 5, 5, -6, 0, 5, 4, -2, -7, 0, 3, 5, -5], [-13, 2, 4, 7, 4, 1, -3, 4, 3, -4, 0, 0, 6, 1, 11, -3, 1, 3, 3, -6], [5, 5, -12, 3, -2, -2, -12, -4, -8, -3, 6, 5, 2, 3, 3, 8, 7, -3, 0, -1], [-5, -4, 4, -4, 4, -1, 3, 5, 3, -6, 0, 0, 1, 4, 1, -2, 0, 2, 5, -4], [-2, 4, -4, 8, 4, 1, -10, 2, -8, -1, 2, 1, 6, 2, 2, 2, 2, 1, -6, 2], [-5, -2, -1, 2, -2, 0, -1, -3, 0, 5, 4, -2, 1, -4, -3, -5, 3, -4, 4, -1], [8, -5, -3, -2, -3, 2, 6, -7, -3, -2, 4, 3, -5, 0, -1, 6, -3, 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3], [13, -1, -7, 2, 2, -1, -3, -3, -6, -5, 7, 1, -5, 2, 0, 3, 1, -5, -1, 5], [4, 3, -2, -11, -8, -1, 9, -4, 2, 2, -3, 4, -6, -1, -3, 5, 2, 1, 0, 1], [9, 2, -4, -5, -6, -6, 0, -5, -6, 6, -4, 3, -3, -5, -4, 8, -2, -2, -10, 6], [-5, -2, -10, -1, 1, -3, -9, -1, -5, 0, 5, 5, 3, 1, 1, 1, 0, -5, -4, 3], [-5, 8, 12, 5, -1, 5, 1, 4, 1, 1, -5, -6, 3, -6, -2, -4, 6, 1, 1, -2], [23, 3, -12, -11, 10, -5, 0, 1, -4, -7, 3, 8, -6, 5, 0, 11, -4, 3, -6, 10], [6, 6, -3, -4, 9, -1, -1, 5, -2, -10, 0, 0, 1, 5, 1, 4, -6, 3, 1, 4], [-3, 4, -1, -2, 6, 4, 1, -1, -4, 2, -3, 3, 6, 9, 4, 3, 0, 4, -1, 0], [4, 3, -5, 8, 5, -3, -12, 8, -3, -6, 5, -4, -2, -4, -3, -3, 4, 2, -4, 1], [23, -9, -1, -1, 0, -2, -7, 1, -3, -5, -1, -3, -3, -1, -7, 1, 1, 3, -5, 3], [-14, -2, 10, -2, 2, 2, 4, 3, 1, 3, -1, -2, -4, -6, -9, -7, 0, -1, -5, 0], [-21, 10, 11, 2, -2, 2, 5, -2, 5, 8, -3, -1, -2, -6, -2, -2, -4, 2, 7, -5], [6, 3, 11, 5, 2, -2, -1, 7, 2, -2, 1, -7, -1, -8, 0, -9, -1, 2, 4, -5], [11, -1, 8, 1, -2, 3, 4, 3, 8, -4, -9, -6, 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0, 0, -3, 0, -4, 9, 5, -4, -2, 4], [-4, 6, -10, 13, -2, 0, -12, -1, -3, 4, 5, 1, 3, -2, 2, -2, 6, -6, -6, -3], [12, 1, 0, 2, 2, 4, 1, -4, -3, 5, 4, 2, 1, 6, -3, 2, 5, -3, 1, 5], [0, 8, -7, 3, 4, 3, 5, 3, -3, -5, 1, 8, 3, 5, 4, 7, 0, 0, -1, 5], [-19, 2, -6, -3, 0, 1, 3, 0, -2, 8, 0, 7, 0, 1, -3, 4, -3, -3, -5, 4], [-10, 1, -6, -2, 0, 2, -3, -6, -4, 5, 5, 2, -5, -6, -5, -2, 3, -1, -2, 0], [15, -3, -4, -4, 1, 2, -2, -1, 5, -2, -3, 1, 1, 1, 0, -2, 3, 3, 0, -3], [10, 1, -4, 7, 5, 2, -6, 0, -1, -9, 8, -1, 1, 6, 5, 1, -2, 0, 6, -3], [-8, 0, -1, 5, 0, 3, -8, -1, -6, 7, 2, -1, 1, 5, -2, -3, -4, -2, -1, 2], [2, 1, 2, -3, -4, 0, 0, -6, -4, 6, 0, -3, 0, 0, -6, 2, 0, -2, 0, 2], [10, -3, -6, 0, 1, -5, -7, 5, -1, -2, -2, -1, 1, 0, -3, -1, 3, 3, -2, -4], [-11, 4, -3, 1, -4, 4, 5, -2, -3, 5, 1, 4, -5, -1, -3, 7, 2, -5, -4, 4], [-14, 2, -4, 9, -2, 1, -11, 1, 0, 2, 4, -3, 0, -1, 1, -4, 1, -7, 0, -5], [-9, -1, 9, 2, 2, 7, -1, 2, 4, 6, 3, 0, 4, 5, -1, -8, 4, 2, 3, -3], [8, 3, -13, -6, 2, -4, 0, -4, -8, 2, -1, 8, -7, -4, -6, 9, -6, -4, -18, 14], [1, -1, -11, 0, -4, -1, -4, -4, -3, 2, 7, 2, -5, -1, -5, 0, 4, -7, -1, 4], [24, -1, -3, -11, 1, -2, 7, -3, 0, -6, -7, 0, -5, 1, -4, 11, -1, 2, -1, 7], [0, 0, 10, -4, -1, 1, 9, -3, 0, 0, -5, -4, -7, -6, -8, -1, -8, -2, -3, 4], [-8, 0, -7, 5, 6, -4, 0, 2, 0, -3, 6, 2, 0, 0, 3, 0, -7, -5, 0, -2], [3, 5, 0, 3, -1, -8, -5, -4, -5, 5, 2, 2, -8, -9, -7, -4, -5, -3, -10, 4], [10, 4, -3, -2, 2, 3, 10, -2, 3, -3, -3, 6, -3, 3, 6, 10, 5, -3, -1, 6], [-13, -4, 2, 2, 0, -4, -6, -2, -3, 4, 4, -2, 5, -2, 0, -7, 0, 2, 8, -3], [2, 1, 6, -4, 3, 1, 2, 1, 3, 3, -5, -2, -1, -1, 0, 1, -11, 2, 0, -1], [19, 6, 8, -1, 2, 2, 7, 6, 8, -6, -7, -2, 0, 4, 4, 0, 3, 3, 0, -1], [0, 6, -11, 0, 1, -3, -6, -1, -4, -1, 7, 1, -3, -2, 2, 0, -6, -2, -1, -3], [-3, -6, 5, -3, -2, -1, -2, 0, -4, 9, -6, -5, -3, -9, -10, -5, 1, -1, -2, 1], [-7, 4, 4, 3, 5, 4, 2, 5, 5, -1, -2, 3, 8, 5, 5, 3, 2, -2, 6, -1], [-27, -1, 10, 3, -8, -1, -2, 2, 7, 3, 1, -5, -5, -11, -2, -16, 1, 1, 8, -10], [1, 1, -19, 0, 4, 0, -8, -3, -3, -3, 5, 2, 3, 3, 9, 6, 2, -2, -1, -1], [-22, 2, 18, 3, -5, 7, 3, 2, 9, 9, -7, -8, 2, -10, -4, -17, 3, 2, 8, -11], [16, -1, -1, 2, -3, 1, -7, 1, -6, 10, -5, -3, -2, -11, -11, -4, 0, -2, -15, 6], [-1, 6, 1, -13, -3, -2, 13, -5, 4, 2, -7, 3, -3, -1, -4, 4, -5, 3, -2, 7], [-13, -6, 11, -2, -4, 4, 1, 0, 1, 6, 0, -4, 5, 1, -4, -6, 2, -2, 9, -4], [9, -1, -12, -3, 6, -6, -1, -1, -4, -2, 5, 13, 2, 6, 6, 5, 1, 0, 4, 0], [25, -6, -7, -12, -7, -2, 5, -5, 0, -2, -6, 5, -6, -2, -4, 7, -2, -1, -7, 7], [-7, 4, 3, 9, 0, 5, 3, 0, 4, -1, 3, -2, 4, -1, 2, -7, 3, 1, 4, -5], [1, 6, 11, 3, -1, 2, 5, 6, 1, -3, -1, -11, 1, -4, -1, -9, 2, 5, 3, -6], [11, 9, -7, -4, -2, 3, 1, -8, -1, 5, -1, 3, -1, -1, -6, 4, -5, -1, -3, 0], [-6, 1, -4, 4, 1, -3, -13, 1, -3, -1, 9, 0, 3, 1, 3, -5, 4, 3, 1, -8], [2, 4, 4, -6, -1, 0, 9, -2, 6, 3, -2, -2, -4, -4, -6, -8, 1, 7, -6, 1], [14, -4, -6, -6, -2, -3, 3, 0, 3, -1, -2, 7, 3, 6, 6, 4, 2, 6, 8, -3], [-12, -1, 2, 10, -7, -2, -10, 3, 0, 2, -1, -3, 2, -7, 0, -10, 1, -1, 3, -12], [14, -13, 2, -5, -2, 2, 1, 3, 5, 2, -4, 1, 4, 3, 0, 2, 6, -1, 9, -9], [-6, -4, -5, -7, 0, -6, 5, 3, 1, -6, 2, 3, -1, 2, -2, 5, -9, -2, 5, 2], [-4, 2, 0, -4, -1, 0, 0, 3, 2, 2, -1, 1, -9, -6, -2, -3, -4, -4, -7, -1], [-21, 4, 4, 12, 1, 4, -14, 5, 3, 4, 2, -6, 4, -5, 2, -10, 3, 0, 2, -13], [2, -2, -11, -15, -2, -1, 5, -3, -5, 7, -1, 6, -3, 1, -8, 3, 2, -4, -3, 7], [-12, 6, -1, 6, -5, -6, -12, 5, -3, 8, 2, -2, -4, -13, 2, -9, -5, -2, -6, -5], [-17, -12, 6, -4, -3, 5, 6, -3, 4, -3, 1, -3, 1, 2, 0, 2, -2, 0, 12, -5], [-25, 3, 7, 6, -1, -6, -12, 4, -3, 2, 1, -7, -1, -6, -4, -10, -4, 4, 0, -8], [2, -3, -6, -14, 2, -4, 4, -4, -2, 4, 3, 3, -5, 2, -6, 4, -2, 1, -4, 6], [-2, 15, -14, -7, 1, -8, 2, -3, -7, -1, 2, 6, -5, -4, -1, 4, -1, -2, -8, 8], [6, 6, 7, -8, 3, 3, 7, 0, 0, 1, -2, 4, 5, 7, -4, 7, 2, 10, 3, 5], [-10, 4, 3, 6, -6, -3, -7, 2, 5, -1, 2, -11, -8, -11, -2, -13, 1, -1, 0, -2], [6, -8, 5, -8, 3, -3, 2, 7, 0, -5, -3, -2, 0, 0, -2, 2, 3, 3, 5, -1], [12, 1, 2, -7, -1, 1, 10, 3, 7, -3, -4, 0, -6, 4, 2, 0, 6, 4, 5, -5], [7, -3, 11, -1, 0, 9, 7, -2, 3, 10, -11, -4, 1, -5, -9, 2, 1, 3, -7, 6], [16, -4, -3, 0, 1, 1, -3, 3, -5, -10, 3, -3, -6, 2, 1, 4, 3, -5, 2, 0], [17, 1, -9, -4, 1, -1, 1, -1, -4, -1, 4, 5, 1, 11, 4, 9, -7, -2, -5, 6], [10, 7, 1, 3, 0, 4, 0, -5, 1, 2, 1, 3, 4, -2, 0, 9, 3, -1, -2, 2], [2, 6, -13, 9, -1, -7, -8, -8, -4, -3, 7, 5, -1, -2, 6, 6, -3, -2, -8, 6], [20, 8, -18, -4, 7, 0, -2, -5, -4, 3, -1, 8, 1, 4, 2, 5, -4, 1, -8, 4], [-13, 1, -2, 6, 1, -1, -11, 4, -3, 7, 0, 2, 8, 1, 2, -5, 2, 2, 4, -8], [9, 0, -13, -3, 7, -1, -2, -3, -7, -1, 4, 10, 0, 6, 6, 10, -2, -2, -4, 1], [-11, 9, 18, 4, -4, 5, -2, 5, 5, 7, -3, -9, 6, -3, -2, -12, 8, 3, 8, -6], [30, -3, -13, 2, -4, -5, -9, -3, -2, -5, 1, 7, 2, 7, 8, 12, 6, -2, 0, 4], [24, 0, -7, -5, 3, -5, 7, 4, 2, -7, 0, 3, -3, 1, 5, 10, -9, 2, -3, 2], [8, 1, -8, -3, 3, -2, 9, -1, 3, -8, 1, 4, -5, 3, 0, 10, -3, -2, -5, 9], [20, 4, -18, -6, 0, -3, 4, -9, -5, -2, 7, 8, -1, -1, -1, 9, -1, 0, -5, 5], [-17, 5, 3, 0, 1, 4, -7, -2, -5, 5, 1, -5, 3, -5, -5, -9, 7, 1, -2, -2], [3, 4, -4, -1, -3, -5, -4, 0, -7, 2, -3, 3, -5, -7, 1, 0, -9, -1, -13, 0], [-11, 2, 1, 5, 5, 0, -8, -3, 1, 1, 8, -4, 6, 1, 3, -8, -4, 1, 8, -9], [5, -4, -3, 2, 0, 6, 8, -2, 7, -9, 1, 2, -2, 8, 1, 2, 3, -2, -3, 5], [-10, 7, -10, 7, 0, -4, -17, -3, -4, 7, 8, 3, 2, -6, -3, -10, 0, 0, -7, -5], [0, -7, 0, -5, 7, -2, -2, 8, -1, -11, 4, 1, -3, 0, 1, -3, 0, 4, -2, -3], [29, 4, 8, -3, 6, 6, 6, -1, 0, -3, 2, -2, -8, 0, -3, 0, -1, 2, -8, 7], [1, -1, 4, 0, 0, 3, 0, -3, -7, 3, -2, 0, 0, -1, -6, 2, 4, 0, -5, 3], [7, -6, -1, 2, 5, -6, 1, 5, 1, -9, 4, -3, -1, 8, 4, -1, -8, -1, 9, -1], [14, -5, -5, -10, -5, 3, 8, -9, 0, 9, -2, 6, -4, 3, -2, 8, 0, -3, 1, 3], [10, -2, -4, 3, 1, -1, 0, 1, 1, -5, -2, -4, -5, 2, -4, 6, 4, 0, -2, 0], [-7, -8, 7, -1, -2, -6, 0, 10, 5, 1, 2, -6, 1, -2, -1, -12, 3, 1, 7, -9], [8, -6, -6, -10, -1, 1, 8, -3, -3, -1, -6, 2, 0, 7, -4, 14, 1, 0, 2, 7], [3, -2, 2, -4, -3, -5, 2, -1, -3, 4, -3, -1, -7, -5, -11, 0, 0, 1, -4, 1], [-11, -6, 10, 1, -3, 0, 3, 5, 4, -2, -2, -5, 7, -1, -2, -11, 3, 5, 13, -6], [9, 10, 3, 6, 2, 5, 5, -5, 0, -3, 2, -2, 2, 0, 2, 3, 3, 2, -5, 3], [-8, 0, -11, 4, -1, -3, -10, 1, -1, 2, 2, -1, -6, -4, 3, 2, 1, -4, -8, -3], [8, -5, 10, 4, -2, 1, 4, 4, 4, -4, 0, -2, 1, 3, 9, 0, 2, 3, 3, -3], [-13, 3, -5, 6, -3, 2, -5, -1, -1, 2, 4, 7, 6, 3, 8, 3, 5, -2, 6, -5], [9, -4, -11, -11, 3, -4, -6, -6, -11, 0, 8, 10, -3, 8, -6, 12, -9, -3, -6, 9], [-16, 9, 3, 1, 0, 0, 0, 3, -3, 1, 1, 1, 4, -2, -1, 1, 8, 3, -3, -2], [-4, 4, 27, -9, -3, 10, 25, 0, 16, 3, -5, -3, 7, 8, -1, -8, 4, 7, 18, -6], [19, -5, -16, -10, -1, -4, -3, -9, -8, -3, 2, 7, 7, 3, 3, 14, 2, 2, -2, 4], [22, -3, -13, -10, -2, 2, 2, -13, -4, 1, 2, 2, -4, 2, -7, 11, 0, -3, -6, 15], [1, 6, 6, -1, -2, 2, 4, -3, -2, 1, -2, -12, -6, -4, -7, -4, -1, 0, -4, 5], [2, 11, 9, 1, -2, 1, -3, 2, 6, 4, -9, -3, 6, -3, 5, -2, 2, 2, 4, -7], [9, -3, -5, 9, 2, -5, -10, 7, 2, -8, 6, -1, 0, -2, 11, -10, 0, 5, 7, -9], [7, 8, 3, 4, 0, -3, -3, 0, 3, -2, 0, 2, 0, 1, 3, -2, 0, 6, 6, -6], [12, -14, 6, -8, -1, 2, 10, 2, 1, -6, -8, -1, 1, 4, -4, 8, -1, 5, -5, 11], [-5, 10, 12, 5, -3, -3, -5, 3, 2, 6, -3, -6, 1, -4, -5, -9, 7, 1, 0, 1], [0, -1, 7, -6, 0, 4, 5, -6, -3, 11, -3, -4, -4, -2, -7, -1, -1, -1, -3, 5], [35, -4, -2, 10, 2, 5, -5, -1, -7, -6, -1, -1, -1, 3, -1, 9, 7, -2, -13, 5], [7, 1, -2, 3, 2, 7, 6, -6, -5, -12, 0, -4, 8, 9, 5, 10, 2, 4, 1, 4], [-16, 10, 0, 7, -5, 1, -13, -3, -5, 9, 6, -7, -1, -12, -6, -12, -2, -2, 3, -6], [-13, -3, 8, 4, 0, 0, -7, 6, 0, 7, -1, -3, 1, 0, -1, -7, -5, 0, -1, 0], [-33, 10, 13, 11, 2, 1, -7, 2, 0, 9, 7, -6, 2, -3, -2, -15, -3, 5, 6, -5], [-12, -3, 6, 4, -1, 2, -1, 3, 4, 4, 4, -1, -1, -3, 4, -9, 7, 0, 7, -12], [7, 15, 5, -8, 2, 1, 11, 4, 3, -2, -8, -3, -4, 3, -1, 3, -1, 1, -3, 6], [-1, -4, 0, 6, 5, 2, -4, 5, -5, -4, 0, 4, 5, 3, 0, 8, 2, 3, -3, 0], [3, -4, 3, 1, 1, 2, -1, 8, 6, -5, -3, -2, 4, 3, 6, -1, -4, 3, 7, -6], [19, -2, -5, 2, -2, -1, -6, -6, -5, -3, 3, -1, -5, -5, -8, 7, 0, -2, -13, 10], [8, 2, 3, -2, 1, 1, 4, -1, -6, -3, 1, -6, 7, -1, -8, -2, -3, 6, 4, 5], [21, -6, 11, -9, -9, 1, 13, 1, 10, -1, -6, -3, -7, -3, -6, -6, 7, 3, -1, -2], [-6, 7, 0, 4, -3, 3, 2, -5, -4, 1, -2, -2, 0, -5, -2, -1, -3, -2, -12, 1], [-2, 7, 14, 6, 6, 7, 1, 5, 3, 0, -2, -1, 4, -3, 5, -3, 3, 2, 7, -7], [-21, 0, 9, -1, -6, 5, 8, 2, 0, -2, -11, -4, 2, -1, -2, -3, 3, 0, -2, 2], [-1, 0, -2, -11, 5, -5, -2, 8, -4, -1, -4, 2, -4, 0, -10, 4, -1, 5, -13, 6], [-7, 11, -5, 5, 6, 1, -3, -3, -1, -1, 4, -1, 4, -1, 10, -1, 1, 1, 5, -1], [-20, -6, 18, -10, -9, 1, 16, -1, 13, 8, -6, -4, 1, 0, -7, -13, 1, 1, 15, -8], [-37, 4, 3, 6, 0, 0, -9, 7, 11, 2, 6, -4, -1, -6, -2, -15, -1, -2, 0, -6], [3, 3, 1, 7, -7, -6, -7, -1, -2, 4, -1, 2, 2, -8, -2, 0, 2, 1, -6, 2], [-1, -14, 16, -4, -5, 0, 5, 6, 9, 0, -8, -6, 2, 5, -6, -5, 8, 6, 11, -8], [6, -4, 1, -12, 2, -5, 2, 2, -2, -3, -5, 4, -4, 5, -4, 7, -1, -1, 0, 1], [5, 2, 4, -10, 4, 6, 9, -6, 1, 12, -2, 1, -1, -1, -12, -5, -6, 3, -1, 4], [0, 9, 1, 5, 0, 1, -1, 2, 2, 0, 5, -1, -6, -5, -3, 0, 2, 0, -4, 1], [-5, 4, 2, -3, 0, 6, 5, -8, -7, 8, 3, -3, 3, -3, -8, -4, -3, -4, 1, 4], [7, 3, 10, 13, -3, -2, -5, 5, -1, -5, 6, -6, -5, -6, -4, -13, 6, -1, -7, 3], [-16, 6, -14, -6, 5, -3, -1, -5, -3, 2, 2, 4, 0, -1, 1, 3, -8, -1, 0, 1], [-19, 11, 14, 10, -11, 6, 4, -7, -2, 9, 0, -8, -3, -11, -7, -10, 2, -4, -5, 0], [34, -2, -18, -6, 1, -11, -11, -1, -14, -5, 4, 9, -6, 4, 2, 14, -5, -5, -15, 15], [5, 4, -1, -9, 5, 1, 1, 4, -5, -4, -6, 3, 1, 0, -1, 3, -4, 3, -8, 3], [-19, -5, 1, 11, -2, 0, -8, 2, 0, 0, 4, 2, 3, 2, 10, -1, 0, -1, 4, -11], [-19, 11, 2, 16, -1, 4, -1, 0, 0, -3, 8, 0, -4, 0, 9, 1, 4, -5, 1, -10], [28, 3, -2, -9, 5, 4, 7, -2, -2, -6, -3, 2, 4, 15, 3, 11, -7, 3, 3, 6], [-5, -7, 10, -7, -2, -3, 9, 3, 13, -4, -4, 1, -1, 0, 3, -4, -2, 4, 15, -10], [-24, 1, -6, 5, 4, -3, -9, 4, 1, 4, 3, -2, 1, -7, -2, -11, -3, -1, 1, -11], [-11, 0, -9, -3, -2, -2, 10, -3, 0, -2, -3, 5, -3, 4, 2, 4, -1, -3, 2, 7], [15, -9, -3, 0, 6, 5, -4, 1, -1, -1, 0, 3, 8, 9, 0, 3, 0, 1, -1, 5], [26, -2, -2, 12, 2, 4, -3, 3, 0, -6, 9, -5, 4, 2, 6, -3, 6, 3, 0, -3], [-14, -4, 13, 9, -5, 8, 0, 3, 1, 10, 1, -4, 5, 1, 0, -11, 7, -1, 2, -3], [-6, -2, 2, -4, 7, -4, -4, 9, 1, -8, 3, -2, -6, 2, 0, -1, -6, 3, 4, -2], [5, -8, 8, -10, 2, 0, 6, 5, 5, 2, -4, 2, -1, 2, 2, -2, 1, 1, 7, -8], [-2, 4, 5, -9, 3, 4, 3, -3, 2, 5, -3, 7, 1, 10, 3, -1, -1, 0, -1, -1], [-15, -4, -13, 1, 2, -3, -5, 4, 0, -4, 3, 8, 2, -1, 3, 1, 4, 1, -1, -1], [25, -2, -6, -2, 4, 2, 9, -2, -4, -6, -5, -1, 1, 4, -2, 13, -12, -2, -6, 14], [18, -7, -6, -3, -5, -2, 4, -10, -4, -5, 6, -1, -7, 0, -3, 5, 1, -4, -3, 10], [5, -8, -13, 4, 2, -4, -7, -7, -11, -1, 9, 8, 1, 9, 3, 15, -5, -7, 1, 7], [8, -5, 9, 11, -3, 2, -10, 8, 5, 0, 4, -3, 5, 0, 3, -6, 8, 2, -1, -6], [1, 17, -3, 4, 2, 0, -1, -4, 1, 0, 4, 0, -3, -4, 2, 4, 2, 1, -7, 3], [-1, 0, 13, -3, 0, -3, 2, 10, -1, -4, -8, -8, -3, 2, -5, -5, -7, -1, 1, -1], [-7, -8, -2, 0, 2, -4, -9, 4, -10, -4, -1, -3, 3, -4, -8, -7, 4, 6, -6, 0], [0, -7, -4, 8, 2, 2, -8, 12, 1, -9, 5, 2, 4, 4, 5, -10, 5, 2, 2, -7], [2, -9, -17, -7, 0, -1, 3, -3, -5, -7, -3, 5, -4, 2, 2, 15, -1, -4, -9, 6], [-9, -3, 16, 5, -4, 4, 3, -1, 5, 8, -1, 1, 2, -4, 0, -6, 0, 2, 5, -8], [-1, -1, 8, -3, 0, 6, 10, -1, -3, 1, -4, -2, -5, -2, -5, 7, -2, -6, -6, 8], [-2, -6, 14, 12, -9, 6, 3, 7, 7, -3, -10, -5, 6, -1, 11, -3, 8, 0, 4, -9], [-9, -2, 2, 7, 1, -3, -6, 9, 0, -6, -3, 1, -3, -6, 1, -1, -1, -1, -3, -7], [-5, -1, -4, -7, -4, 5, 4, -11, -4, 6, -4, 8, 6, 6, -3, 13, 8, 0, 5, -3], [4, -7, -2, -8, -1, 5, 5, -5, -6, 4, -3, -2, 2, 3, -5, 3, -8, -3, 3, 8], [7, 2, -11, 6, -7, -3, -4, -4, 2, -6, 6, 8, -2, 8, 13, 9, -5, -6, 3, -2], [11, 6, 0, 1, 3, -1, -9, 3, -4, 8, -3, -4, -8, -15, -11, -5, 0, -2, -9, 4], [-17, -5, 12, 7, 2, 3, -3, 13, 4, -7, 0, -8, 8, 7, 10, -7, 10, 6, 10, -11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4010_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4010_2_a_m();".
// 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_4010_2_a_m(:prec:=20)
    chi := MakeCharacter_4010_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_4010_2_a_m();".
// 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_4010_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4010_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![104, 1972, 7920, -8984, -68876, 37365, 211360, -189098, -150697, 190617, 34362, -83577, 2776, 19157, -2591, -2392, 460, 154, -35, -4, 1]>,<7,R![814592, -1700672, -9615552, 35241520, -27706332, -26591471, 41762271, 2277150, -21177395, 3269369, 5124325, -1295078, -634313, 211579, 37502, -17399, -594, 703, -31, -11, 1]>,<11,R![-24378448, -182436384, 705716048, 821789228, -2275086887, 152229420, 1250476406, -289821722, -290696103, 92360842, 33806267, -13789522, -1902776, 1114552, 29044, -49510, 1845, 1122, -85, -10, 1]>],Snew);
    return Vf;
end function;