
# q-expansion of newform 4028.2.a.f, downloaded from the LMFDB on 24 April 2024.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_data]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 4028
weight = 2
poly_data = [139, -454, -4299, 5074, 30973, -17877, -73816, 23819, 72868, -18074, -36567, 8368, 10113, -2310, -1554, 364, 124, -30, -4, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 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, 0, 0, 0, 0, 0], 1], [[-4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[9549212368394962571709, 90955650844496841415443, -95863405384260650578292, -486508306277487080639637, 199437779162941782881844, 628050699125868748022486, -173273403885715628219601, -386387527001845234846065, 78774350849770433685435, 134639173642389041239377, -21632651398845249336587, -27975363851656900935375, 3904702873986854643991, 3427742929074332447811, -464929689331149538134, -228419041336493582073, 32803688712224653227, 6383747831994437549, -1013664970932510359], 3478013382692242784767], [[-705048640521550753042, -2596169087667389090749, 2463678410242470840173, -17339171104576617579725, 1526271337661236731546, 75554189902004790177990, 1974045361748312551583, -80261934709718916524839, -2126285279123905811034, 39188082322529543612125, -101997297017966586210, -10269789484355555618923, 425689341513369750520, 1491567589135895220569, -122100450800907992989, -113206383041965395179, 13579116821905178591, 3504924600062958270, -539666137422440460], 267539490976326368059], [[-27761816022030137605193, 30696628439091594640205, 409776759302680392011964, -227160207345741293466219, -1251133748221990524239808, 498719127530284526140679, 1313330090608220013976793, -560547808941881549889402, -642414053222074731899579, 328269697028020825287738, 156111636410667783119248, -103019545288071940947301, -16346936089987529154161, 17414925175315486603474, -106264229225035104222, -1492920352180888609775, 149810450509960698612, 50880960665791710281, -8082393497238969918], 3478013382692242784767], [[-10377632966827837230239, -31727649377226224441820, 208367847416842091060431, -192745004320722586135416, -596066650789253892258349, 823567850258607734691640, 706422386390206413846371, -929821432914397749569504, -377008214805426624184168, 492197278713807875634399, 91128082993182265376372, -140062845135953860708398, -6799678529958853409030, 21925206662121998889480, -955354606015150023233, -1774315165620840355805, 192444453247398587102, 57926039052261232522, -9012255479615042506], 3478013382692242784767], [[15374816921928772387641, -57209576941038616681447, -234975891591382458211335, 519748650162003555440563, 625724025506759966537372, -1321405476326058704896156, -713224542559588122446075, 1334621888720261551464630, 369120574768379920994051, -677647246369639428662873, -81682667251389768988723, 188792773780458340703538, 2617817530125671121032, -29172991237797969942832, 1830443128106006764405, 2340433431752486897347, -279042796602546905350, -75947048383905243843, 12279512726675507013], 3478013382692242784767], [[-15853588812280295893235, 23840706565177952489302, 267273604049339077042987, -268415003301992744582801, -785844487179013523942315, 636223365416096458777228, 688097937337630306295553, -686917593354695955136902, -230459265695755928256657, 380259478726129276646478, 9828945992025776641311, -113935358339789835760057, 12819124770709822621272, 18593376947389501907842, -3348151150958097388752, -1550580703664022927300, 335701280791985868818, 51660710244299223716, -12314233074030226871], 3478013382692242784767], [[-28540922552046344408613, 93150844803515409831409, 350995415744919634018826, -622937693458533872437694, -946705048039643406597082, 1056358072940628201207339, 772620603092657335620073, -917172825624740629980752, -230905172735736471947896, 453114600578601704464099, -3016754497444860404492, -128638473769920492132139, 17408020904773011271906, 20488657664749485257454, -4076081956140805156372, -1691009143344334751529, 392201561392315144161, 56121672911821323774, -14065420372976239962], 3478013382692242784767], [[262481099435760130156, -139956340876172755996760, -39479411448802007040965, 968395259204709981854911, 265372142123995710209070, -2016364357746995762333272, -448004696282821758586931, 1836166095699660343346776, 250297060329871709304802, -880863656053346109871933, -42092416564172393413605, 237678327165159094674093, -6868373979917272840906, -36042418775253329863245, 3252798257142975761126, 2857419119741421916394, -394095446186709103493, -91966583363222535462, 16072902599680908164], 3478013382692242784767], [[11724294891590491667200, 49020417227303367772001, -179606709602492481236308, -760900293840186507497893, 791993991768000869119075, 2180901680454689586443078, -1058318120853383259078135, -2239724891565664824511173, 744046916765897486514004, 1128173850045330519126837, -313689368711917707970084, -310036125983548622083787, 80170999974298067452935, 47245043857622210665744, -11963509787421702685666, -3740690915509558295648, 950561701022030194182, 119883384573197873174, -30920828473910836670], 3478013382692242784767], [[1702381593629708054002, -3661243503266635980568, -9804675742813541314583, 59232482816746354496069, -2496555985410708691581, -172099709947074829946152, 26888038421775232406730, 182933524201420133860161, -33475877988139306145394, -95597437847383147171360, 19341402264776327604609, 27142175615340616021689, -5923377110657869476954, -4248080377837881910399, 981957540622806796858, 343539561760627625000, -83094520879379915030, -11193644850226234750, 2810991684136139728], 267539490976326368059], [[-677863154453489493139, -7840357659589812655154, 19433210783559378052114, 80504940697230352291768, -71019286251562866791270, -211203652985939143543248, 87958646698080945085998, 212283232873006321803260, -60921824645513997515995, -106323157980190747086538, 26237875672467871584328, 29205864935468996547239, -6918026893091333773760, -4456501003206093014746, 1061232596093721452725, 353490448330280482628, -86108385019162773642, -11349244696320364505, 2843677740234168613], 267539490976326368059], [[18658611605802252907286, 23092845362882775090641, -236425795844853268020665, -762538587432516170978728, 820001318022588237116666, 2523077391398104754291646, -972178791369964142248127, -2735691555041965902330833, 670278700785601694237276, 1421761132329473018744018, -299779358367535526932361, -398916321013084576590828, 83153170959761190732914, 61679278046621658447683, -13340827181414440749721, -4935523460323896609933, 1120729210703677838193, 159449436967766400861, -37978614188213183709], 3478013382692242784767], [[556041491076014525149, -113067419744114310145875, -134077632441870097460973, 1146406281163259773760714, 291400103820138825132342, -2983358528152073364113123, -118537219926767515035213, 3047221868344953263091880, -170599429571660391077600, -1557247265878266347682706, 181268757821956946544273, 435638704099058343964752, -69043082073825151131539, -67477650887082219075842, 12745454425647953208988, 5416654855583962870974, -1146351490718162460997, -175567919116157259598, 40260786094638775650], 3478013382692242784767], [[15780177101141905520027, -183519035076096841260150, -243669449289305650873372, 1557597897636070494502675, 487122365658243491064730, -3747246844666897741476244, -283478766103769053494878, 3727403505749017453402842, -116988033924023668491712, -1890216667950960941706585, 187312962802747769677365, 528414033628651687262510, -77276600504063208667028, -81978738033841550109093, 14770922372314263261637, 6594657794653130538136, -1354654963745146272946, -214183933471934477069, 48161166165251651829], 3478013382692242784767], [[-1760874733024935089596, -41897785519851591635960, -31233698366619744425441, 859923878877118650833277, -6987864269487478484626, -2646330299676823810790046, 240849246121361947629705, 2962668592805649058520512, -408904352130513796813489, -1614284798142203414385807, 279312317316166220169993, 473026411609922601696842, -94140475891623742215449, -75769169127617525350201, 16540275904376039157620, 6232225832453837317484, -1452874990210943747302, -205635435554699590005, 50398135494972326796], 3478013382692242784767], [[-25629137576845305236733, -140612089369578959957683, 327642677232636835042171, 1757835191063551020382195, -1282054769126796767569368, -5009650161716357552256666, 1732825690188432852621703, 5253982706384872413598083, -1319563541418099291012535, -2703288083749504306601356, 614094686929332164171070, 755874140813028078170070, -170512673658002814491048, -116715759953417738577203, 27005489836129121170361, 9332831270331460449619, -2234608040469850149678, -301318492504557167969, 74710486437446437192], 3478013382692242784767]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -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, -1, 0, 0, 0, 0, -1, 1, 1, 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, -1, 1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0], [1, -1, 1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0], [1, 0, 0, 1, -1, 0, 1, -1, 2, 0, 0, -1, -1, 0, 2, 1, 0, 1, 0], [-1, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, 0, 2, -1, -1, -1], [2, -2, 1, -1, 1, -1, 0, -1, 0, 1, -1, 1, 1, 1, 0, 0, 0, 0, -1], [1, 0, 0, 0, 1, 0, 0, 1, -1, 0, -1, 0, 1, 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, 1, -1, 0, 1, 0, -1, 2, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 1], [1, 1, 0, -1, -1, 0, 0, -1, 1, 0, 1, 0, 1, -1, -1, 1, -1, 0, -1], [5, 1, 1, 1, 1, 1, 0, 1, -1, 0, 0, 0, 1, 2, 1, -2, 0, 0, 1], [0, 1, 0, 1, 1, -1, 1, 1, -1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0], [-1, -1, 0, 0, -2, 0, -1, 1, 0, -1, 0, 1, -1, 1, 2, 0, 0, -1, 1], [1, -1, 1, -1, 0, 1, -1, 0, -2, -1, -2, 2, 1, 2, -1, -2, 2, 0, 1], [2, -2, 0, 0, -2, -1, -1, -1, 1, 1, 0, -1, -1, 1, 2, 0, 0, 2, -1], [0, 1, 0, 1, -1, 0, -1, 0, 0, -1, 0, 1, 0, 0, 1, 0, 0, 0, 1], [0, -2, 0, -1, 0, -1, 0, -2, -1, 0, 0, 0, 1, 1, -1, 0, 1, -1, 1], [1, -2, -1, 0, 2, -1, 0, -2, 1, 3, -1, 0, 0, 1, 0, -1, 1, 2, -1], [2, 2, -2, 1, 1, 0, -1, 1, -2, 0, 2, -2, 0, 0, -1, -2, 0, 0, 1], [1, 0, -1, 2, -1, 0, 1, 1, 0, 1, 0, -2, -3, 0, 4, 0, 0, 0, 1], [3, 1, 0, 1, 2, 1, -2, 0, 0, 0, -2, 1, 0, 2, 0, -2, 0, -2, 1], [-1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, -1, -1, 0, -1, 1, -2, 0, 0], [0, 4, -3, 1, 0, 0, 0, 0, 0, -1, 2, -2, -2, -5, 0, 3, -3, -1, 0], [-1, -3, 2, -1, -1, 0, 1, 0, 0, -1, -2, 0, 1, 1, 0, 1, 1, 0, -1], [0, 2, -2, 1, 1, -1, 1, 1, 1, -1, 1, -1, -2, -1, -1, 2, -1, -1, 0], [3, -3, -1, 1, -1, 0, -1, -1, 1, 0, -1, 1, 0, 0, -1, 0, 3, 0, 0], [1, 0, 0, -1, 0, 2, -2, 1, -3, -1, -2, 2, 0, 1, -2, -1, -1, -1, 0], [1, 3, -1, 0, 1, 2, 1, 1, -2, -2, 1, -1, 1, -1, -4, -2, 0, -1, -1], [2, -5, 1, 0, -2, -2, 0, -3, 2, 1, -1, -1, -1, 0, 3, 1, 1, 3, -1], [4, -2, 1, -1, -3, 1, 0, -1, 1, -1, 0, 0, -1, 0, 1, 2, -1, 0, 1], [-1, -3, 2, 0, -3, -2, 2, -5, 3, 1, 0, 0, -1, 0, 1, 3, 1, 2, -1], [0, 2, -1, 0, 0, 0, 1, 0, 0, 0, 1, -2, -2, -1, 1, 2, -1, 0, 0], [0, -5, 1, 2, -1, -2, 2, -2, 3, 1, -1, -1, -1, 1, 3, 1, 2, 3, -2], [3, 1, 2, -1, 3, 2, 0, 2, -3, -2, 0, 2, 5, -1, -4, -3, 1, -2, 1], [6, -1, 1, 2, 2, -1, 1, -1, 2, 1, 1, -2, 1, 2, 2, -1, 0, 1, 1], [-2, 0, -1, 0, -1, -1, 0, -2, 1, 0, 1, 0, -2, -1, 1, 1, 1, -3, 0], [1, 0, 0, 2, -2, -1, 1, -1, 0, -2, 1, -3, -2, 0, 1, -1, 0, 2, 1], [0, -2, 1, 0, -2, 0, 1, 1, -2, -2, -1, 2, 0, 2, -2, 1, 2, -2, 1], [-1, 0, -1, 0, -2, 0, 1, -2, 4, -2, 0, -1, -4, -2, 2, 1, 0, 2, -1], [2, 1, 0, 0, 0, -2, 1, -4, 3, 2, 2, -2, 0, -2, 0, 4, -2, 1, -2], [3, -2, 0, -1, 3, 1, 0, 0, -2, 2, -1, 3, 5, 1, -2, -3, 0, -1, -1], [6, -2, 2, -1, 0, -2, 0, -3, 0, 0, -1, 1, 0, 3, 0, -2, 2, 0, 2], [1, -7, 5, -1, 1, 0, 0, -1, -3, 0, -3, 4, 4, 3, -2, -3, 3, 0, 2], [4, -3, 2, 0, 0, 0, 0, -3, 1, 0, -3, 2, 0, 2, 0, -1, 1, 1, 1], [-3, -1, -2, 2, -4, -3, 0, -1, 3, 2, 1, -4, -5, 0, 8, 0, 0, 3, -1], [-1, 1, -1, 1, -1, 0, 3, -1, 2, 1, -1, -3, -1, -1, 4, 0, 0, 4, -1], [1, -6, 1, 0, -1, -2, 0, -2, 5, 2, 0, -1, -1, 2, 4, 0, -1, 3, -1], [2, -3, 1, 2, 0, 0, -2, 2, -1, -2, -2, 1, -1, 3, 0, -1, 1, -1, 1], [4, 2, -2, 1, 0, 1, 0, -1, 1, 2, 0, 0, 1, 1, 3, 0, 1, -1, 1], [6, -6, 2, 0, 0, -1, -3, -1, 3, 1, -1, 2, 0, 2, 3, 1, 1, -1, -1], [4, -5, 3, 0, -1, 2, -1, 1, -1, -1, -1, 2, 2, 3, -2, -1, 0, -3, 0], [-4, 1, 0, 0, 0, 1, -1, 2, -3, 0, -1, 0, -1, 3, 0, -3, 0, -1, 1], [5, -4, 1, -1, 0, 0, -2, 1, -2, 1, 0, 2, 2, 2, -1, -3, 1, -1, 1], [-1, 0, 0, 1, 1, -1, -1, 0, -2, -2, 1, 1, 2, 1, -2, 2, -1, -3, 1], [5, 3, -1, 0, 4, 1, 0, 2, -2, -3, 0, 2, 2, -1, -6, -2, 1, -1, 0], [1, -2, -4, -1, 2, -2, -2, 3, 1, 2, 0, 0, -1, 0, 1, 2, -1, -3, -1], [-5, -1, 0, 0, -1, 2, 2, 4, 0, -2, -2, 3, 2, -2, -3, 0, 2, -1, -1], [1, 2, -1, -1, 3, 1, -3, 1, -2, -2, 2, 1, 2, -1, -5, 0, -3, -1, 1], [0, -3, -3, 0, -3, 0, -1, 2, 3, -1, 0, -3, -2, -4, 3, 3, 0, 1, 1], [2, -8, 3, -3, -3, 1, -2, -3, 2, 1, -4, 5, 2, 1, -2, 0, 2, 0, 0], [-1, -4, 1, 1, -1, 1, 0, 0, 2, -2, -3, 0, -2, 0, 3, -1, -1, 1, 1], [3, 1, -1, -1, -1, -2, 2, -4, 3, 0, 0, -1, 0, -4, 1, 3, -1, 2, -1], [2, -4, 2, -3, 6, 0, 2, 0, -2, 3, -1, 1, 6, 0, -5, -1, 2, -1, -1], [-4, 2, -3, -2, -1, -1, 2, 1, 0, 3, 3, -3, -3, -3, 1, 2, 0, 4, -2], [6, -1, 2, 4, -2, 2, -2, 2, 0, -1, 1, 2, -1, 3, 0, -3, 0, -3, 2], [1, -2, 3, -1, 0, 1, 2, 2, -3, -2, 0, 0, 0, -1, -2, -1, -2, 1, 0], [5, 7, -4, 4, 4, 4, -2, 4, -2, -1, 0, 1, -3, 3, -1, -3, -1, -1, 2], [-3, 0, -1, -1, -1, 2, -1, 2, 2, -2, -3, 3, -1, -2, 1, 1, 2, 1, -2], [4, 2, 0, 2, 0, 3, 0, 5, -1, -3, 2, 3, 2, -3, -4, 1, 0, -4, -1], [-2, 0, 4, 1, -1, 1, -2, 5, -4, -2, -3, 4, -1, 5, 1, -7, 3, 1, 3], [-6, 1, -2, 1, 1, -1, 2, -2, 3, -1, 0, -2, -1, -2, -2, 5, -3, -1, -1], [-3, -2, -4, 4, 0, -3, 1, 2, 5, 3, 4, -5, -4, 0, 6, 2, 1, 2, -1], [0, 2, -4, 0, 0, 0, 3, 1, 2, 0, 0, -1, 0, -1, -1, 3, 0, -2, 0], [5, 1, 1, 0, 0, -1, 2, -3, -1, 1, 1, -1, 1, 2, -1, 1, -2, 2, -1], [0, 4, 3, -3, -1, 2, 1, 0, -3, -3, 1, 2, 2, -1, -3, 3, -3, -1, 1], [-7, 2, 2, 1, -1, -3, 4, -5, 1, 1, 0, -2, 0, -1, 0, 3, 1, 3, -3], [2, -2, -2, 2, -2, -5, 1, -5, 1, 3, 0, -4, -2, 4, 7, -4, 4, 4, 1], [-2, 2, -2, 2, 2, 0, 1, 3, -1, -1, 2, -2, 0, -1, -3, 1, -2, -2, 1], [0, -2, 1, 1, 0, 3, -4, 6, -5, -4, 0, 2, 2, 3, -2, -2, 0, -3, 0], [1, 7, -1, -2, 4, 3, 0, 2, -4, -1, -2, 1, 1, -2, -7, -2, -4, -3, -2], [-6, 1, 1, 3, -2, 2, 2, -1, 3, -1, 0, -2, -3, -3, 4, 2, 1, 2, -2], [4, 7, 1, -1, 1, 2, 1, -2, -5, -2, 0, 1, 2, -3, -5, 0, -2, 0, -1], [0, 0, -3, 0, 0, -1, -3, 0, 3, 0, 5, -2, -4, -5, -1, 3, -5, -1, -2], [5, -4, 3, 0, -1, 1, -3, 3, 0, -1, -4, 1, 0, 6, 5, -7, 4, 2, 5], [10, -3, 0, 0, 3, 0, -1, 2, -1, 3, -1, 0, 1, 3, -1, -5, 1, 3, -1], [-3, 2, 1, -1, -1, 2, 1, 4, -2, -2, -1, 3, 3, -2, -3, 1, -1, -3, -2], [-1, -5, 1, 1, 0, -1, 3, -3, 3, -3, 0, -2, -4, -1, -1, 4, -1, 2, -1], [-7, 4, -2, 1, 3, -2, 1, 0, -1, -1, -2, -1, -2, -1, -1, 3, 0, 1, -1], [-8, 9, -4, 3, -1, -2, -2, -2, 3, 1, 2, -7, -6, 1, 7, -1, -4, 4, -1], [-7, -4, 1, -2, 1, 0, -1, -2, 0, 3, 1, 0, 2, 0, 0, 0, -1, 0, -2], [1, -2, 4, 0, -6, 1, 0, 3, 5, -3, -2, 1, -3, 0, 8, -1, 2, 4, 1], [5, -3, 0, -3, 1, -1, -3, 3, -2, -1, -1, 2, 2, 1, -1, -3, 0, -2, 3], [-2, -4, 1, -3, -2, -4, 1, -5, -2, 5, 3, 2, 1, -1, -1, 6, 1, -1, -2], [4, -6, 4, -2, -2, -1, 0, -7, -1, 5, 0, 2, -1, 6, 3, 1, 2, 0, -1], [8, 7, -1, 5, 7, 2, -1, 5, -3, -2, 0, -2, -1, 4, 1, -6, -2, 0, 3], [-6, -2, 5, -1, -3, -3, 1, -5, 1, 1, 2, 1, -1, 1, 2, 1, 0, 4, -1], [7, 3, 3, -2, 1, 3, 1, -3, 2, -4, -1, 3, 1, -1, -4, 0, -1, -1, 1], [3, -3, 3, 0, -1, 3, -2, 1, -3, -1, 0, 4, 2, 2, -2, -4, 2, -1, 3], [0, 5, 0, 2, 3, 1, -5, 7, 1, -2, 3, 1, 0, -1, 1, -2, -4, -5, 2], [0, -6, 1, 1, -2, -2, -1, -3, 2, 6, 0, 0, 0, 4, 4, 3, 1, 2, -2], [-7, 5, -5, 1, 4, 0, -2, 4, -3, -3, 0, 2, -1, -4, -5, 0, -4, -4, 1], [1, -6, -3, -3, 1, -1, 1, -1, -1, 4, 0, -1, 1, 0, 1, -2, 3, 1, 1], [-4, 6, 0, -1, 3, 3, 0, 4, -5, -5, -1, 3, 3, -1, -9, -1, 0, -7, 0], [4, 1, 3, 2, -1, 2, 2, -5, -1, 1, 2, 0, 0, -3, 0, 0, 1, 3, -1], [3, -3, 1, 3, -3, 0, 2, 0, 2, 3, 0, 1, -5, 5, 8, 1, -2, 4, 0], [-3, -5, 0, 1, -1, -4, 2, -2, 0, 3, -1, 0, 1, 3, 2, 2, 0, -2, -2], [1, -2, 1, -4, -3, 0, 0, -1, 0, 1, -3, 5, 4, 1, -1, -1, 3, 0, 0], [-2, 0, 3, 1, -2, 1, -3, 5, 1, -1, 2, -1, 0, 2, 2, -1, -1, 1, -1], [-4, 2, -5, 1, -7, -3, 2, -6, 4, 2, 2, -5, -7, -5, 7, 6, 1, 1, 1], [0, 2, -4, 1, -5, 1, -1, 2, 0, 0, 3, -1, -2, -2, 2, 1, 1, -2, 0], [3, 12, -5, 5, 4, 0, 1, 0, 1, 2, 2, -4, -3, -3, 1, 0, -3, 1, -1], [2, 5, 0, 1, 0, 0, 0, -1, 1, 3, 0, -1, -2, 3, 5, -3, 1, 4, -2], [-2, 10, -1, 2, 4, 2, 2, 2, -4, -2, 1, 0, 0, 4, -3, -6, 0, 0, 2], [-2, 7, 3, 3, 3, 3, 0, 6, -3, -8, 2, 2, 2, 3, -8, -2, -2, -4, 3], [-4, -10, 1, 1, -4, -7, 2, -10, 8, 2, -4, -5, -4, 0, 7, 5, 2, 6, -4], [-6, 0, 0, -1, -1, -2, 1, -5, 2, -4, -1, 3, 1, -3, -2, 7, -2, -5, -1], [3, 0, 0, -1, -2, 0, -1, -2, 2, 1, 2, -4, -6, -4, 4, 4, -3, 1, -2], [-9, -5, 0, -5, -3, -1, 3, -2, 2, 1, -4, 3, 3, -2, -1, 5, 3, -2, -1], [1, 11, -2, 2, 0, 3, -1, 6, 1, -4, 6, -3, -4, -5, 1, 3, -10, -3, 0], [0, -3, 0, -4, -3, -5, 2, -12, 6, 6, -2, -3, -1, -2, 5, 1, 3, 9, -1], [-3, -6, 1, -2, 2, 1, 0, 4, -5, -1, -2, 1, 0, 2, 0, -3, 1, -2, 2], [-3, 0, 4, 1, -1, -2, 1, -5, 8, -6, 4, -2, -2, -3, -2, 7, -4, 1, -2], [3, 0, 4, -1, -2, 3, -4, 0, -4, -5, 0, 1, 2, 0, -5, -4, -2, -3, 2], [4, -4, 2, 0, -2, -2, -3, -4, 6, 0, -3, 0, -4, 0, 6, 5, -2, -2, -3], [-1, -4, 3, 0, -1, -1, 2, 1, 2, -3, -5, 0, -1, 1, 4, 4, 2, -2, 2], [-3, 5, -6, 5, 3, 2, -1, 0, 0, 1, 0, -3, -3, -2, 1, 1, -2, 2, -3], [5, 4, 3, 3, 4, 2, -2, 4, 3, 2, 1, -2, -2, -1, 4, -1, -4, 1, -1], [5, 5, 1, -1, 3, 3, 0, 0, -2, -3, 1, 2, 5, -3, -7, -4, 2, -2, -1], [1, 2, 1, -1, 0, 4, 3, 2, -5, -2, -3, 1, 3, -2, -1, -5, 7, 0, 0], [2, -8, 2, -3, -5, -1, 5, -8, 8, -1, -1, -1, -3, -4, 1, 6, 1, 7, -2], [2, -1, 3, 2, 4, 0, 1, -2, 0, -3, 1, 2, 7, 5, -5, 0, -2, -7, 3], [4, 2, 6, 3, 1, 3, -1, 6, 0, -4, -2, 2, 1, 7, 3, -4, 0, -3, 4], [4, -10, 0, 1, -1, -3, -4, -4, 1, 3, -2, 1, 0, 1, 3, 3, 2, 1, -1], [-9, 7, -4, 0, 2, 2, -1, 1, -2, -1, 0, 0, -4, -2, -5, 2, 0, -3, 1], [6, 14, -6, 1, 4, 5, 1, 1, -2, -1, 4, -2, -2, -6, -5, 7, -10, -5, -1], [0, 6, 0, -4, 5, 3, 0, 0, -6, -3, 1, 3, 7, 0, -10, -7, 1, -2, 1], [3, -6, -2, -1, -5, -5, 1, -3, 3, -1, 3, -3, -2, 0, 2, 4, -1, 1, 1], [-2, 9, -1, -2, 0, 1, 3, -7, 4, -3, -1, -3, 0, -5, -3, 3, 0, 1, -1], [-2, -2, -2, -4, 5, -2, -2, -3, 0, 7, -1, 4, 7, -3, -5, -1, 1, 0, -3], [3, 4, -3, 6, 3, 4, -5, 12, -4, -1, 3, -2, -4, 4, 2, -4, -4, -4, 2], [4, 4, 0, -1, 4, -1, -4, 1, 4, -3, -3, 3, 3, -1, -2, -2, 0, 1, 0], [-12, 1, 1, -4, -2, -2, 0, 1, 0, -1, -1, 3, 1, -4, 1, -3, 0, 3, -1], [-4, 1, 1, 0, 5, 6, -3, 5, -8, -2, -4, 2, 6, 2, -5, -7, -1, -3, 0], [5, -3, 0, 2, -5, -3, 0, -7, 4, 2, -3, -2, -6, 6, 10, -1, 3, 3, 2], [9, -7, 2, -1, -5, -4, 4, -4, 8, 5, -1, -2, -2, 0, 8, 3, 6, 6, 0], [0, 4, -1, -3, 1, -1, -2, 0, -5, 0, 6, 3, 5, 1, -11, 1, -3, -5, -1], [0, 2, 2, -6, 2, -1, 0, -2, -1, 1, 1, 2, 8, 2, -7, 1, -1, 0, -4], [-8, 10, -5, 1, -5, 0, 6, -2, 9, -1, 5, -7, -10, -6, 4, 6, -5, 5, -1], [6, 0, -3, 2, -2, -2, -2, -4, 2, 0, -2, 4, -5, 2, 3, 6, 0, -4, -1], [-7, -4, -4, 0, 2, -5, -2, 0, 4, 6, -2, 1, 2, 1, 6, 1, 3, 2, -2], [-5, -1, -3, 0, 1, 2, -7, 2, -3, -1, 6, 1, 0, -2, -3, 0, -4, 1, 1], [-5, -2, 4, -3, 0, 1, 1, -3, 0, 1, 0, -3, 3, 0, -3, -1, -1, 3, -2], [7, 1, 2, 4, 3, 1, 3, 2, 2, 1, 3, -2, 1, 2, -1, -2, 2, 3, -1], [-14, 14, -7, 5, -1, -1, 2, -4, -1, -5, 3, -4, -5, -4, -1, 1, -3, -3, 1], [-4, 4, 0, -3, 7, 3, 0, 5, -8, -1, 1, 3, 8, -3, -13, 2, -6, -10, -2], [-1, 9, -1, 0, 3, 4, 1, -3, -1, -1, -1, -2, 3, -6, -6, 1, -1, -1, -2], [-15, 2, -5, 0, 1, 0, 7, -3, 3, 3, 0, -5, -3, -4, 1, 2, -1, 4, -3], [-15, 3, -1, 4, -2, 0, 0, 0, 1, -2, -3, 0, -4, 1, 7, 1, 2, 1, 0], [-2, 2, -5, -1, -5, -5, -1, -3, 4, 6, 4, -2, -10, -3, 10, 8, -6, 1, -3], [10, -11, -4, 3, -6, 1, 4, 6, 3, 3, 2, -2, -6, -1, 8, 4, 2, 2, -2], [-6, -7, -1, 3, 0, -1, -1, 0, 1, 3, -6, -3, -5, 1, 8, -3, 1, 5, 0], [-10, 5, 6, 3, -1, 2, 4, -5, 1, -1, -5, 2, 0, 0, 4, 1, 1, 0, -2], [3, -6, -1, -2, -1, -2, -2, 2, -2, -5, 0, -2, 2, -2, 2, -1, -1, 0, 3], [-5, 2, 0, 0, -2, -3, -3, 3, 5, -5, -2, 0, -6, 1, 5, -1, -2, 1, 2], [-10, 0, -1, 6, -3, 1, 5, 0, 3, 2, 5, -3, -2, -2, 2, 2, 1, 3, 0], [-10, 0, -3, -1, -5, -2, 5, -4, 1, 3, 3, -4, 0, -3, -1, 9, -1, -3, -3], [-2, 5, -5, -1, -7, -3, 1, -9, 7, 2, 0, -3, -6, -7, 5, 12, -2, 0, -1], [2, 5, 1, -1, 4, -1, 0, -1, -10, 3, 0, 0, 5, 2, -1, -4, 2, -1, 3], [7, -1, -3, 3, 1, 3, 1, 0, -1, 2, 5, 2, 1, 1, -6, -2, 2, -4, 2], [-1, -11, 1, 3, -1, -1, 1, -3, 0, 5, -2, 0, -5, 6, 5, -6, 5, 6, 2], [0, 1, -3, -1, -3, -3, 1, -5, 4, 5, 0, -6, -6, 0, 6, 1, 1, 8, -4], [-1, 7, -3, -4, 6, 2, 1, 3, -8, 1, 2, 2, 7, -1, -12, -3, -1, -5, 0], [13, -1, 1, 0, -3, -5, -3, -6, 7, 5, 3, -5, -4, 0, 5, 4, -3, 5, -4], [0, -7, -4, 6, -4, -2, -4, 1, 5, 2, 4, -7, -8, -5, 12, 7, -5, -1, 2], [5, -1, -4, -1, -1, -3, 5, -10, 5, 5, 0, -7, -4, -5, 1, 7, -4, 0, 1], [-11, 2, 3, 0, -4, 5, -1, 6, -5, -9, 1, 3, 0, 2, -5, -4, -1, -4, 3], [2, -7, 0, -3, -5, -1, 5, -4, 0, 3, -2, 3, -1, -6, -3, 4, 4, 4, -3], [4, -9, 3, -4, -5, -2, -2, 1, 2, 0, -2, 3, 0, -3, 5, 2, 4, 5, -2], [-6, -7, 0, -6, 2, -4, 0, -2, 3, 4, -4, 0, 1, 0, 6, -3, 6, 5, -2], [15, -11, 10, -3, -2, 4, -2, 2, -2, 0, -2, 8, 7, 5, -3, -6, 4, 4, -1], [8, -9, 8, 2, 2, -2, -1, -1, 2, -1, -1, 4, 4, 5, -3, -8, 7, 3, 0], [-10, -1, -7, 1, -6, -1, 0, -1, 8, 0, 4, -4, -9, -9, 8, 7, -2, 10, -2], [-2, -12, 7, 2, -5, -4, 3, -5, 2, 5, -3, 1, 0, 12, 9, -4, 11, 4, 2], [-2, 2, -4, -4, 1, -3, 3, -1, 2, -1, 1, -2, 2, -6, -7, 7, -1, -2, -5], [3, 0, 3, 4, 0, 3, 3, 8, -2, -6, 2, 2, 2, -5, -7, 5, -1, -4, -1], [2, 11, 0, 1, 6, 1, -1, 0, -1, 1, 2, -5, 1, -1, -1, -2, -6, 4, -2], [5, 3, -4, -1, 2, 5, -1, 1, -6, 2, 3, -1, 4, 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