# q-expansion of newform 144.16.a.u, downloaded from the LMFDB on 25 October 2026.

# 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 = 144
weight = 16
poly_data = [-22, 0, 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], 1], [[0, 1920], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0], [0, 0], [9170, 19], [340200, -203], [-38860316, 8646], [-103940666, -11546], [836656846, 200918], [1120443556, -653134], [8327900776, -1498154], [69746987946, 4485641], [21566811712, 1001617], [-728552772498, -4401848], [688961918886, 25160866], [1381827248828, -6760490], [-3698220841200, 315053666], [2912330130626, -426498403], [-11396392387724, -2961839352], [7311708697558, -2851127524], [-34207175494892, -5296855880], [32202111651128, 1039362654], [114740075268730, 4532958624], [4451077951472, 19271905877], [-119736316709092, 15802156222], [-236302624469226, -37801246524], [1016068325597474, -58194785300], [-55863874634382, 166349576277], [1043765176771592, -63800637445], [-1482105211453884, -58873939516], [-379972560986138, 319388450258], [2401196839213806, -139146057580], [-6568415971342624, -391284609229], [5758548053872556, 218900053520], [-1587596353605498, 408754458470], [-19412560575090852, -405874444096], [13765445593433314, 1812662525171], [-23511810148814984, 370954670485], [-6127298985416714, 473447481816], [-36925940412725196, 3319501307286], [-4191340950014952, 883368510120], [33979505622012282, 5230146589895], [-79459835815071492, 5088141617368], [-32330726617640034, -4462446263830], [-37723454590564800, -12115517127560], [42512777691941186, -6893476093568], [-203355826346204334, 447575234041], [-28203974028904984, -31180993501065], [55123528472682020, 51384929292520], [134895862982892224, -13597930674961], [-340846602328683636, 44460373867866], [-685642617344998802, -16093204142866], [-460678435201757530, -2853507762620], [-221557010579434864, -103693688517244], [-356708766361656942, -38404051204508], [-276703495442589580, 24468841439622], [-366270144951558914, -9405000944288], [-1482618656229877064, 85440661645488], [714057126280235290, 179228129601193], [-900685852670662928, -154449733661881], [-1742821190898448322, -267384064465542], [-2598753216779337386, -8280355444400], [-3782265622944030356, 64743720751520], [-11227685790315726, -289155837095347], [-4182346883416501884, 100215769070104], [-3253400858758157304, 379389895630812], [5308847051937214602, -321146952645456], [5228441949130603338, -410024237446475], [2117778424517741980, -1378689776815556], [5279337498897072306, 435964821531392], [-14285751584194609644, 29082741024014], [-9081570348575597130, 1220847136032348], [-6353676746695393826, -1728564741738572], [-60383048587322856, 1326885084887166], [5267380617473584016, 308388238823465], [-13311156524464476130, 2175518349165252], [10361886937580708620, -954006625240878], [18165495584021212800, 2353226858690112], [2426084226432875858, 723673784910739], [-34939435128683903482, -2825122123887180], [-27943400970194821170, -656126978602734], [6579714821619490986, 6888327166569076], [21808913090673041868, 263245356484258], [-32546417324733414610, -1516586644344630], [34019614234472263952, 3110806814770366], [-59827945309784565422, -3226193375029656], [64867997011354752600, 2026909718685555], [53852022631142343860, -4893993437000866], [74798661807739938686, 6387263855288670], [51222328174738573946, 776889612561788], [70635229721720008346, -10832193693468705], [27566231623484802736, -2127837007220481], [17258730312902105628, 16979038544013314], [2133087850968997856, -2767186316035730], [17720207829622527944, 9141339307440461], [9832421701330860228, -21823095263714812], [-43065758721481715388, 6114456321976576], [-13720902504111744888, -3898512630101018], [75626262486013189386, -12922928135362707], [24244694447211776134, -11638257058593702], [-56380614901226090212, -373736921244382], [-65412011910930198858, 11408965386979578], [100066658634995742772, 28219305706716074], [8227520298942620282, 27337347039016803], [-216677699862174235076, 21613454774752270], [242713648585420605110, -33923449042026078], [92335178554288173324, 19608131870727500], [-333197039598103199806, 27252170504857936], [-53357080242508520348, 14659908259782876], [16838236344078122702, 55798367424998868], [588453209000755780776, 30615199827891514], [-577638677620380486038, -50772577937057000], [70961896080293433280, 79172007617769099], [288063986749131323054, 38594864075486012], [-386305911224917119834, -27708499211672728], [-550019464417326002884, -15911505270941112], [1021841164680553889624, -54426558479898603], [119645813956859695934, -119206053244687146], [-252738540768650366316, -24941289655721358], [-288519126427097348104, 24751720199177894], [-1389812274738584746278, 40777958066474475], [-227579642102860489572, -144796524638198388], [-237984142553993307010, -180948860995408408], [-791527083559309693598, 58018129126874176], [-1192793017267795265806, 55062698536538783], [-1058138034922061416508, 190015898858162578], [2085774285693618237124, -56890740299190186], [713139184669099899594, -174904631494632859], [414876307502337357710, -83293742742671090], [725012250403327165552, 16961790248138594], [-51908948342788675656, -262313069015907867], [1373324521983219311222, 393789323584179374], [-4455409648520865006284, 199426366552755900], [3191938943054031207256, 182719272449765908], [-4525864409414443930224, 33293336758128301], [-438472872714292531234, -280438025808523634], [5512405833379462556662, 170298085717445766], [1731626666429405669378, -400713262046376216], [4669957565829663889554, -308428156194334207], [2666145361889444799524, 72651509311989718], [7675363500064496510186, -100921104960721189], [1398754274135300903206, 144434675797191766], [3655910760773567785916, -442191680721299514], [3779395444416116181570, 511194375894149873], [653023689266419383448, 249163575750647001], [-5223270806754563624716, -487504551762127028], [-3628549612134353619050, -513382851689030254], [5450507698483691602104, 1121712566352543250], [-3302731886153262340930, -1395324705626188896], [-4108483202865547825194, 140771285903745858], [2909839670952076168428, 1580199560870357210], [14933432712515618736544, -386287469463320056], [-1041967286669444835546, 1265972558183617636], [-12253089532999210063250, 599327349842133368], [-3651932805044885027708, 205276366519385902], [-5485222146642608667064, -1325873765216053472], [3812013322570758626972, -708958440386200542], [12741506181368020842736, -357960496278440436], [11020894569932222819576, 469310944608323433], [-1956359837924832860450, -2772188369544493734], [-5796376972924784851686, 194476414247694960], [22651904521411088238714, 371766302976970183], [-21492662978602944906756, -787321460379656776], [-4605268600539788113098, 1321825262154484986], [11455599188812815293032, 3336892773320920835], [-22221150920564662719324, 1806903251947255210], [27826190590088555581710, 2235005930264956138], [41675262010035228156648, 790797129356034936], [-18304953239875935706240, 1726687606416375581], [-9655651366009393847762, 2295204644636263908], [-7474348180838862404974, 496701796918009444], [-10262740312577790418750, -2301116659160682769], [-7571976364867797633996, -4766226293100143588], [-398404402633588062570, -679516765250367072], [-40886772980394343504840, 1508136789816217172], [38345110440457288236730, 260998228021970304], [-19551374738944227212240, -1448640322285237479], [1295133748316369699926, 3578522821693855696], [-8843389694015252141972, 5335308106867262144], [-24789959992248999030030, -182039647936942189], [-14959787950442221244728, 394255967935183827], [51137123406492108873446, 2034861767526120248], [-13377369423323994203936, 10078101135897711525], [-75728425910601537065812, -4737836909825168250], [91416945851161915825422, 803194325198497606], [-16702090367630284065402, 60303215121625888], [2342664612043707125424, -435640599760845312], [-78940505522329303835358, 350498943652276295], [-54586485408158539939850, -3536321380024608730], [15867084452135213075508, -2549823341717319288], [-58893139085880175087910, 7088969667737108308], [24737257242149325782400, 871872215519794832], [26187100717017689009794, -13857461491082192488], [119879203341120976676004, 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-4402222306202641441], [147619392470723603679416, 2991824774519573755], [116509231145725381344404, -25063563192501130448], [-294167204918393771259688, 12601849419339123442], [141638687369430701484698, -38563279084635875112], [103835196439994875875280, 9436841926387284717], [-293032377298943377139698, 15058527762769548310], [-95171890786291624362648, -38409727863368098856], [40571818184575247085738, 11201591861682518135], [-80651116180504567749010, 6943271031311320018], [197894479819214599839000, 58073359188292884459], [-81312810566328333586242, 72995553248514348586], [-373895072681776263457424, -28325081034534783337], [-586402298781190510508580, -13448724329437760242], [-321452057201713830428290, 49973206606516035400], [-386497140179941232085162, 22269261783474975708], [-127187509219270691133408, 34563574813856657998], [616339457354798763649224, -13593387110685529001], [694126029897691791637252, 19999280245901446820], [-300403308738389393319258, 47973760143385577698], [169503051452584214438468, 88405672422874818138], [149281158359621373396640, 18381137586897735303], [106218531189372098230790, -50369844624187273416], [-254365790587169003722020, 20672958537176845282], [234011106943214683926512, 62691458180463683414], [-6937167925072117833550, -13148028357604837068], [-357477333623478328323550, 99507993131308678339], [-280655542112959812023020, -28830152846424030646], [-30741926913584450701160, -134123812397965502826], [-325902541225796431263108, -13604155505388400162], [-686306068174719971626292, -26627988674049521204], [219297604584563717123048, -90851486813557252515], [163247107388590042440454, -50729055703932571648], [165535894496227416535566, -92312462001851721168], [-132706593689908884641560, 95908276962401352218], [-709522439324328998634688, 101218197745438916425], [232803826383718823461260, -62184602383346298620], [-38760183486537405800068, -63808376356850047652], [-42794480387773661019568, -102137739243813984852], [1110830744476753334862486, 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[829453916641243827876550, -74471050209160263756], [-182630499856677158736476, -185207601234676446156], [-791313632500876142908694, -216689032009100400384], [-2384119939045762113228160, 101861311138197970813], [1658194985307703682362642, -49366419016646099936], [65216315555109589272472, 221266552622346481395], [3790522345123885841457396, 60383473506105921402], [-1544355972174162504122602, 182543184220290125596], [-388726229241611491813190, 104186595137166071624], [3692197611740084366055196, -30230765045096935324], [600340538240194709414560, 196100112651600183726], [1337616918886243937006024, 68273988946438183763], [-1153865795508427652317496, -87725829945265113374], [-1093467237126329718215218, -175367890596567271132], [2216890711797967396417756, 242427961018832537062], [719171177719935079211056, -169341743355847393564], [-3867322386332025795340750, -19035862464778484148], [-1983751757050903227340254, -256369850359175408073], [3616013275005287460447544, -97674116946345000723], [-1574052138231602869249058, 301085284668052799092], [2636781368375375001213626, -283096774920022483189], [-993076707485432636126916, 178076166505713728800], [-3934169862184632944223946, 148454173838015694076], [5267470376113450453352868, 16323112996492306432], [-1495548464898197499457914, 455010343322879037736], [-1537392843818931714404374, 248997370657483858543], [-9218774169596964995509760, 23680219850756239347], [-694848395430287538317502, 169009239593091095285], [1126498703888952051971892, 850056606080463700898], [-1053851864278773299263148, -454414155773808616050], [6844295841865694610879482, 195654118145550486616], [-2413964729998730550138854, -274008439941254989093], [-1912104522873805472468240, -793088034624341147471], [-4294537659456444262317860, -249860625216658630840], [-5765422471158624371916564, -378388131932094731018], [-749680047556704328772446, -306382452414956389340], [-7263657043473562195885308, -412612223269290884186], [4803484668547233122725094, 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