# q-expansion of newform 5.22.a.b, downloaded from the LMFDB on 20 September 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 = 5
weight = 22
poly_data = [239341586400, -743130000, -1929606, -1, 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], 1], [[0, 2, 0, 0], 1], [[187959204, -6622602, -3089, 5], 324], [[287631324, -1174314, -877, 1], 324]]

hecke_ring_character_values = None
aps_data = [[728, -1, 0, 0], [20803, 14, 0, 1], [9765625, 0, 0, 0], [128229189, -151718, -240, -3741], [8426594482, 10314300, -34960, -498310], [215800602474, 164888472, 10880, -2463340], [4424319060766, -1294061352, -1490560, 56964212], [16304015379096, 763721112, -3945616, 93015260], [76520469215315, 24630667030, 76636400, -418790491], [-296443394191386, 952540949168, -306462784, 78166280], [2031846005744342, 133752737900, 1773573920, -862528630], [12678019030524750, 8600426719280, 7046988160, -190415090072], [-37800336566436498, -20801622292600, -18359135680, 580236129020], [48792380862265727, 35817392485846, 50846202400, -186104750483], [-115510891753802843, 61133418909546, -16982582800, 1761380347691], [253393522035018698, -160292367443896, -37579912320, -4654113643524], [2499469711860952868, -1816200651268864, -78619775088, -10107368119560], [-2704962433906538338, 1208712183964000, 1085335436800, 36191725358800], [5490782175513793701, 1878089789248978, -3277891990560, 13649810007327], [12820470449883872422, 6658993564529500, -2368785369600, 122110301527650], [18680267051247552550, 8041828828308920, -7145920103040, 99802721989764], [-15242079871567732836, 16637645067077848, 23120794903296, -591598647271500], [-69659091046984109349, -46943043913465122, 20433394056000, 42495668251833], [-80868888287293415358, -88135702035718896, 34085851966848, 1609362983646840], [77141812121329572982, 287086760138387176, -276035270648320, 1333589101939436], [-182336878023443824258, 303324567468835600, 237730946377280, 8530185975789080], [-731634721067279947377, -129549488691498706, -198544578705680, -2418651913220039], [-620098036432681826111, -592776825777825958, 323641692728320, -2829852386722901], [-1397826007655287364762, -369173710012598384, 454293234117632, 5807934049182200], [-1247947958023965935366, 778087310952321552, 420076688286080, -18546639595438600], [-1637225844753112224067, -463597203135024326, 3093631175091760, 27940882680632035], [2508420848872650019902, -2836106412971360700, 1459754182773840, 146478625191237990], [-4515461559920586289270, 2012348730140520640, 7674041769687680, -239202765993549472], [21232340689004043260612, 7757545350653647584, -4012090955981232, -105128636450314200], [-12622608913115667839010, 18956859546767686640, -10670173617614080, -196045255036546360], [11711431261914799087742, 43925878166973213100, -16156202786736320, -47239962485903270], [54023592756755353717254, -33172973567012128528, -11868568817121920, -85425064826377976], [15709452660330079591519, -79034340554550472058, 3714477751509440, -293224137289420795], [155389754464115956982561, -55824455880899910062, 14323248214823120, -22160763955217113], [-165397989879283513950250, -37468645558132125200, -118248983195474560, -1589752072191997816], [-471003941903972312375116, 134029681252639893008, 57242335992219696, -433591279975381920], [130460343604121513336982, -288514883223692092000, -131918633894321600, -2023545947358047600], [-571149583939918648435118, 54868302685051464100, 64720461759053280, -1914410767744633170], [-272384648664385117660898, -56140919710523064584, 126199274570875520, 5604874560566275172], [413230127403368606720762, 142989536060720726536, 697962048347340800, 4293360199859228636], [1003090434687050586321320, 125630101275648471520, 65116095335150560, -2050856378039264480], [1617861953486587304230102, -12479574600293009900, 918069770194036880, -7341145729393081570], [2333824957779819797387015, 157593395017686731790, -51074746502381040, 15151417177682491449], [1879120002114128613166133, -783235308779563471566, 1263357011977748320, -8843631765885340625], [-3110174033296696423768906, -1735844813564402758912, 1587341452754204096, 4119358413180444320], [-1132818665415854427639874, -245530999353415022552, -354819590343239680, -31124297280535589812], [3815118546863178259334932, 2344950088217393556744, -1665310165355621632, -3406296912608944420], [-6608962467000327729749658, 902729599011715017000, 590612856879953600, 6667658530744117100], [1861366922465114769304002, -1825268488481801218500, 897464019228654000, 59193574689606332250], [-12842931005939104336730406, 903047422623547402032, -6393334945022186880, 37954832276374704744], [15545184833103858215569379, 9111706838817004064022, 2248469138436981040, -86699786790368748155], [-17308158679165223650283714, -2046366027096742101888, 4079393959247569664, -92250760234737472960], [-5287443367906965586904718, 6199513005209254368900, 3611002151207810720, -255395673718312711330], [29001239264558033961655358, -11800715933036445145936, -9589961312700678400, -90001404627195581080], [-18162368963898182434552058, 10070697070164499981400, 12855056555111005120, 20164727855792796820], [34450861473906949569049431, 12800178693681125979238, -4189132322124206880, 380740289427137024133], [-23543843708077203519339418, 35044059919589310044896, 1780419683486168960, 23875568126397910352], [30228603809081946216413589, 14185018063842709916562, -5005768881127444960, 392182034270709038879], [-73931684299117126762716478, -3501625497197567497100, -39385305427506820480, 261615283633120764470], [96870982834962494439443314, 49975316875157049028912, -38469935124141760000, -456434549607925543000], [-151578061134137070646174134, -39658463763058181416072, 31893233637794405120, 97382040241205045732], [42811318265504669826449662, 18257686176965213807700, 36545135917092790960, 2277339197750523315310], [-90578193371886634043823590, -135770964729867973744240, -5615783118214986240, -33784196355194881032], [-319595627549524757360503023, 91485165767052703275226, -19545242999873190080, 639974028210865125451], [-28824296673616071597791810, -67469712682022714004640, -30482843712283229760, -3621767636123474910480], [198808242597851256972360938, -13135561251512064281456, 196442540848426265600, 2413758284316113866616], [-228627879487672696839586472, 251141271204107767615616, 43464419966800758112, -4206438070251327102320], [-96246391692452738776672519, 151431737036534112371858, -74699566446732823120, -2430889824423947759593], [349669596035737513558657230, -101905452639013285017840, -142697957498073015040, -5595425710479549036616], [-247278428912900936831165776, -624371978000112407021672, 213914582784681689296, -6640306983484635973460], [-55275491983945082643330889, 420090216705425174892718, -603995928682749021040, -2575312007795188451287], [-305105494313049191652761738, 440623561463499307505984, 268328158337483987968, 8723827646349889583200], [-612117917430653667857068918, -326931081626603016661544, 115039066545729919360, 2433922305247628435956], [-767122461420708262900595278, 146280428309075940692800, -1225885212037271647360, 916464866385919771040], [3445635322182593575686321958, -384501837914840137334504, -228019094035105450048, 6157304500263372414260], [-2199977579202403788022996404, -777445423815761531892448, -899714800496375861776, -10630907746650554238280], [-21869769423117647285516818, 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-2651503188212923213600, 56192329513597516925507], [-2330353883966513778041211796, 1877106143440469957332888, 4800652135255561697216, -83955129100812639451660], [18168213055911122690453274665, 1402955000506530261473250, 265081094723242766160, 11864185642094201546463], [9251577516848109729280405922, 3434573861250795160052700, 2003400230225638731760, 14582648142146375493610], [810514634236799758590238840, -5602145442803206887802600, -4228253025645357365520, -108097998007588809734820], [15851281480074130166472562423, -4782243415400158004585186, 2195928999539518440240, -55769159413090867123359], [21124951102301499053392611438, 3295968339270186617405056, 10328319136122513383872, -46623595014737022705440], [-9821988844395319501475144918, -13734380496465860663225600, -3659900157624349831680, -48917399455400321508480], [-4031608694731685608116606585, -3478853823163039972212570, 2762162483137606580000, -81141188039030114901691], [16189278725371468988519061862, -6716714136435048585829200, 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[-312762588089896153142887587863, -104159537727665415568083294, 33549183969437985037520, -1465697334000147657811009], [-14961196057773858683554005982, 182383396962191837777819784, -148459802596686870617600, 725315075617957148443036], [74721113313192997047220800788, -109211940817852721099265904, 82818242522562857614512, -1591155360504952204334880], [318627035773926556176035661622, -17393497837026551909701600, -51182721379500884628480, -1374589685451971574683280], [167142074383889696776444581710, 32971382664741190126933400, -13737862411344281021440, 3294588848889861103771604], [-64850113656081672362201475582, -72840185022593288287206696, -68311133060228804259840, -2021556331837765579428300], [669607090753041112232240438631, -161327644091264222307025242, -63027775662497240216160, -264503037920820030363387], [53743505028142329372799645482, -281896937279466544939872900, -131871180157797672297520, 1078439964465014377407530], [-351275590699810585310749332138, -162948555162438676247527600, 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