
# q-expansion of newform 304.5.e.f, downloaded from the LMFDB on 20 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 = 304
weight = 5
poly_data = [34273490123096064, 0, 34336742150504448, 0, 10365057927217152, 0, 915101881952256, 0, 37911701888064, 0, 861580608848, 0, 11414409521, 0, 89661524, 0, 408854, 0, 996, 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, 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, 0], 1], [[76456656887018284541945328020963796713472, 0, 40244551137216999503608585033015821139968, 0, 4353828936944978660336592305349348956160, 0, 204916299226419925363498434451520373312, 0, 5087982217402337932788988323231547792, 0, 71688787769461832364673746163452457, 0, 587790123766048552377507145752628, 0, 2762453645622673838324578791526, 0, 6877057136442363798278307204, 0, 7015591482736478697048953, 0], 57579639647349195649389006725245304832], [[70698692922283364977006427348439266230272, 0, 40186971497569650307959196026290575835136, 0, 4353828936944978660336592305349348956160, 0, 204916299226419925363498434451520373312, 0, 5087982217402337932788988323231547792, 0, 71688787769461832364673746163452457, 0, 587790123766048552377507145752628, 0, 2762453645622673838324578791526, 0, 6877057136442363798278307204, 0, 7015591482736478697048953, 0], 57579639647349195649389006725245304832], [[58090622717697780565593410239714409054208, 0, 34960748522456071674010884825018728644608, 0, 4357172388027681444166990554565277116416, 0, 232436557284241687911141751343697094080, 0, 6348511732789677193101321543582364784, 0, 95555342365459310547859333569292151, 0, 819248809664970853004336184445004, 0, 3969323674535653396041654595418, 0, 10093865115949232203052077308, 0, 10455416804468796827256871, 0], 57579639647349195649389006725245304832], [[-9398646894548195730109601850383344336896, 0, -5277393403135015289567544539634859573248, 0, -519193082079399066235804477752172296192, 0, -22528723381674710988047329081062250560, 0, -526273740917645490897535374871063696, 0, -7101429163025354149803831900898849, 0, -56499853652885085742913376980116, 0, -259993917040271693274733929142, 0, -637580256755232951101678052, 0, -643312172309955074890481, 0], 3030507349860483981546789827644489728], [[-3955144028429171873970587124361719152640, 0, -2203607703171522522839869736233199622144, 0, -226929819092999084567903673382426546752, 0, -10273561523110636706368379785287355344, 0, -248350846952221396150448127622415497, 0, -3438018037140310115727303132963703, 0, -27862778610788824562264188853698, 0, -129922657869745666360708045222, 0, -321672121357164915533942613, 0, -326858064707069045267699, 0], 899681869489831182021703230081957888], [[380377480317245530016520067415011954262016, 0, 204926522714218558841067465735083517149184, 0, 19100515377661570656509089954774492237824, 0, 785408101220240112321856690491480183360, 0, 17469228706108904660959373065879595504, 0, 226347285305475167107772619976906865, 0, 1744536228692261486845893770227988, 0, 7835356762863573048101522959382, 0, 18863078260110627805736419716, 0, 18765241665041430289977025, 0], 28789819823674597824694503362622652416], [[-322250322855149837384824380948872964341760, 0, -156342028813703659660921085022631958347776, 0, -1471744953938229746830417462633450924032, 0, 554392432559250613299773920959651654720, 0, 25580063507462217511580478649168985872, 0, 479054233294477260474837587630685073, 0, 4606325962317218964453922339110484, 0, 23883932186162958841250597643862, 0, 63431418977529811606584931620, 0, 67669402085462715252962657, 0], 57579639647349195649389006725245304832], [[-2486395785099896932935915635491112963014656, 0, -1369635074731126216680733994200038630162432, 0, -130950749090552781663472710565650872567808, 0, -5491696571065831659876244594123487144640, 0, -123928134836511151628463012123413744816, 0, -1622614672608434508954758940494290379, 0, -12600755952704034603443858077324508, 0, -56905001939647026820676839758866, 0, -137545212550475095237072867020, 0, -137240621828284286426961595, 0], 57579639647349195649389006725245304832], [[0, -70468374363693968194408871321538285010944, 0, -40186971497569650307959196026290575835136, 0, -4353828936944978660336592305349348956160, 0, -204916299226419925363498434451520373312, 0, -5087982217402337932788988323231547792, 0, -71688787769461832364673746163452457, 0, -587790123766048552377507145752628, 0, -2762453645622673838324578791526, 0, -6877057136442363798278307204, 0, -7015591482736478697048953], 57579639647349195649389006725245304832], [[0, -33103571513741873814383925595772966535168, 0, -8603216675179839177307242219963959205888, 0, 262711646049447987545955679869328945920, 0, 54203015331270985665855140593184110464, 0, 2081934755712703856445202528146229724, 0, 36423013119501783511210139625351659, 0, 338160805602856785259840544633060, 0, 1717351867500227807304918591074, 0, 4500420772583206945097889144, 0, 4757507011033628632965739], 32388547301633922552781316282950483968], [[0, 9410768923947637666035789009693922295808, 0, 5277393403135015289567544539634859573248, 0, 519193082079399066235804477752172296192, 0, 22528723381674710988047329081062250560, 0, 526273740917645490897535374871063696, 0, 7101429163025354149803831900898849, 0, 56499853652885085742913376980116, 0, 259993917040271693274733929142, 0, 637580256755232951101678052, 0, 643312172309955074890481], 3030507349860483981546789827644489728], [[0, 1993204154558833521222249060232778861248512, 0, 1144268233184962788658118007184657299603456, 0, 112014512994327709551867824101809028872192, 0, 4806122491228111583350687714922428161216, 0, 110827155338421622564330993768844607472, 0, 1478068970425135060871726718929145187, 0, 11648138211081514032945870078408508, 0, 53204034049504430945268855975202, 0, 129726445798216282516267191276, 0, 130314032331384017607966419], 518216756826142760844501060527207743488], [[0, 1856441532162924812077690391101440, 0, 1035521777419187138093400013799424, 0, 102698886644271549417058259963904, 0, 4481857070326235379379724216640, 0, 105004302444107628034030635568, 0, 1418607308373456959300506669, 0, 11290193140987889665694404, 0, 51949520568053402047678, 0, 127364482636332577140, 0, 128472238193535293], 489305261028270253253023236096], [[3396760456325830704956610267043442511052800, -3336877101768509122411872022072807422689280, 1971082819490521371673497742276095075287040, -1835024744762747502017004013896272477749248, 213767163452324164287355771532702758096896, -177736384736686419055298524862396799473664, 10067556742550107115247062221980737741184, -7566018479420387457857630946671621946816, 250270845195032230074264095160069882720, -173286581159122358912523025054857733936, 3530090551240008804435854378739159942, -2298273089724429172361720769983694055, 28966237799985711984331693035354744, -18033841597222665526797307801015180, 136200991851480339701727952415940, -82101334648667974579891775458810, 339170327404441280638325520984, -199688104766568244479139705980, 346060871401547756311157862, -200210600552828147661370775], 345477837884095173896334040351471828992], [[3396760456325830704956610267043442511052800, 3336877101768509122411872022072807422689280, 1971082819490521371673497742276095075287040, 1835024744762747502017004013896272477749248, 213767163452324164287355771532702758096896, 177736384736686419055298524862396799473664, 10067556742550107115247062221980737741184, 7566018479420387457857630946671621946816, 250270845195032230074264095160069882720, 173286581159122358912523025054857733936, 3530090551240008804435854378739159942, 2298273089724429172361720769983694055, 28966237799985711984331693035354744, 18033841597222665526797307801015180, 136200991851480339701727952415940, 82101334648667974579891775458810, 339170327404441280638325520984, 199688104766568244479139705980, 346060871401547756311157862, 200210600552828147661370775], 345477837884095173896334040351471828992], [[0, -60047837806541658463437230735853237633024, 0, -33369741090976583217740716963051308398592, 0, -3239537350237690091105302813972978443072, 0, -138283683002747593502755136004565159824, 0, -3176315381116064873338789709726693197, 0, -42235227375263661857820924817733257, 0, -332098957116204928345990160763778, 0, -1514390436249429079633618897186, 0, -3687971553150258167061489921, 0, -3701240537101493545783253], 2699045608469493546065109690245873664], [[0, 156288098797745245063055883287489138393088, 0, 86481873506557731474767753474810644463616, 0, 8676395530632321361793554972273649068032, 0, 382839349800472436520069213796808318784, 0, 9051020523449177721835247968625761424, 0, 123124481428295173942584040944498221, 0, 984878840499038787219994418783492, 0, 4548496388545692553789759981886, 0, 11181902569463184778985391060, 0, 11302110714735679635473341], 5822660189057783829713495062103457792], [[0, -24244918478947441021245712897205773283622912, 0, -13449774943388775506925322098877529727369216, 0, -1317109410863299598962976095493158704064512, 0, -56717491026505220960927230734516393393600, 0, -1312802631246926504637542608690968487792, 0, -17562698831876423820043913344275387799, 0, -138731991940756946637712262099617996, 0, -634783652976433794767906566229530, 0, -1549799964337629468243093550716, 0, -1558346050875267305097550919], 518216756826142760844501060527207743488]]

hecke_ring_character_values = [[191, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [229, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [97, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[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, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, 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, 0, 0, 1, 0, 0, 0, 0, 0, 0], [11, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [30, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-29, 0, 0, -3, 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, 0, -1, 0, -1, -1, -1, 0, 0, 0, 0, -1], [0, -20, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, -2, -2, -1, 1, 2, 0, -1], [0, -16, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 2, 0, -1, -1, 1, 0, 1, -1], [0, 26, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, -2, 4, 2, -1, 1, -1, -2, 0], [61, 0, 1, 17, 0, 4, -2, 3, 1, 2, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0], [191, 0, -4, 1, 0, -7, -6, 4, -2, -1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0], [0, 17, 0, 0, 0, 0, 0, 0, 0, 0, -10, -3, -2, 0, 0, -5, 5, 4, 1, 2], [0, 43, 0, 0, 0, 0, 0, 0, 0, 0, 6, -1, 5, -2, 0, -1, 1, -5, 0, -1], [25, 0, 9, 38, -1, -20, 11, 3, 1, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 0], [0, 85, 0, 0, 0, 0, 0, 0, 0, 0, -10, -1, -6, -14, 0, -5, 5, 0, 0, 3], [0, -70, 0, 0, 0, 0, 0, 0, 0, 0, 14, 3, -6, 10, 0, -1, 1, -4, -4, -1], [62, 0, -13, -42, -11, 18, -3, 6, 1, -2, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0], [0, 36, 0, 0, 0, 0, 0, 0, 0, 0, -7, -6, -2, 20, -14, -2, 2, 6, 8, -2], [834, 0, -27, -2, -12, -30, 33, -1, -3, 2, 0, 0, 0, 0, 0, -3, -3, 0, 0, 0], [0, 84, 0, 0, 0, 0, 0, 0, 0, 0, -8, -13, -2, -8, -20, 1, -1, 9, 2, -6], [0, -130, 0, 0, 0, 0, 0, 0, 0, 0, 1, 7, 3, -23, -7, -7, 7, 16, 3, 1], [-1440, 0, 60, 46, 12, 114, 40, -4, 6, -2, 0, 0, 0, 0, 0, -6, -6, 0, 0, 0], [0, -106, 0, 0, 0, 0, 0, 0, 0, 0, 26, 13, -10, 22, 26, -1, 1, 0, 4, 5], [0, -63, 0, 0, 0, 0, 0, 0, 0, 0, 10, -15, -13, -6, -10, 5, -5, 1, 8, 5], [0, -399, 0, 0, 0, 0, 0, 0, 0, 0, -7, -12, 1, 9, -3, -12, 12, 22, 2, -7], [0, -324, 0, 0, 0, 0, 0, 0, 0, 0, 16, 15, -2, 16, 2, 9, -9, -1, 0, 2], [0, 300, 0, 0, 0, 0, 0, 0, 0, 0, 7, 15, -22, -50, 8, 9, -9, 0, -8, -13], [1243, 0, -105, 231, -24, 0, -58, -15, 9, -6, 0, 0, 0, 0, 0, 6, 6, 0, 0, 0], [-1320, 0, 73, 216, -3, -170, 25, -8, -7, -6, 0, 0, 0, 0, 0, 22, 22, 0, 0, 0], [-2839, 0, 54, 67, -32, 210, 43, 18, -10, -6, 0, 0, 0, 0, 0, 10, 10, 0, 0, 0], [1252, 0, -42, -363, -78, 168, 46, -12, -6, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -496, 0, 0, 0, 0, 0, 0, 0, 0, 20, -20, 20, 8, 10, 16, -16, -8, 16, 0], [-1585, 0, 43, 239, -37, 158, 27, -27, -1, -10, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0], [-3792, 0, 196, -310, 92, -294, -12, 48, 2, -4, 0, 0, 0, 0, 0, 7, 7, 0, 0, 0], [0, -308, 0, 0, 0, 0, 0, 0, 0, 0, 37, 2, -12, -52, 2, 8, -8, 4, 0, 30], [0, 799, 0, 0, 0, 0, 0, 0, 0, 0, 73, 34, -22, -9, -59, 22, -22, -40, 6, 5], [0, -102, 0, 0, 0, 0, 0, 0, 0, 0, -26, 16, 19, -80, 66, 17, -17, 15, -8, -28], [0, 611, 0, 0, 0, 0, 0, 0, 0, 0, -55, 4, 32, 5, 129, -36, 36, 14, -2, 13], 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