
# q-expansion of newform 384.10.a.f, downloaded from the LMFDB on 17 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 = 384
weight = 10
poly_data = [561570, -13768, -2070, 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], 1], [[1012188, 76928, -100, -88], 483], [[-1362252, -15488, 1396, -8], 483], [[-3249600, 246656, 4736, -160], 483]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0, 0], [81, 0, 0, 0], [-60, 0, -1, 0], [1210, 1, 2, 1], [-24916, -5, 13, -4], [-15210, -5, 23, -19], [-108738, -41, -83, 10], [157944, 55, 29, -30], [187348, -146, 216, 174], [-1977136, -556, 41, 202], [2837810, 663, -1072, -171], [-3398230, -1571, 1123, -2949], [-4709722, -1171, 9263, 1018], [3544480, 1509, -8681, -30], [-9444780, -6662, -2440, -3450], [28834128, 1480, 31753, -2334], [-28757020, -4356, -19436, 1924], [43307162, 5041, 38587, 4043], [-57946276, -17388, -34324, -42716], [49369052, 51562, 20804, 39230], [11157350, -8672, -38004, 33290], [-88943646, -7939, 98534, -25099], [-151903332, 20983, -79319, 60312], [-289286606, -6142, -68962, -114996], [-399884118, 49562, 41750, 36136], [-27112240, -79510, 25479, -37042], [170255422, 284405, 232832, 195243], [-535364772, 58578, 746966, -134184], [-26037506, -30853, -524501, 25945], [-642590110, -133734, 642662, 28400], [-657660314, -244631, -703948, 137519], [-1554670476, -108768, -874832, -59912], [-808941666, 77655, -818507, 499238], [-321069404, -703368, -76016, -518764], [-3010667404, -589584, -775449, -1056968], [609215234, 77863, 1171484, 432613], [-1838027366, 656057, 2421775, 255007], [-4321488872, 236133, 995871, -23006], [-4526867408, -1527020, -1288380, -1829100], [-3352423436, 1458190, 1244069, 943692], [-5740706004, 876364, -2927180, 2652828], [-5263001362, 968527, 6673527, 1637581], [12796187208, -519880, 907584, 1970672], [-11773066226, -1469112, -4300116, 1887198], [-7822634760, 1026144, -12046995, 2152982], [1541315766, -1491839, 7357264, -815105], [-14863339956, -628892, -3677884, -919356], [17215626422, -3364325, -18158210, -6008273], [-9308273476, 4510461, -12360605, -3383224], [-18181098298, 1091619, -18333745, -5161759], [-14895570406, 4978754, 17342838, 3311552], [32736926176, -4117776, -13897884, -7634140], [-7693345618, 3680318, -34924842, -5522898], [-32040297668, -2520605, -869259, 7415644], [-48202795054, -4258160, -20226504, -2008632], [18782101576, 8269136, -25615728, 11547756], [-20205809792, 8036534, 27907247, 6062110], [30349120982, -12490653, -47149762, 2698351], [-82524342242, 2180711, -28516801, 6375269], [-82877951566, 19302232, 16817640, 15582324], [-66182437364, 12844904, 11774960, -893704], [31967513984, 36782036, 5947581, 9935218], [-36439162644, 34611140, -20977348, 10543380], [10324291056, -32146212, 41808944, -11521176], [-160279668362, 14466960, 50767908, -10064538], [-48234532024, -11138798, 78669575, 5219306], [-177830071988, -39497214, 27418126, 26072792], [-2919707090, 46048, 64652300, 24268138], [-118158048252, 656239, -51808095, -27268072], [-51178665494, 32020373, -41976937, -18192725], [94501698050, 48023466, 81720910, 35820008], [153065080396, -9030642, 36501312, -41549226], [18020703250, -2024557, -144131188, -19016915], [-162717234806, 56740265, 51768275, 7616215], [-269564962144, -46243001, -27408155, -119981566], [-210757154840, -88479908, 46091348, -11300592], [-348248629548, -45765606, -5897299, 24100252], [-74707052702, -30109343, -234866053, 49121883], [38682668742, -28479827, -21154033, -115813010], [-57251988334, 39854502, -61655222, 122602824], [17803910876, -7409351, 8243783, 44000968], [-438285689090, 41724043, -34727653, -51835531], [267765041484, 67540122, -26383836, -52978366], [590641522162, 20638468, -122750076, 1570780], [239415770382, 51332737, 200117860, 61960155], [-173606269836, -8504441, -61372503, 72739520], [287594166158, -58518485, -332051807, 174411786], [67910413682, 31333218, -58854050, -79303672], [983022049956, 67258586, 144626041, -28942772], [-264865861330, -43117011, -200542620, 16345443], [46774195180, -64325487, 408663807, -147184968], [-309928954220, -13658998, -332570684, 167309986], [735124610730, 240614553, 188136634, 149116353], [655946912964, 161709034, -201138050, -47543380], [305158990596, -151561336, 208908576, 52788600], [-1417430369796, -133944946, 4147608, -64023274], [1315983538424, -188657780, 93508685, -25374194], [-513642431090, 81267625, 950688955, 252021286], [83149092368, -216150361, 480933213, -248449646], [855357539654, -81044489, 480041583, -257535007], [625099554640, 77984163, -189784815, -160499058], [2214506395132, -278501684, -450458797, -146739860], [58971649812, 158258223, 384476025, 158290476], [-1276019231610, -96991683, 181762639, -593543598], [1710078249476, -128599190, -1139325930, 111284804], [2923508939830, -29994440, -248864436, -271492914], [854661933316, -472345618, -436407414, -10510372], [1021989280194, -121609894, 703277654, -145816992], [-1387573843636, 420203166, -1430727588, 288677266], [590190570286, -44568892, -1890836896, -52444642], [399092993462, 294891205, -111286544, 661576127], [-703377017638, -21936907, -323636609, -181209265], [430343765778, 321702756, -1341704556, 703247452], [2890734740284, 392008732, 845596084, -262176504], [-332562205478, 404242697, -1077023998, -320685407], [2162569142838, -685379721, 670437381, -114314442], [1502709366128, -429117273, 601595157, -94537250], [-4715151538044, -594440878, -126023196, 197024830], [3532026295340, -648622358, -691262379, -524769480], [-901008996196, 103110318, 1295150970, 325223236], [-5446545952238, 310807733, -686118237, -22405597], [4613608104258, -466964696, 1537177856, 35524892], [1036008063756, -445628104, 1022583867, 664354652], [-909974890140, -803181567, 1920390095, -677882584], [1142415225080, -1305381323, 52838223, -961035030], [1596269610072, -107953350, 2440463815, -996910966], [-4862393482890, 201337079, 3082392907, 777105681], [2705860062380, 305944062, -469989708, 189840166], [3783645081186, -525156285, -955849464, 1215393477], [-1158254214226, 57657915, -4143484921, 698601693], [6022490720908, 202625554, -1550497346, 218360420], [-996701280168, -286673656, 3422733292, -1929470136], [5654448492610, 280872589, 3183127190, -326462747], [-10358596784458, 815356363, 1407876231, 574322161], [-5309808964906, -179163369, 1137347925, 753320594], [6155037219870, 332622372, -1052916848, 451553766], [4883735309976, 866412934, -334299097, -988428686], [8611681927904, 605530893, -3653642825, -108558746], [2340711569004, -560883464, 2999458551, 1479708596], [-2790002375346, 1386298511, -3176883403, 363130494], [-4206013138520, -788027987, -1296974105, 933141950], [-5114035203464, -383211806, 1470191299, 1525918306], [7312790389378, -1162033487, 1139674358, -2409546563], [-6187739347636, 1695940894, 9542476890, 465728584], [-11492889307810, 739203491, 2042912979, -1103728655], [-1458742047724, 1230453886, 3533428004, 2716961930], [-12005881165822, 728226301, -5781571757, -152020189], [-5254081343370, 1882396365, 6064152439, 1159316650], [3323541069032, 2069602747, 7138694337, -1497758690], [-4856187622728, 65202480, -142702136, -343043424], [-1917868481078, 2152775605, 54671487, -1028507197], [-6742213963374, 1934360972, -4996032260, 2702817664], [-4322382699528, -1593929967, -5775035733, -1404045782], [-4117928429816, -1798451624, -6805534936, -22885212], [-2827372970088, -1065049817, 7247893365, 1391633998], [11965308271264, 1486869636, -4875415976, 2754984612], [19159410003602, -1455477351, -1480822338, -485622179], [-7829771275434, 3779970801, 1548962707, 3732915550], [4534882661402, -2598883400, -260176216, 937755792], [4805621525220, 1185227758, -9654605875, 716547228], [-6437378507796, 2967857900, -6386739420, -446509828], [-7441183333762, 4460252841, 8790787163, 3702676822], [18616115994502, -1141881805, 6050997626, -2655395249], [5396930797300, 321930533, -17616131333, -5758475376], [-14816430995706, -1728497975, 8019778123, -2266706226], [-1252413615976, 4607573648, -8530376072, -210158564], [17026166600866, -1132556439, -1446832342, 1043838261], [5903825706258, 2319091893, 14248202703, -4962626865], [-6739785607478, -5194479778, 6841789010, -5547913016], [-5481528136052, -489206778, 3276035661, -522954824], [2417610482036, 3034769622, -2043644814, 7567771516], [-23732046741094, 1801720677, -5114249125, 2311098279], [-11070090533256, -4188885548, -10944894848, -487816048], [28417163233114, 1196552592, 4250173696, -3867897184], [17656135084826, 441237197, -2320166374, 3056274153], [-7955907916478, -11561329832, -3292023704, -8044934596], [25711724486924, -2061268264, 20081830000, 1176532728], [14873105622132, -3546405264, -164516809, 3292574592], [12801218783118, -2884263999, 16088040132, 4924913491], [-21387124691374, 9374347109, 15304976731, 7340379079], [-9050136331438, -1284554187, -3709642786, -9147875779], [-24415310568276, -1236170789, 12904793573, -8618461976], [1699300175598, 387126323, 3343537631, 8073126201], [25035403096258, -3424987248, -27469899232, -3598160236], [23126291482072, -4165771244, -28911003684, -6683484296], [45645536196140, 7195760906, 8802751937, 2375481688], [-10000291508546, 10206978687, -6796572789, 3209711781], [16981606696324, -7560441244, 7495416420, -8947278064], [36594454800534, -4043837098, 6299779470, -6673227706], [-6478333468320, -3171821300, -23552551740, 10387163608], [26039760149682, 7332863524, -8954974644, 12862031424], [11151669709092, -17691666232, -1796948504, -15294035620], [-11311506407324, -15308549200, 2712390664, -13801616968], [31299282427596, 7415418894, 591364137, 2865373144], [-2992562263388, 1109727184, -4800255760, 13896004720], [26766455896562, 737522588, -23322480340, -8672117748], [-195291638010, 1510448304, 10153484068, 5218786638], [23190773210574, 752257203, 3666155879, -5299396307], [-29164824050102, 1017714764, 1402353204, -5452336516], [-54642357270608, 510090584, -27586792452, -4845435168], [8651330542212, -4741232796, -3954804593, -12200248864], [-10856803655554, 6403642837, 18447017860, -8704746469], [30523021028790, -14657020941, -18221407861, -13431928451], [-15725385825938, -3494529512, -48917634732, -5107898110], [8122429979484, -15942774527, 371556903, -390319108], [6596609620440, -5882767956, 20478899621, 19636545902], [-87781677215874, 13610192887, -47244797850, 13536085675], [19667129878980, 14456522963, -13559743563, 6345288524], [10296937967610, -6945435570, -23498282878, 9070594480], [-46923749024536, -13543416585, -5563811059, -11739681146], [7757459295214, 259393060, 34277772224, -16558905130], [25468908878556, -22211605400, 21103394827, -9855365452], [15391052991838, 1606566717, -11487838272, 23845916299], [36264382572020, -311566224, 20097723776, -19920629060], [-12783506192796, 7177463030, 41223818300, 15183359322], [58054185209050, -8122663172, 637939276, 9488159148], [-49821068261510, -18163522709, -19672874624, 9732407897], [11431561915958, -4015054553, 18234975085, 2385294830], [-1401634274592, 19674112256, -14143019912, 9104390516], [-14502024490140, 924457780, 1553030703, -22378050040], [12732315382870, 13294487843, -32338942653, -7553768963], [99305484727302, 29446014603, -11919874558, -1461969865], [-1745857944898, -18619813655, -52141476565, -5888182390], [-1051514882138, 6296366331, -87578922466, -15564699401], [-39602474537924, -3640854993, 37819193889, -1545850920], [101075327643378, -7829116599, 3324520959, 13700408707], [-56765833724606, -31035107762, 19937372522, -3261486796], [-16160411097516, -5391359914, -15121732760, -23197492662], [402177144398, -13237313637, 15366429598, 10056929035], [55947145388052, -25561698198, 14800725758, -4791500316], [-63156384675490, -15089316129, -31254316617, -8196826015], [27554858397872, -11003869365, 92215308121, -2917499498], [-57620713718770, 18045366461, -82863875108, 6226040275], [21303827880610, 18230099644, -64236249276, -10879991884], [-23803792216, 34143970383, 64387440917, 18310906262], [156169272917412, -18689768850, -30720188968, -3565311926], [22029231768714, 30752690022, 62313779034, 42910207808], [948383291012, -7954624556, -57047438525, -34000980256], [108685491720548, 5001134829, 25158715075, 26956254128], [-38932217969452, -14372114582, -27891709044, 8734345230], [-65064355533804, -3710646681, 27049031953, -4814695268], [6282991100044, 35453108114, -57261903146, -1861524720], [49957825008274, 7325724979, -73176445744, -2340649099], [4570443823514, 11503290481, -114651980129, -7745201673], [127769185403354, -4050266680, 1223502192, -12576308532], [-95685994541628, 4989381514, -133491818480, 12544732898], [-149294543679222, 24299954341, 150243138370, 12380995633], [-108663717741980, 38352271474, -66036718634, 48374390192], [-215886225315172, -15025715446, 9613166190, -41975666592], [338496443190280, -13497167680, -71623870468, -10171060348], [107564017212462, 10965433787, -7100839109, -20171764519], [11441831956434, -19150975094, 93148815190, -6764567312], [126712741814868, 41119246530, 2646433740, -4857319114], [-222495091469302, 18255716444, 80474586708, -6125228400], [-54003461214880, -30432531568, -35743214683, 20391830010], [10670220707700, 79972240378, -49933721858, 30567650048], [-24910159082474, 16457859911, -46960111065, -23230273427], [-384872049479500, -20468015198, -19416685098, -2551693380], [-114784586558984, 38451437716, -34002320519, 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