# q-expansion of newform 392.6.a.g, downloaded from the LMFDB on 01 May 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 = 392
weight = 6
poly_data = [144, 0, -110, 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], [[0, 2, 0, 0], 1], [[0, -214, 0, 2], 3], [[-220, 0, 4, 0], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0], [-88, 0, 0, 3], [0, -12, 11, 0], [0, 24, 22, 0], [0, 75, -28, 0], [492, 0, 0, -9], [1010, 0, 0, 0], [0, 438, 88, 0], [2626, 0, 0, 18], [0, -696, 170, 0], [7184, 0, 0, -27], [0, 594, -64, 0], [-7074, 0, 0, 114], [0, 1755, -220, 0], [0, 1128, 365, 0], [-18144, 0, 0, -123], [44184, 0, 0, 126], [0, -4032, 0, 0], [33992, 0, 0, -24], [0, 3171, 1432, 0], [0, -3240, -312, 0], [0, -192, -334, 0], [0, -6972, 1353, 0], [0, -6438, 336, 0], [56904, 0, 0, -147], [-19126, 0, 0, 174], [117790, 0, 0, 513], [252596, 0, 0, 63], [0, 2595, 2488, 0], [16022, 0, 0, 240], [0, -7551, 1220, 0], [371606, 0, 0, -324], [-243596, 0, 0, -483], [0, 24156, -2081, 0], [-170056, 0, 0, 1569], [0, -22290, -3104, 0], [0, 26844, 6275, 0], [-352208, 0, 0, 705], [0, -4224, 291, 0], [-387264, 0, 0, -678], [392566, 0, 0, 399], [-262274, 0, 0, 342], [0, -24450, -1800, 0], [-177904, 0, 0, 3285], [0, -24396, -7480, 0], [0, -51105, 14004, 0], [0, 21132, -3813, 0], [97094, 0, 0, 4044], [52348, 0, 0, -6723], [0, -85032, 8578, 0], [0, 333, 16808, 0], [0, -75384, -360, 0], [-297544, 0, 0, -5238], [0, -26892, -14777, 0], [0, -20952, -7752, 0], [1263974, 0, 0, 2208], [-1475082, 0, 0, 2148], [0, -7833, -25624, 0], [0, -99396, -2891, 0], [0, -23133, 18240, 0], [0, 25200, -8600, 0], [0, -34392, 12738, 0], [-403098, 0, 0, -846], [-1546016, 0, 0, -1491], [-1444462, 0, 0, -2469], [-296144, 0, 0, 8817], [0, -155736, -6465, 0], [0, -128184, -10820, 0], [-501396, 0, 0, 6615], [0, 53340, 24912, 0], [358550, 0, 0, 18780], [-1001968, 0, 0, -18363], [0, 38466, -38984, 0], [-1039582, 0, 0, 13170], [0, 208032, 78099, 0], [139050, 0, 0, -20523], [0, -201792, 22794, 0], [0, -142479, -84848, 0], [870534, 0, 0, 16260], [-1873596, 0, 0, 8655], [0, 28800, 31538, 0], [0, -104544, -66336, 0], [1161664, 0, 0, -8769], [3759554, 0, 0, 8016], [2179950, 0, 0, 16659], [0, 275880, -95453, 0], [4559728, 0, 0, -8526], [0, -95817, -38052, 0], [0, -332394, 57104, 0], [786876, 0, 0, 22779], [-405344, 0, 0, -393], [6147016, 0, 0, 5907], [0, -497916, 47968, 0], [0, -99480, -58877, 0], [0, -408, 35686, 0], [0, -146421, -30596, 0], [3513710, 0, 0, -36960], [-4723048, 0, 0, 7161], [-4394106, 0, 0, -35826], [0, 208437, 51468, 0], [-5393454, 0, 0, 25617], [9063088, 0, 0, 13905], [0, 298680, -28332, 0], [0, -12603, 2124, 0], [0, 279864, -191932, 0], [-2657632, 0, 0, -35904], [0, 278856, -66224, 0], [0, 447216, -105160, 0], [-812262, 0, 0, 56964], [4176330, 0, 0, 2343], [0, 378987, -126944, 0], [3525152, 0, 0, -54708], [-2963350, 0, 0, 3849], [0, -925803, 115716, 0], [0, 668190, -197928, 0], [-13136510, 0, 0, 22584], [-15456600, 0, 0, -12747], [0, -557604, -262663, 0], [-5581794, 0, 0, -30048], [0, -40296, -124917, 0], [6060296, 0, 0, -20523], [0, -682509, 75400, 0], [444130, 0, 0, -31740], [2614994, 0, 0, -1170], [0, -390018, -154168, 0], [0, 418122, -259280, 0], [0, -1278432, 147993, 0], [18251160, 0, 0, 22785], [4670636, 0, 0, -11445], [11185764, 0, 0, 87843], [-15754694, 0, 0, -52920], [0, 1237680, 118838, 0], [0, 413400, 111966, 0], [0, -432696, -203707, 0], [0, 616911, 248676, 0], [0, 1383612, 268551, 0], [696510, 0, 0, 24183], [0, 1364547, 378472, 0], [-11899530, 0, 0, 49572], [12023864, 0, 0, -70038], [-23456104, 0, 0, 26025], [0, -307584, 317989, 0], [0, -1119198, 295096, 0], [0, 204708, -337125, 0], [0, 525192, 252658, 0], [0, 1226625, -265536, 0], [-2579520, 0, 0, -40134], [26875514, 0, 0, -16758], [0, 1340712, 33144, 0], [-17074280, 0, 0, -88221], [0, 128970, 497968, 0], [22453808, 0, 0, -46425], [26875916, 0, 0, 65721], [32007880, 0, 0, -56382], [0, 729600, 225566, 0], [0, 931560, -513872, 0], [0, 851052, -172519, 0], [-31997112, 0, 0, -27075], [16257978, 0, 0, 31800], [34758540, 0, 0, 70995], [0, 2025915, 19084, 0], [8707662, 0, 0, 84456], [0, 1628058, -185240, 0], [-2993128, 0, 0, 104076], [0, -492456, 54613, 0], [-1695482, 0, 0, -225897], [0, -2142804, -53153, 0], [-39797712, 0, 0, -104613], [0, 2040720, 460231, 0], [10141912, 0, 0, -47190], [25885910, 0, 0, -105504], [0, -896562, 275696, 0], [0, 2337504, -535660, 0], [-3902792, 0, 0, 16929], [7782426, 0, 0, -186108], [0, -2917020, -624792, 0], [0, -154380, 84189, 0], [48799892, 0, 0, 3843], [0, 1319937, 136148, 0], [-12433854, 0, 0, 268290], [0, -2833656, 599948, 0], [-44594948, 0, 0, -97935], [0, -1253292, -417641, 0], [23819190, 0, 0, 110358], [0, 2486691, 428492, 0], [69032022, 0, 0, -24], [0, -3432390, 803680, 0], [0, -2593584, -68228, 0], [-1584408, 0, 0, -24471], [8572760, 0, 0, -123825], [0, 3765132, 824635, 0], [-27874296, 0, 0, 98619], [0, -2144352, -339260, 0], [36428898, 0, 0, -26865], [27289474, 0, 0, 160140], [0, 2010072, -698816, 0], [0, -2827464, 76184, 0], [2998662, 0, 0, -146130], [0, 4447314, -1030128, 0], [0, 3692736, -49431, 0], [0, -4237608, 688944, 0], [0, 1770171, -1077724, 0], [0, 34980, -275155, 0], [0, 3649866, 447760, 0], [-41935008, 0, 0, 283431], [-11049882, 0, 0, -215787], [0, -3720135, -116720, 0], [2800922, 0, 0, 78393], [0, -5238984, 17357, 0], [21995540, 0, 0, -101523], [0, 5153607, -952016, 0], [0, 4605024, 1005808, 0], [0, 1659600, 1512278, 0], [64485296, 0, 0, -108042], [0, 2543400, 740406, 0], [-3803056, 0, 0, 465948], [34551042, 0, 0, 58524], [-17699350, 0, 0, 360504], [0, -5688624, 1250752, 0], [-77946334, 0, 0, -118680], [17503484, 0, 0, 33357], [0, -5315253, -1405940, 0], [-108974150, 0, 0, -31800], [0, -1786176, -1201552, 0], [-60118548, 0, 0, 257877], [0, 3855900, -245736, 0], [-1558456, 0, 0, -669033], [54258866, 0, 0, -268956], [0, -895875, 217040, 0], [-37466560, 0, 0, -235158], [13129386, 0, 0, 55440], [0, 2543763, 1355208, 0], [0, -7187280, 446632, 0], [0, -7869264, -212442, 0], [-115011946, 0, 0, -219588], [-116062848, 0, 0, 13875], [0, 2531292, -1668368, 0], [6777648, 0, 0, 120459], [0, 848667, 1422516, 0], [0, -5252076, -1751512, 0], [-27253578, 0, 0, 410826], [0, -230448, 259992, 0], [0, 846438, 1139344, 0], [0, -695304, -1259352, 0], [0, -4250457, 746920, 0], [25212952, 0, 0, 129351], [-21930888, 0, 0, -218208], [46895038, 0, 0, -536868], [0, -3720336, 1748772, 0], [-19175184, 0, 0, 484608], [0, 2293920, 1571616, 0], [0, 778704, 231365, 0], [70046032, 0, 0, 273141], [-64888070, 0, 0, -188712], [0, -2935011, 362100, 0], [0, 2123040, -1254243, 0], [0, -2052432, -1470414, 0], [101987676, 0, 0, 193101], [-24103872, 0, 0, -148353], [0, -2066448, -178203, 0], [0, -6366864, -390143, 0], [0, -997464, 1516578, 0], [0, 406437, 1110012, 0], [68831838, 0, 0, 690408], [0, -10196256, 505640, 0], [164016376, 0, 0, -145371], [49918386, 0, 0, -66258], [0, 9068028, -2095445, 0], [14026944, 0, 0, -683085], [0, 7689576, 2233940, 0], [24137668, 0, 0, 224907], [0, -2548200, -53904, 0], [0, -881364, 143024, 0], [96449464, 0, 0, 22587], [-88717370, 0, 0, -477486], [105467110, 0, 0, 15333], [0, -6346713, 948944, 0], [0, -2706780, -1798400, 0], [12749624, 0, 0, 603306], [0, -3138078, -798152, 0], [0, -10481124, -3126267, 0], [0, 1790163, -2675592, 0], [-2905140, 0, 0, 267717], [90364658, 0, 0, 159768], [92905310, 0, 0, -428118], [0, -2054478, 2513112, 0], [0, -6567048, -3060220, 0], [205309110, 0, 0, -103710], [0, 11176575, -382328, 0], [-5012534, 0, 0, 339651], [0, 5503719, 2745920, 0], [141129022, 0, 0, -426288], [0, -8456352, -1418861, 0], [0, -7478280, -227928, 0], [0, 9964404, -3112249, 0], [0, -5151639, -1860416, 0], [0, -377499, -1807544, 0], [0, -5883912, 24874, 0], [75307146, 0, 0, -612282], [-118608096, 0, 0, -286074], [193563632, 0, 0, -401433], [-132731768, 0, 0, 255399], [-9947682, 0, 0, 1368624], [-213486048, 0, 0, -328011], [0, 6592416, 4373025, 0], [-34275916, 0, 0, 571125], [-63448794, 0, 0, 1308012], [0, -4003428, 292368, 0], [718690, 0, 0, 754254], [16739262, 0, 0, -694680], [27857872, 0, 0, 9327], [-172796880, 0, 0, -649560], [0, 8917920, 1125024, 0], [0, 11896557, -3955772, 0], [28829904, 0, 0, -196998], [0, 7339272, 966152, 0], [-30914034, 0, 0, -372456], [0, -7791603, 313872, 0], [8574086, 0, 0, -460080], [0, -2510940, 2291317, 0], [107991864, 0, 0, 722172], [258693838, 0, 0, 645459], [0, 4825944, 2843850, 0], [-118105864, 0, 0, -741123], [0, 12353871, 3369648, 0], [-196844356, 0, 0, 138921], [-153960550, 0, 0, 1005564], [-109328586, 0, 0, 566346], [374243426, 0, 0, 169932], [0, -22309038, 1848424, 0], [0, 2960349, -1127948, 0], [-35462008, 0, 0, 874677], [0, -7514061, 140380, 0], [-229640214, 0, 0, 568242], [0, 4302168, -1985668, 0], [0, -14388072, 1578780, 0], [0, 1678482, -2107056, 0], [61980094, 0, 0, -1014630], [-163907038, 0, 0, 354669], [0, -4506816, -4347077, 0], [-149101168, 0, 0, -835452], [-267303390, 0, 0, 723012], [-129525520, 0, 0, -819129], [0, 23307672, -91113, 0], [-201984800, 0, 0, 107373], [0, -7466148, -4435144, 0], [0, -3446064, 198069, 0], [0, -12151887, 2565608, 0], [-149061878, 0, 0, -937224], [237555318, 0, 0, 1419954], [0, 11971410, 4708720, 0], [79513646, 0, 0, -767994], [0, 18726456, -4206820, 0], [0, -3098874, -1191784, 0], [0, -7542729, -1751708, 0], [-79887630, 0, 0, 1622208], [-195436724, 0, 0, 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-1030248], [0, -8719626, -4690248, 0], [0, 26720496, 9589982, 0], [436704744, 0, 0, 290025], [-85727766, 0, 0, -2309988], [0, 18726108, -1065675, 0], [-251787006, 0, 0, 792624], [-222844400, 0, 0, -276870], [0, -26542488, 3570506, 0], [0, -493332, -5233581, 0], [0, -6789990, 4724672, 0], [-252718034, 0, 0, 2000628], [538685094, 0, 0, -268305], [0, -7333959, 2772792, 0], [0, 28394637, 2407860, 0], [0, -3554304, 2310744, 0], [201590330, 0, 0, 1300368], [406798528, 0, 0, -1279707], [74559368, 0, 0, -1848381], [28853414, 0, 0, 990852], [0, 16392000, 7705215, 0], [734948392, 0, 0, -340284], [0, 12151530, -2315696, 0], [0, 8730336, -7172582, 0], [0, 11681604, -2299200, 0], [-356656070, 0, 0, 1733862], [0, 32313696, -1642651, 0], [-105887864, 0, 0, -1118148], [0, -42551403, 981056, 0], [0, 24519504, 7678106, 0], [0, 25295496, 1121441, 0], [180913728, 0, 0, 1525179], [-109609514, 0, 0, 2445996], [0, 18582429, 9094704, 0], [0, -31434258, -1759048, 0]]