# q-expansion of newform 392.4.a.k, downloaded from the LMFDB on 19 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 = 4
poly_data = [13, -10, -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], 1], [[-1, 4, 0], 1], [[-15, 2, 2], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0], [0, 0, -1], [4, 1, 0], [0, 0, 0], [-5, 1, -3], [25, 1, 6], [30, -3, -10], [-21, -7, 11], [60, -6, -17], [-61, 7, -14], [197, 9, 3], [89, -16, 34], [247, -17, 26], [-92, -8, 24], [31, 27, -19], [71, 10, 26], [148, -40, 51], [165, -48, -2], [-186, 50, 11], [70, 42, 28], [581, 14, 56], [428, -58, 83], [-458, 22, 28], [551, 52, -44], [877, -31, -2], [32, -27, -140], [-154, 0, 59], [123, 21, 63], [842, -33, -76], [37, -23, 62], [-286, -154, -28], [-1094, 10, -59], [1274, 157, -86], [-894, 130, 132], [-1108, 1, -24], [-251, -91, 175], [-823, 76, -118], [-2076, 24, 201], [-1067, 99, -34], [467, 10, 58], [727, 25, 233], [802, 36, 320], [711, 59, 117], [1188, -161, 434], [-65, -349, 74], [-149, -193, 195], [450, -198, 244], [3202, -106, -108], [-2323, 275, -275], [78, 247, -404], [-632, -209, 438], [-2201, 73, -422], [307, -156, 80], [-1441, 37, -30], [-446, -143, -26], [-650, -52, 765], [1337, -28, -366], [5042, 88, -33], [1483, 246, -318], [689, 85, -906], [-1, -331, -147], [-1218, 336, -240], [5108, -256, 328], [5284, -338, -175], [1161, 82, 240], [4659, 66, -30], [-5710, -42, 273], [-3021, -373, -78], [-476, -568, -109], [2187, -225, -798], [4428, -41, 98], [6321, -147, 693], [10638, 136, 49], [-1453, -850, 534], [-2635, 303, -90], [1921, 41, 425], [-6272, -699, -108], [798, -581, -712], [-606, -375, -1458], [-2754, 1, -438], [-7260, -16, -1032], [-6059, -195, -626], [-593, 687, -1697], [4441, 69, 6], [1685, -767, 1237], [-787, 643, -501], [3995, -589, -1054], [-1943, 522, -1160], [-225, 747, 1066], [-2032, -336, -336], [-4732, 560, 1969], [-5576, -274, 13], [2498, -240, -43], [7421, -41, -1130], [-9980, 592, -1615], [-5414, -1098, -508], [2408, 665, -1604], [1799, 1036, -1356], [9783, 505, 487], [-8664, -679, 924], [13047, 1029, -574], [386, 231, -28], [-14090, -278, -621], [-2951, -590, 384], [7407, -251, -2355], [-881, -432, 2656], [-17896, 580, -688], [16692, -713, 1546], [-1124, 422, 785], [9733, -239, 894], [1733, -699, -2751], [8966, -353, 1416], [5407, 1815, 218], [-7586, 1278, 1071], [4182, -438, 2812], [1515, 996, -1448], [4220, 488, -2504], [484, 870, -1559], [-10609, 1406, -494], [-12010, -266, 700], [-5695, -1632, -1290], [2195, 11, 2018], [6586, 1307, -1308], [-6207, 1923, -1415], [6550, -186, -1739], [16450, -124, -2008], [14441, 1264, -1230], [-16758, 896, -1745], [6650, 182, 3172], [18007, -386, 2074], [7169, 959, 1701], [4282, 1078, 84], [-4549, -49, 133], [-21305, 315, -2198], [6478, -1807, 3298], [15863, 471, -1782], [21028, 1729, -1108], [7716, -1900, -785], [12015, 27, 1306], [-7171, 1110, -1272], [1032, -2780, 1504], [3066, -685, -1620], [-20788, -218, 801], [-8096, -692, -2432], [2590, -413, 228], [4568, 400, -464], [1185, -367, 3878], [9877, 518, 32], [8093, 3411, 523], [15468, 2014, -197], [-3076, -1379, 2716], [1406, -2900, 720], [19454, 1022, 1596], [-10345, -873, -1047], [3112, 2620, -3261], [-13472, 800, 1968], [581, 1857, 1541], [-3123, -590, -1660], [-8962, 332, -3416], [21615, -2130, -4122], [-10498, -802, -2459], [9189, -2799, 5982], [-13382, -3242, -1292], [-6334, -3386, 1009], [-17580, 215, 2778], [-4527, -423, 4503], [-1620, 602, -539], [-4055, 3552, -3186], [-6039, 2933, 3206], [6201, -72, -5798], [6128, -3140, -1829], [-25695, 25, 2886], [-27332, -478, 1063], [5819, 840, 476], [-33087, -59, -733], [-5669, 3131, 2194], [21007, 1845, 1570], [-10637, -1626, 2666], [-10430, -658, -1788], [-27188, -371, 1852], [-31244, -714, 427], [-7030, -1502, 3012], [24890, -708, -1040], [14927, 2596, -308], [14046, -1432, -2655], [-36463, 1120, -3194], [-38358, 855, 1516], [33511, -255, 1697], [24800, -3145, -1114], [-30437, 859, -2209], [-5505, -4864, 4480], [-34338, 790, -6276], [-4505, 2829, 963], [22526, -1421, 2912], [-4621, 2101, -2327], [-19988, -2197, 742], [-46160, 2203, 1602], [-22474, 2971, 2672], [71, 807, -5814], [32534, -368, -5573], [21496, -207, 5080], [-36254, 1846, -7772], [-27057, 2082, 5198], [-7790, 401, -234], [10734, 1574, -4356], [32685, -920, 1602], [-24088, 3442, -6285], [31693, -1817, 4583], [-24370, 4156, -792], [11784, -3396, -2045], [24240, 1211, -4494], [41535, 1107, -4374], [25892, 2676, -4360], [-33020, -2620, 3037], [-23235, 4245, 4679], [-13070, 4933, 4834], [-42597, -2925, -745], [59770, -1567, 3774], [-26585, -2233, 5705], [65558, -1600, 5584], [33228, 4121, 944], [-40671, 775, 5574], [-21783, 358, -10020], [24811, -1681, -10091], [-13486, -6742, 52], [-26867, 965, 9166], [19466, 1909, 3850], [55710, 2624, 633], [-38637, 307, 3033], [39546, -3650, 625], [-10319, -1508, -5038], [-1572, -3396, -2561], [59292, -3052, 6104], [17102, 6041, 1106], [19804, 5280, -5368], [-24828, 2522, -12787], [-20436, -2785, 890], [-40442, -4985, 3636], [-59712, 1644, 1760], [34683, 7325, -830], [-26542, 4110, -14213], [7527, 5009, 383], [47502, -3052, 8845], [37273, -3260, 4110], [31918, -1516, 10616], [-57416, -130, 3511], [-53630, 1366, -4812], [9161, 1319, 4453], [-68003, -1641, 3537], [12493, -2845, -4898], [-20171, -2139, 1790], [101411, 1180, -776], [15503, 2699, -8181], [-12295, -4563, 4694], [14275, 1754, 3518], [73621, 1655, 383], [14277, -3908, 5102], [12671, -1755, -1699], [13821, -1387, -642], [-53218, -2591, 1730], [-48462, 4300, 6385], [35984, -2252, -2288], [-56554, -1297, -4600], [-32205, 4295, -7518], [17150, -6139, -1954], [-50431, -1021, 4719], [13101, 989, 1646], [35863, -4385, -3878], [47808, 828, 1520], [44244, 545, -9412], [52072, -303, -7932], [705, -2505, -7171], [-50830, 3165, 9458], [27477, -107, 14209], [-6227, 4913, -5346], [-4634, -3752, -5885], [-19314, -9022, -591], [-84695, 1068, 394], [-29051, -306, -3744], [-48421, 2089, -7381], [28311, 3135, -13049], [59061, 5389, -4505], [31404, 9244, -6920], [-27633, 1091, 6746], [14947, -115, 1903], [-73475, 5557, -12471], [22755, 11732, -1456], [-13809, -4349, -9654], [-50022, 3248, -547], [36125, 4721, -5538], [12292, 7157, -5616], [-10441, -4103, 2089], [47243, 7040, -7876], [-82507, -1697, -3194], [22181, -1867, -12242], [-15098, 7947, -11100], [-14921, -981, 7381], [-41643, -2835, 10254], [-40382, -3218, 6761], [-13381, 2337, 11226], [-50759, -4582, -296], [-75228, -683, -1108], [-1653, 5939, -4153], [8898, -9222, 5140], [10665, 5255, -8205], [62303, -2497, -11094], [82254, -4274, 15951], [-5097, -4685, 3578], [-68377, -9689, 2985], [-48692, -4679, -2672], [41604, 6222, 4323], [-35815, -3592, -4778], [59910, -3619, 18158], [9096, 568, 3343], [-13088, 8696, -10800], [12400, 4647, 6814], [-71626, 710, -1460], [-11425, -9113, 5191], [11779, 11595, -10190], [-29355, -5047, 3822], [-69239, 1711, 6861], [48904, -5433, 1662], [-25109, -609, 11458], [-52921, 9209, 506], [65643, -2792, 13276], [74873, -82, 6628], [43200, -2836, 801], [50450, -5010, 8761], [6150, -9012, 12847], [100973, -987, -6146], [-165, 2882, -2486], [-31884, 10045, -13272], [-14906, -1518, -1108], [10774, -5486, -2149], [10175, -2371, -5963], [5784, 6388, -12816], [61277, -3587, 11342], [-10062, -783, 10998], [294, -5180, 4280], [4270, 4396, 10967], [-26018, -1937, -1728], [-101599, 1141, 4774], [-44667, -7875, 7710], [36022, -7446, 1772], [104045, 4328, 974], [34038, 7654, 10204], [-7359, -15320, 14718], [-18695, 9695, 3339], [90321, -5761, -4842], [38355, 5758, 3974], [-20428, 808, -2039], [-71254, 4573, -4038], [-94757, 5967, -1038], [93048, -8826, 14949], [-25597, 8622, 18150], [-83106, -284, -15904], [116624, -2806, 3875], [-13335, -4053, -9319], [-16606, -3815, 3038], [-118676, -2244, 6872], [-36585, 603, -18742], [-116820, -85, -6258], [11053, -3543, -6661], [49004, -10164, 20349], [-27326, -2922, 5139], [-8666, -5023, -2998], [9349, -475, 9566], [-80813, 2487, 9843], [-91578, -996, 12592], [55480, -2488, -10001], [-43271, 2239, -15445], [60803, -4009, 6021], [-127638, -1444, 3184], [-45975, -3967, -18233], [-25253, -4214, -10430], [-29013, 4533, -5487], [91424, 3760, -7472], [-4380, -2145, -21710], [48173, 4954, -13412], [111533, -2551, 2766], [-36868, -1643, 21496], [-44833, -761, -1014], [58752, -5400, 9648], [-45551, -4930, -14260], [81272, 10924, -3232], [164067, 6519, -2013], [3430, 2880, -26287], [28497, 18816, -15034], [-97468, -1900, 1325], [49696, -12720, 31376], [30470, -7764, -12856], [-217206, 3515, 8404], [-50064, 11940, 12963], [46724, -240, -3407], [-58629, 7891, 15009], [120805, -6730, 20064], [26440, -9266, 14909], [-11701, 6971, 10354], [-95168, -4740, -2048], [80950, -6569, 9648], [37827, -6302, 20114], [-243762, 357, 1806], [174217, 5225, -8367], [-8861, 12168, -4340], [-47634, -12409, 1960], [3084, 4306, 2311], [205495, 182, 1638], [-13053, 1227, 834], [20113, 8103, 20317], [-98630, 6650, -22123], [-6271, -15930, 9684], [-32865, 3714, -23714], [217576, 2564, 7120], [-161215, -5357, 10079], [95175, 4751, -8982], [29617, -9380, -18162], [-120843, 11893, 12033], [-119073, -5333, 10235], [-82961, -12097, 13514], [74035, 4283, -15023], [83897, -14372, 29802], [70177, -3720, 1858], [-208624, -8096, -2592], [-225618, -170, -4708], [40959, -8361, -32390], [-106689, -14914, 14726], [134663, -2547, -829], [-54467, 6713, -24178], [-113422, 12514, 2431], [79673, -1535, 28247]]