# q-expansion of newform 2016.4.a.w, downloaded from the LMFDB on 14 June 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 = 2016
weight = 4
poly_data = [-3, -6, 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], 1], [[-8, 0, 2], 1], [[8, 4, -2], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0], [0, 0, 0], [2, 2, 1], [-7, 0, 0], [0, -3, -7], [-2, -8, 3], [22, 0, 14], [-56, -3, -14], [112, 11, -7], [-30, -4, -2], [-168, 16, 14], [6, 22, -24], [150, 14, 28], [0, 1, 21], [168, -62, -28], [26, -66, -54], [-168, -71, 56], [166, 2, -13], [336, -21, -7], [168, 58, -14], [-78, -104, 4], [-56, -144, -28], [-1008, -29, -42], [-82, -34, -38], [-838, 142, 92], [218, -228, -37], [-336, 76, -154], [-560, 189, -21], [-882, -50, -60], [-418, -123, 89], [224, -119, 7], [-336, 195, 210], [198, 228, -152], [-1848, 15, -182], [-406, 196, 168], [-1120, 275, 77], [-1042, -108, 23], [-1008, 281, 133], [896, 102, 28], [-2414, -64, 129], [0, 313, -161], [-2098, 382, 177], [-168, -206, 322], [-622, -163, -127], [-2822, -218, -130], [-2240, -544, 182], [-3360, 179, -63], [1232, 58, 210], [-728, 523, 406], [3782, 72, 295], [-3258, 308, -28], [-2016, 29, -217], [1130, 10, 544], [448, -211, -490], [1726, 868, 28], [-1064, -78, -294], [2050, 564, -299], [224, -14, -266], [-3050, -568, 304], [-1194, -880, -496], [-2464, -173, -112], [1498, 548, -433], [-784, -507, -770], [1960, 532, 140], [-4582, 590, 120], [706, 278, 230], [-2352, -615, 189], [-478, -471, 461], [-3920, -1421, 119], [-1562, 22, 669], [-882, -778, -214], [5040, 99, 581], [-3808, 738, -630], [-1834, 28, 504], [-5600, -999, -63], [-1176, -1146, -280], [5898, 558, 1140], [5126, -114, -99], [-2706, 385, 1001], [3402, 20, -410], [-5488, 947, -56], [1030, 604, 260], [8400, -225, 245], [-2118, 416, 838], [-4648, 762, 1022], [-2240, 1847, -315], [1134, -1484, -476], [650, -2343, -251], [394, 1570, 1359], [1400, -326, 462], [-840, -789, -1134], [8904, -1084, -490], [-4480, -99, 287], [672, 1583, 1645], [-5040, 1325, 679], [1624, -186, 1638], [-9190, 1414, -217], [-6650, -372, -1026], [4536, 1711, -1260], [-1122, 340, 2032], [560, 1809, 1897], [2178, -1370, -1658], [-6552, 1587, -560], [6022, -2187, -467], [-10192, 2901, 1869], [4450, 1262, -1462], [-10584, 2217, 546], [7486, -1688, 668], [-3640, 1396, 2772], [11538, -346, -894], [3360, 128, -1288], [9894, 940, 386], [9574, 787, -9], [7952, -873, -1260], [13720, -796, -56], [-8226, -167, -2439], [14392, 1633, 1092], [-3248, 2104, -1246], [10370, 1792, 910], [6944, -69, -301], [8358, -1308, 753], [-11454, -676, -688], [-6302, -1306, -2277], [-24864, -691, 623], [-14896, -1703, 490], [2402, 32, 1150], [-14714, 1394, -1004], [11368, -2750, -1792], [560, -1514, -3556], [-15018, 714, -3269], [5264, -1095, 805], [-10752, 647, -567], [-4032, 517, -1673], [-15434, 1320, -1314], [6262, 1674, 508], [-2598, -1442, -560], [-6718, -2194, 289], [-3192, 789, 1582], [-4350, -3632, -87], [-5226, -457, 2547], [672, -737, 840], [8442, 3336, -3008], [10696, 1766, 322], [2576, -3623, 1239], [-218, -2670, 1843], [19712, -876, -1526], [13958, 48, -2377], [5206, 608, -1698], [-6720, -753, -2506], [21784, -1202, 1694], [14190, 3266, -1580], [5294, 3694, -386], [112, -1115, -4221], [24416, 10, 4060], [32816, 593, -1729], [33488, 1209, 63], [8120, 3822, 5082], [-20762, 236, -78], [17874, 308, 1008], [-13950, -676, -821], [16576, 3859, 931], [-7274, 972, -2916], [-9744, -943, -6461], [-11368, -967, -2408], [10190, -4904, -268], [-10416, 2620, 994], [-4312, 636, -4956], [-3218, -738, -3253], [7362, 4081, -2499], [7354, -2596, 1901], [8400, 1219, 1225], [-35706, 1466, -499], [-15848, 1682, 2058], [14010, 380, 5132], [13272, 14, 112], [-23906, 1674, -5022], [-14112, 1367, 2793], [7642, -5084, 2750], [19432, -3552, -2100], [-26450, 308, 2429], [21280, -315, 5271], [23688, 2987, 1400], [-4426, 3778, 5816], [-23762, -2924, 540], [-14784, 205, 3787], [-11478, -4744, 2549], [15870, 1846, -3606], [35000, 2837, 2002], [2490, -8668, -2192], [-5208, 4040, -154], [12450, 3326, 2370], [-7392, 1953, 987], [-18480, -3599, 1071], [-16014, -1500, -239], [8288, -4629, 637], [26030, 3406, -3498], [-17694, 6885, 6897], [47710, -4512, -2746], [32238, -1754, 838], [-18424, 1970, 2870], [34626, 2302, 514], [14280, 896, -6846], [-59970, 2370, -481], [-766, -1658, 6094], [-11368, -1787, -168], [8386, 4628, -7157], [16968, 9932, 742], [-3920, 61, -5299], [-30090, 4649, 6101], [-26656, 2463, 2842], [31922, -1025, -2161], [13778, -3534, -647], [-7616, -5485, 1981], [52528, 4693, -448], [-7672, 798, 4578], [2602, 9386, 1760], [-19544, -2190, 7350], [32646, -6694, 608], [16296, 3228, 6972], [-40654, -6716, -2294], [32182, -852, -4738], [-49000, 6034, 4382], [-9234, 908, -7540], [7392, 8175, -2835], [-9016, 2727, -4046], [-36586, 7128, -2246], [-41234, 5890, -1178], [1792, 13473, -133], [-65800, -8754, 2870], [22400, -9695, -6475], [398, -5312, -6534], [-4592, -7351, 4116], [20664, -4622, 5726], [17558, 500, -5532], [-2352, 3379, -2548], [-5824, -8526, -3570], [-15686, -7392, -10598], [4762, -2552, 6116], [-10640, 1867, 3297], [-28392, -3616, 4788], [8736, -1047, 3549], [-29960, -1817, -868], [-49336, -4882, 1414], [66398, -1434, 3210], [27998, -6234, -2354], [-7840, -9228, -3626], [-560, 15056, 896], [-49952, -4713, 1414], [23184, -3393, 10143], [-12712, -5524, 7252], [30478, -8092, 3416], [-66242, 2726, 4282], [-56392, -1564, -2520], [-846, -8614, -8514], [-19270, -1526, 6601], [-19264, -2259, 6027], [26934, -3152, -7638], [15624, -1509, -4214], [-12958, 4438, 6013], [-46358, 9064, -550], [-560, -11901, 4249], [13440, -7003, 9877], [13390, -602, 7063], [-26026, 6238, 7655], [81734, -734, -3860], [63336, 7765, -3220], [18178, 6624, 708], [-57346, -15650, 5874], [-43456, -677, 357], [-15270, 2686, 2680], [-32514, 18388, 9019], [66080, 8159, 5187], [56002, 358, 7774], [-72800, -567, -10465], [-26750, -8952, -192], [-83944, 4848, 924], [-94640, 2373, -1043], [73150, 4722, 3978], [22730, 3743, 6579], [83328, 99, -490], [31136, 17146, -6958], [-30632, 1010, -6370], [42000, 13206, -6272], [32486, -14164, -12423], [-15344, -7785, -2968], [-33600, -1315, -8197], [8354, 508, -4436], [23658, 4634, 3346], [-74032, 8586, -1344], [-3746, 12614, -1974], [64226, -8150, -9794], [30576, -19137, -5166], [49046, -5873, -11109], [27328, 5369, 7784], [-49138, -12404, -5236], [29978, -146, -1697], [6272, 8530, 8470], [-102886, -4304, -9075], [26096, 11453, -7770], [-8624, -19407, -8400], [-86086, -188, 9062], [6306, -7690, 9224], [17528, 11654, 10794], [83664, 9781, 6293], [-23968, -7313, -497], [-43102, 8736, 6748], [35504, 12351, -1869], [-3082, 1766, -9477], [-85680, 16829, 7959], [-55130, 8260, -7420], [109984, 4082, 7434], [15402, -458, 8164], [51102, 8908, 7856], [-60592, -6285, 3675], [5208, -10764, -1792], [46786, -20050, -2486], [68488, -11737, 532], [-151368, 3966, -630], [-21614, 14438, 2158], [-63058, 5388, -11460], [-41664, 2825, 3836], [-16374, 22480, 14572], [-19486, -3216, -6977], [-12488, -18848, -924], [-76010, -2061, 4111], [-91734, -14456, 2502], [33936, 12749, 14917], [9520, -6185, 7000], [74480, -6815, -3073], [60938, -10792, -4626], [28798, -10762, -14698], [68498, 9548, 14266], [-6104, -1474, -6636], [166152, -2409, -2576], [-9744, -19013, -12187], [-103096, -12687, -3780], [55278, 1758, 340], [16974, 5254, 3754], [54450, 792, 13136], [75768, -9460, 1918], [54246, -7142, -13966], [-4314, -19667, -7331], [-139598, -5586, -8085], [112504, 1896, -8876], [79650, 15160, -7554], [20384, -2955, -17479], [74766, 7266, 17549], [46592, -1023, 6545], [49840, -6994, 4158], [-67166, 11854, 1321], [1680, -20499, -4060], [130138, -7520, -2808], [34770, -23242, -15646], [-44968, 6120, 266], [-80170, -9414, -4182], [-87282, 11290, -494], [4536, -14192, -20510], [-39368, 13969, 20048], [112894, -2992, 16032], [79968, 9451, 3745], [54934, 23496, 1668], [45822, -6880, -10804], [98448, 6763, 4669], [-11760, -16801, 18809], [2632, 13841, -8890], [2586, 9401, -13979], [42786, -5052, 4901], [-29008, -10255, -13125], [-179110, -6431, -5011], [74816, -11713, -14623], [-60648, -19737, 13370], [124712, -8124, 10332], [-5110, -8960, -10444], [13160, 1540, 20916], [14878, 22520, 18428], [-81704, 20671, 13762], [181776, 12797, -4137], [40018, -18722, -9410], [19846, -11746, 16576], [58266, 292, 1294], [-26742, -10306, 8791], [-83594, -9313, -19045], [77000, 12994, 5558], [-50658, 7712, 552], [-26152, 17337, 4732], [106456, 3270, 6034], [-13944, -17780, -2296], [-82146, 21582, 1495], [-36288, -17339, -15897], [75432, -11496, 8974], [-62382, -26283, 2353], [-45102, 16618, 17801], [38080, 16577, 19047], [99904, 11537, -9772], [1232, -5863, 14959], [101290, 7316, -2992], [30184, -9628, 5362], [-12522, 1908, -1818], [37408, 743, 10073], [106554, 11208, -10006], [-26226, 23184, 8505], [-57074, 4192, -9832], [57400, -11170, 18858], [115638, 5248, -2178], [20090, 14988, 17301], [43456, -32780, -14322], [-69810, 14800, -1000], [29758, 18443, 3975], [73248, 3989, 24024], [29736, -12125, -10752], [-69806, 26970, 11098], [-40486, -30476, 11642], [-212240, 6631, -329], [32928, -2923, -16737], [-14278, 18300, -14412], [-183126, 6310, 10771], [39480, 16696, -9044], [118384, 10462, -5572], [-5130, -6034, -1120], [-134456, 954, -11102], [45202, 366, 11544], [-11266, 13426, -12089], [101528, -37568, -17472], [-206640, -443, 1834], [-136070, 12718, -7648], [-27462, -5978, -9807], [74704, 13465, -10899], [-90026, -17664, 4328], [28896, 18827, 8582], [105840, 16354, 9016]]