
# q-expansion of newform 544.4.a.m, downloaded from the LMFDB on 22 October 2025.

# 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 = 544
weight = 4
poly_data = [1028, 1978, -1476, -964, 383, 116, -34, -4, 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], 1], [[842, -3082, 4033, -2156, -785, 286, 28, -8], 2601], [[-20, -2, 2, 0, 0, 0, 0, 0], 1], [[-455, 36, 2, -74, 32, 6, -2, 0], 17], [[-939, -682, 610, 142, -66, -6, 2, 0], 17], [[-77338, 139688, 43837, -25496, -5495, 2002, 196, -56], 2601], [[54004, -88826, -55936, 31964, 7850, -2860, -280, 80], 2601], [[-7624, 31292, -45488, 21610, 12730, -4406, -686, 196], 2601]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [2, 0, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 1, 0], [0, -1, 0, 0, 0, 0, 0, -1], [-7, 0, 1, 1, 0, 0, 0, 0], [17, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, 0, 0, -1, -1, 0], [0, 4, 0, 0, 0, -3, 2, 1], [-16, 0, -3, 4, -2, 0, 0, 0], [0, -8, 0, 0, 0, 3, -4, -3], [-14, 0, -5, 4, 4, 0, 0, 0], [166, 0, -4, -2, -2, 0, 0, 0], [0, 15, 0, 0, 0, -9, 1, 1], [0, -15, 0, 0, 0, -9, 1, 3], [152, 0, 2, -6, 12, 0, 0, 0], [0, -11, 0, 0, 0, 1, 15, -5], [94, 0, -3, 2, 2, 0, 0, 0], [0, -43, 0, 0, 0, -5, 3, -4], [0, -47, 0, 0, 0, 2, 11, 6], [494, 0, 26, -2, -2, 0, 0, 0], [0, -66, 0, 0, 0, -9, -12, 3], [0, -15, 0, 0, 0, -7, 31, 7], [288, 0, 22, 23, 7, 0, 0, 0], [570, 0, 26, 10, -18, 0, 0, 0], [61, 0, -27, -7, -20, 0, 0, 0], [0, 6, 0, 0, 0, -14, -6, 8], [0, 13, 0, 0, 0, -4, 26, -2], [54, 0, 67, -6, -6, 0, 0, 0], [726, 0, -42, 0, 20, 0, 0, 0], [0, 80, 0, 0, 0, 22, 12, -18], [0, -95, 0, 0, 0, -4, -10, 12], [596, 0, -64, 9, -47, 0, 0, 0], [0, 81, 0, 0, 0, -16, -36, 35], [-50, 0, 84, -64, 72, 0, 0, 0], [0, 8, 0, 0, 0, 10, 96, -26], [-1156, 0, 96, 42, -24, 0, 0, 0], [0, -223, 0, 0, 0, -10, -12, 27], [0, 11, 0, 0, 0, 24, -33, -2], [-88, 0, 83, 2, -8, 0, 0, 0], [0, 51, 0, 0, 0, 87, -9, -8], [-1582, 0, 5, -18, 90, 0, 0, 0], [0, -15, 0, 0, 0, -65, 45, -9], [906, 0, -20, -90, 74, 0, 0, 0], [-584, 0, -9, -32, -54, 0, 0, 0], [0, -220, 0, 0, 0, -33, -98, 37], [0, -453, 0, 0, 0, -56, 118, 24], [0, -31, 0, 0, 0, 29, 41, 55], [0, 25, 0, 0, 0, -94, -8, 45], [-2619, 0, 75, -13, 10, 0, 0, 0], [514, 0, 148, -130, 114, 0, 0, 0], [0, 97, 0, 0, 0, 19, -171, -19], [1522, 0, 226, -20, 28, 0, 0, 0], [0, 67, 0, 0, 0, -71, -95, 52], [-530, 0, -38, 95, 1, 0, 0, 0], [0, -738, 0, 0, 0, -104, 50, 44], [-66, 0, -161, 86, 38, 0, 0, 0], [0, 85, 0, 0, 0, 23, 161, 1], [-2264, 0, -315, -24, -86, 0, 0, 0], [-362, 0, 102, -74, 62, 0, 0, 0], [0, -641, 0, 0, 0, 8, 172, -77], [-1554, 0, 136, 100, -108, 0, 0, 0], [0, 1025, 0, 0, 0, -45, 5, 3], [0, 49, 0, 0, 0, 134, -19, -104], [-2142, 0, 442, -68, -44, 0, 0, 0], [4940, 0, -53, -124, 26, 0, 0, 0], [0, -745, 0, 0, 0, 47, -155, 11], [1810, 0, -138, -30, -98, 0, 0, 0], [0, -67, 0, 0, 0, 134, -24, -83], [1784, 0, -38, -38, 196, 0, 0, 0], [-762, 0, 282, 102, -106, 0, 0, 0], [0, -651, 0, 0, 0, 25, -183, 129], [0, -951, 0, 0, 0, -156, 211, -20], [-197, 0, 19, -227, 218, 0, 0, 0], [0, 17, 0, 0, 0, -104, -96, -100], [0, 648, 0, 0, 0, 62, -248, -32], [6551, 0, -47, 7, 52, 0, 0, 0], [-2498, 0, -301, 156, -280, 0, 0, 0], [5850, 0, -50, 16, 16, 0, 0, 0], [-1518, 0, -328, 160, -120, 0, 0, 0], [0, -99, 0, 0, 0, 66, 122, 78], [-5633, 0, -209, 133, 30, 0, 0, 0], [0, 912, 0, 0, 0, -107, 6, 73], [2206, 0, -152, -307, -29, 0, 0, 0], [0, -449, 0, 0, 0, -60, 45, -74], [0, 1309, 0, 0, 0, 75, 455, -48], [290, 0, 172, 140, -76, 0, 0, 0], [362, 0, -328, -87, 243, 0, 0, 0], [6228, 0, 42, 222, -236, 0, 0, 0], [0, 200, 0, 0, 0, 108, 56, 58], [0, -331, 0, 0, 0, -51, 73, 312], [0, 730, 0, 0, 0, 41, -148, -129], [0, 1281, 0, 0, 0, -148, -149, 200], [0, -697, 0, 0, 0, 7, -249, 58], [0, -87, 0, 0, 0, -18, 148, -104], [0, 99, 0, 0, 0, 102, 205, 152], [7710, 0, -644, 140, -12, 0, 0, 0], [2098, 0, -676, 22, -182, 0, 0, 0], [0, -169, 0, 0, 0, -155, -143, 80], [8658, 0, -455, 214, -38, 0, 0, 0], [0, -635, 0, 0, 0, 412, -122, 72], [3079, 0, -593, -187, 46, 0, 0, 0], [0, 313, 0, 0, 0, -47, -161, -51], [-6194, 0, -42, 46, -18, 0, 0, 0], [0, -935, 0, 0, 0, 174, -182, 166], [1118, 0, 630, -267, -109, 0, 0, 0], [0, 341, 0, 0, 0, 85, 141, -402], [5174, 0, 478, 178, -6, 0, 0, 0], [0, 710, 0, 0, 0, -226, 214, -274], [-11378, 0, 292, 344, 4, 0, 0, 0], [0, -62, 0, 0, 0, -267, 250, 1], [1166, 0, -384, -96, -472, 0, 0, 0], [1374, 0, 132, 238, -266, 0, 0, 0], [0, 1013, 0, 0, 0, -306, 36, 307], [0, -2079, 0, 0, 0, 101, -499, 95], [1502, 0, 1226, 250, -310, 0, 0, 0], [0, 497, 0, 0, 0, 48, 448, -222], [0, 2787, 0, 0, 0, -87, 511, -9], [-3180, 0, -1193, -142, -52, 0, 0, 0], [0, 2193, 0, 0, 0, 47, 297, -147], [-5358, 0, -706, 296, -348, 0, 0, 0], [-1338, 0, 424, 504, 244, 0, 0, 0], [9004, 0, 721, 242, 164, 0, 0, 0], [0, -1467, 0, 0, 0, -368, 276, -34], [0, 635, 0, 0, 0, -168, 540, 455], [12033, 0, -307, -295, 352, 0, 0, 0], [-10388, 0, 83, -418, 336, 0, 0, 0], [0, 1439, 0, 0, 0, 180, 77, 266], [0, 2587, 0, 0, 0, 29, -245, -45], [-16458, 0, 958, 24, -8, 0, 0, 0], [0, -645, 0, 0, 0, -41, 267, -448], [0, -2280, 0, 0, 0, -141, -504, 101], [0, -1789, 0, 0, 0, -436, 309, 88], [-12601, 0, 141, -165, -384, 0, 0, 0], [-4780, 0, 424, -371, 333, 0, 0, 0], [-6456, 0, -1432, 13, 273, 0, 0, 0], [-9563, 0, 859, -285, -134, 0, 0, 0], [0, 485, 0, 0, 0, 4, -240, 113], [10678, 0, -970, 704, -80, 0, 0, 0], [1830, 0, -858, 718, 230, 0, 0, 0], [0, 843, 0, 0, 0, -454, -332, 29], [8280, 0, -167, -134, -240, 0, 0, 0], [0, 1424, 0, 0, 0, -51, -930, 303], [0, -515, 0, 0, 0, -242, -460, 382], [-28228, 0, 288, -246, 296, 0, 0, 0], [0, 381, 0, 0, 0, 328, 189, 142], [-946, 0, 893, -688, 468, 0, 0, 0], [-3298, 0, 890, -640, 404, 0, 0, 0], [0, -331, 0, 0, 0, 313, -473, -413], [0, 922, 0, 0, 0, 186, -542, 584], [-1194, 0, 1627, -298, 270, 0, 0, 0], [10138, 0, 246, 602, -866, 0, 0, 0], [0, -3441, 0, 0, 0, -95, 789, 28], [0, 3994, 0, 0, 0, 517, -240, -25], [0, -1235, 0, 0, 0, -430, 114, -55], [0, -4137, 0, 0, 0, -514, -133, 296], [0, -2509, 0, 0, 0, 105, 739, 31], [27822, 0, 1878, 60, 576, 0, 0, 0], [20214, 0, 1180, -68, 184, 0, 0, 0], [-13972, 0, 85, -330, 1304, 0, 0, 0], [0, -411, 0, 0, 0, 108, -384, -107], [4762, 0, -424, 919, -275, 0, 0, 0], [0, -3699, 0, 0, 0, 137, 465, -443], [0, 1717, 0, 0, 0, 637, -753, -739], [1342, 0, -266, 342, 398, 0, 0, 0], [0, 3215, 0, 0, 0, 84, -1015, 302], [0, 2531, 0, 0, 0, 600, -423, -538], [-10408, 0, 1315, 236, -578, 0, 0, 0], [3850, 0, 836, 1044, 276, 0, 0, 0], [1710, 0, 317, 290, -118, 0, 0, 0], [0, 325, 0, 0, 0, -721, 453, 493], [16617, 0, -1451, 957, -508, 0, 0, 0], [0, 1790, 0, 0, 0, -45, -630, 211], [628, 0, 754, -441, 51, 0, 0, 0], [0, 1944, 0, 0, 0, -94, 1880, 360], [-4454, 0, -218, -498, -78, 0, 0, 0], [0, 847, 0, 0, 0, -366, -1432, 175], [7466, 0, -1119, 98, 350, 0, 0, 0], [0, 397, 0, 0, 0, -75, -787, 691], [8341, 0, 831, 171, -534, 0, 0, 0], [0, 3914, 0, 0, 0, 582, 278, -292], [0, -1461, 0, 0, 0, 370, 162, -407], [-4382, 0, -1411, -238, 182, 0, 0, 0], [-2754, 0, 1056, 28, -464, 0, 0, 0], [0, 194, 0, 0, 0, 800, 454, 152], [3287, 0, 227, -159, -294, 0, 0, 0], [8478, 0, -2553, -16, 436, 0, 0, 0], [0, -1965, 0, 0, 0, 129, -649, 535], [-4262, 0, -642, -106, 922, 0, 0, 0], [0, 964, 0, 0, 0, -221, -856, 33], [1042, 0, 1054, 698, 182, 0, 0, 0], [0, 4479, 0, 0, 0, -536, 1200, 198], [0, -1407, 0, 0, 0, -267, 945, -178], [22885, 0, -1585, 451, 338, 0, 0, 0], [0, -4129, 0, 0, 0, -234, 404, -255], [27058, 0, -2400, 24, 336, 0, 0, 0], [39734, 0, 476, 250, 122, 0, 0, 0], [-42456, 0, 461, -778, -100, 0, 0, 0], [25086, 0, 1506, 182, -834, 0, 0, 0], [0, -8012, 0, 0, 0, 1148, 328, -684], [13534, 0, -735, -490, 1038, 0, 0, 0], [0, -3062, 0, 0, 0, 879, 160, -1087], [-40146, 0, 2028, 328, -416, 0, 0, 0], [1738, 0, 970, -1541, 905, 0, 0, 0], [0, 3661, 0, 0, 0, -409, 877, 515], [117, 0, 3327, -349, -798, 0, 0, 0], [0, -604, 0, 0, 0, -632, 1288, 608], [0, -1221, 0, 0, 0, 1167, 659, 194], [10074, 0, 1518, -138, -486, 0, 0, 0], [0, -283, 0, 0, 0, 471, 2381, -57], [19454, 0, -1246, 814, -778, 0, 0, 0], [26645, 0, 411, 731, -126, 0, 0, 0], [0, -6442, 0, 0, 0, 155, 2254, -567], [0, 243, 0, 0, 0, -13, 1439, 532], [0, -1766, 0, 0, 0, 115, -922, 257], [-414, 0, -602, -1144, 952, 0, 0, 0], [0, 600, 0, 0, 0, -592, -188, 1340], [-12858, 0, 940, 1508, 264, 0, 0, 0], [0, -4460, 0, 0, 0, 697, 1852, -951], [-58531, 0, -135, 57, 676, 0, 0, 0], [20452, 0, 516, 830, -1756, 0, 0, 0], [0, -2157, 0, 0, 0, 1132, -289, -438], [410, 0, -1148, -1516, 468, 0, 0, 0], [0, -6281, 0, 0, 0, 452, 1399, -752], [0, 985, 0, 0, 0, -345, -473, 196], [16607, 0, 591, -295, 1130, 0, 0, 0], [-20918, 0, -2346, 38, -54, 0, 0, 0], [0, 5195, 0, 0, 0, -1098, 797, 1188], [0, 1346, 0, 0, 0, 1264, -1182, 920], [0, -983, 0, 0, 0, -294, 2542, 624], [-56414, 0, -2490, -264, 748, 0, 0, 0], [0, -1049, 0, 0, 0, -166, 374, -944], [0, -4127, 0, 0, 0, -1015, 505, -29], [-3244, 0, 2618, 1489, 1, 0, 0, 0], [0, 8479, 0, 0, 0, -603, 525, -100], [0, 4453, 0, 0, 0, -747, 105, -251], [44594, 0, -1628, -1774, 1078, 0, 0, 0], [42106, 0, 2079, -1336, 1608, 0, 0, 0], [0, -137, 0, 0, 0, 814, 452, 268], [0, 4005, 0, 0, 0, 25, 985, 443], [0, 5215, 0, 0, 0, 700, 2140, -724], [0, 6047, 0, 0, 0, 709, 2243, -269], [0, -6361, 0, 0, 0, 369, 1511, -753], [-82791, 0, -1341, 435, -386, 0, 0, 0], [-67506, 0, -1026, 936, -1164, 0, 0, 0], [0, -1327, 0, 0, 0, -898, -287, 848], [0, 34, 0, 0, 0, 945, -1738, -687], [0, 1571, 0, 0, 0, -1040, 248, -134], [0, -5651, 0, 0, 0, -91, -2509, -433], [0, 733, 0, 0, 0, 121, 2429, -779], [-6380, 0, 2736, -226, 972, 0, 0, 0], [9542, 0, -5028, 946, -574, 0, 0, 0], [0, -4, 0, 0, 0, 106, 1792, -2], [25742, 0, 1762, 522, -1694, 0, 0, 0], [-10937, 0, 5245, -217, 236, 0, 0, 0], [0, 4183, 0, 0, 0, -627, -307, 92], [-27812, 0, -3973, -484, 874, 0, 0, 0], [0, -543, 0, 0, 0, -400, 1892, 1476], [-6668, 0, -551, 254, 416, 0, 0, 0], [5348, 0, 2588, 1215, -877, 0, 0, 0], [0, 7827, 0, 0, 0, -316, 2067, -440], [0, -7905, 0, 0, 0, 1541, -5, -317], [3024, 0, 1277, 1930, -664, 0, 0, 0], [-5600, 0, -2861, 14, -1276, 0, 0, 0], [-37202, 0, 618, -6, 250, 0, 0, 0], [0, 4733, 0, 0, 0, 1787, -277, 150], [-10006, 0, 3294, -684, -488, 0, 0, 0], [-22282, 0, 3316, -428, 80, 0, 0, 0], [0, 2605, 0, 0, 0, 98, -408, -561], [18976, 0, -3148, -738, 88, 0, 0, 0], [12680, 0, 6047, -338, 316, 0, 0, 0], [0, -4449, 0, 0, 0, 1921, -1005, -19], [22302, 0, -2114, -102, 1650, 0, 0, 0], [0, -2911, 0, 0, 0, 1681, -711, -1065], [-25978, 0, 448, -320, 1324, 0, 0, 0], [0, -2821, 0, 0, 0, 267, -2101, 777], [0, -4951, 0, 0, 0, -55, -2021, 685], [19767, 0, 6619, -371, 374, 0, 0, 0], [24380, 0, 2706, -253, 959, 0, 0, 0], [0, -1131, 0, 0, 0, 369, 1459, -769], [0, 5705, 0, 0, 0, 79, 2873, 485], [0, 10485, 0, 0, 0, 962, 593, -844], [0, 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