
# q-expansion of newform 1024.4.a.m, downloaded from the LMFDB on 25 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 = 1024
weight = 4
poly_data = [-32, 0, 420, 0, -1380, 0, 405, 0, -36, 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, 0, 0, 0, 0, 0, 0], 1], [[0, 4528, 0, -10042, 0, 2823, 0, -250, 0, 7], 208], [[17840, 0, -70882, 0, 21595, 0, -2018, 0, 59, 0], 416], [[-24496, 0, 180342, 0, -55785, 0, 4982, 0, -137, 0], 416], [[0, 18896, 0, -278022, 0, 85849, 0, -7750, 0, 217], 832], [[-18656, 0, 112562, 0, -33651, 0, 2994, 0, -83, 0], 208], [[35856, 0, -277098, 0, 86919, 0, -8042, 0, 231, 0], 416], [[0, 34432, 0, -206166, 0, 62573, 0, -5622, 0, 157], 416], [[0, -13680, 0, 73594, 0, -22135, 0, 1978, 0, -55], 64], [[0, 224336, 0, -1348970, 0, 407255, 0, -36458, 0, 1015], 832]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [-3, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 1], [0, 0, 0, 0, -1, 0, 0, -3, -1, -1], [0, 0, 2, 1, 0, -2, 1, 0, 0, 0], [0, 1, 0, 0, -5, 0, 0, -2, 2, -1], [-29, 0, -2, 2, 0, -1, 3, 0, 0, 0], [0, 3, 0, 0, -1, 0, 0, -18, -2, 2], [-36, 0, 3, -2, 0, -3, -3, 0, 0, 0], [0, -2, 0, 0, -1, 0, 0, -29, 1, 1], [1, 0, -1, 4, 0, 14, 1, 0, 0, 0], [0, -20, 0, 0, -12, 0, 0, -1, 0, -4], [-90, 0, 7, -10, 0, -5, -3, 0, 0, 0], [0, 3, 0, 0, 1, 0, 0, -35, -13, 7], [0, -41, 0, 0, 18, 0, 0, 5, 14, 4], [0, -19, 0, 0, 1, 0, 0, -41, 9, -11], [0, 50, 0, 0, 11, 0, 0, -10, 12, 1], [-347, 0, 10, -2, 0, -15, -3, 0, 0, 0], [-29, 0, -11, 1, 0, 16, -6, 0, 0, 0], [-446, 0, -12, 8, 0, 6, 16, 0, 0, 0], [0, 71, 0, 0, 48, 0, 0, 9, 22, -2], [15, 0, 25, 1, 0, 8, -26, 0, 0, 0], [4, 0, 14, 3, 0, 14, -29, 0, 0, 0], [0, -38, 0, 0, 11, 0, 0, 73, -13, 3], [-691, 0, 0, 10, 0, 43, -25, 0, 0, 0], [0, -110, 0, 0, 45, 0, 0, -10, -12, 23], [0, -6, 0, 0, -11, 0, 0, 88, -32, -8], [78, 0, 60, 14, 0, 24, -10, 0, 0, 0], [-744, 0, -11, -30, 0, 79, 3, 0, 0, 0], [0, 154, 0, 0, 23, 0, 0, 50, 4, -3], [165, 0, 63, 2, 0, -42, 3, 0, 0, 0], [0, -206, 0, 0, -86, 0, 0, -35, 20, -6], [0, 93, 0, 0, -1, 0, 0, 204, -16, 28], [-917, 0, 24, -74, 0, -123, -7, 0, 0, 0], [0, 143, 0, 0, -59, 0, 0, 153, 7, -37], [0, 238, 0, 0, -136, 0, 0, -59, 20, 28], [-1059, 0, 27, -40, 0, 52, 52, 0, 0, 0], [0, -58, 0, 0, 47, 0, 0, 193, 75, 11], [0, 181, 0, 0, 12, 0, 0, 137, -6, 38], [0, -149, 0, 0, 71, 0, 0, 260, -24, -60], [-1208, 0, 75, 14, 0, 177, -51, 0, 0, 0], [-652, 0, -90, 79, 0, 14, 31, 0, 0, 0], [0, 256, 0, 0, -159, 0, 0, 36, -60, -36], [-1107, 0, 49, 52, 0, -74, -2, 0, 0, 0], [0, 179, 0, 0, -182, 0, 0, 63, -50, -68], [-948, 0, -137, -10, 0, -183, 25, 0, 0, 0], [0, 6, 0, 0, 132, 0, 0, 97, 44, -72], [0, -334, 0, 0, 101, 0, 0, 18, 14, 22], [407, 0, 97, -47, 0, 72, 14, 0, 0, 0], [-1578, 0, -116, 56, 0, -14, -32, 0, 0, 0], [-144, 0, 10, -99, 0, -210, 29, 0, 0, 0], [0, 107, 0, 0, 92, 0, 0, -221, -130, -38], [1066, 0, 148, -134, 0, -144, -62, 0, 0, 0], [-731, 0, -100, -46, 0, -105, -37, 0, 0, 0], [0, 96, 0, 0, -251, 0, 0, -260, -20, -28], [74, 0, -55, 122, 0, 69, 131, 0, 0, 0], [0, 333, 0, 0, -59, 0, 0, -317, 85, 9], [751, 0, 61, 107, 0, 288, 88, 0, 0, 0], [0, -73, 0, 0, 81, 0, 0, 298, 70, -19], [0, -614, 0, 0, -103, 0, 0, -112, -48, 16], [0, -169, 0, 0, 45, 0, 0, 384, 102, 49], [227, 0, -85, -156, 0, -2, -34, 0, 0, 0], [685, 0, 291, -120, 0, 158, -135, 0, 0, 0], [0, -819, 0, 0, 251, 0, 0, -120, -52, -32], [0, 192, 0, 0, -185, 0, 0, -32, 72, -111], [-18, 0, -8, 144, 0, -56, -88, 0, 0, 0], [0, 673, 0, 0, 158, 0, 0, -451, 106, 20], [0, 1067, 0, 0, 151, 0, 0, 142, -138, -14], [-318, 0, -44, 118, 0, -144, 198, 0, 0, 0], [525, 0, -164, 66, 0, -33, 139, 0, 0, 0], [3830, 0, -153, -282, 0, -117, -51, 0, 0, 0], [0, -533, 0, 0, -343, 0, 0, 429, -213, 167], [0, 399, 0, 0, -262, 0, 0, -619, 134, -28], [888, 0, -106, 172, 0, 10, -102, 0, 0, 0], [0, 1080, 0, 0, 335, 0, 0, 347, 81, -71], [0, 200, 0, 0, -471, 0, 0, 206, 242, -22], [-160, 0, 42, -315, 0, -242, -107, 0, 0, 0], [953, 0, -417, -160, 0, -82, 101, 0, 0, 0], [0, -1080, 0, 0, -179, 0, 0, -264, -64, 19], [0, -1078, 0, 0, -331, 0, 0, 1062, -174, 98], [918, 0, 312, 288, 0, 54, -28, 0, 0, 0], [680, 0, -86, 245, 0, 182, 117, 0, 0, 0], [2587, 0, -459, 376, 0, -188, 68, 0, 0, 0], [0, 993, 0, 0, -47, 0, 0, 926, 66, -23], [-2860, 0, -290, -77, 0, 78, -221, 0, 0, 0], [-2803, 0, -5, 108, 0, -322, 221, 0, 0, 0], [0, 200, 0, 0, 729, 0, 0, 1914, -114, -98], [5290, 0, -226, 220, 0, 156, -138, 0, 0, 0], [0, -1479, 0, 0, 213, 0, 0, 572, 434, -43], [28, 0, -257, 70, 0, 305, -15, 0, 0, 0], [6661, 0, -109, -488, 0, -380, -36, 0, 0, 0], [0, 1420, 0, 0, 6, 0, 0, -1675, 288, 10], [0, -187, 0, 0, -70, 0, 0, -1373, -86, 264], [1123, 0, 170, 122, 0, -509, 167, 0, 0, 0], [0, -1643, 0, 0, -775, 0, 0, 735, -311, 61], [3997, 0, 15, -118, 0, 334, -165, 0, 0, 0], [0, 36, 0, 0, 685, 0, 0, 338, 56, 11], [0, 1975, 0, 0, -327, 0, 0, 1603, 117, -151], [0, -280, 0, 0, 267, 0, 0, -486, -248, -275], [0, -194, 0, 0, -165, 0, 0, -393, -163, -163], [0, -641, 0, 0, 423, 0, 0, 1526, -594, 111], [-2491, 0, -441, -230, 0, -194, 179, 0, 0, 0], [0, 17, 0, 0, 1053, 0, 0, 1378, 290, -131], [-2206, 0, 192, -236, 0, -232, 476, 0, 0, 0], [0, 1340, 0, 0, 253, 0, 0, 234, -256, -13], [3350, 0, -136, -68, 0, -104, 92, 0, 0, 0], [-3341, 0, 73, -48, 0, 728, -192, 0, 0, 0], [-1761, 0, 465, -155, 0, 440, -54, 0, 0, 0], [1452, 0, 68, 408, 0, 616, 132, 0, 0, 0], [0, -1746, 0, 0, -83, 0, 0, -622, -218, -154], [-3081, 0, -403, 307, 0, 264, 176, 0, 0, 0], [0, -642, 0, 0, 843, 0, 0, -1512, -308, -335], [-5, 0, -332, 78, 0, -727, 141, 0, 0, 0], [1644, 0, -506, 391, 0, -682, 87, 0, 0, 0], [0, 1014, 0, 0, 455, 0, 0, 1682, 68, -195], [-8387, 0, 318, -42, 0, 197, 41, 0, 0, 0], [0, 296, 0, 0, -611, 0, 0, -472, 112, 392], [0, -1069, 0, 0, 308, 0, 0, -563, 318, -30], [0, 1609, 0, 0, -225, 0, 0, 512, -652, 72], [44, 0, 694, -345, 0, -698, -601, 0, 0, 0], [0, -2314, 0, 0, -801, 0, 0, -1581, -79, 17], [0, 256, 0, 0, 172, 0, 0, 2759, 368, 292], [0, 677, 0, 0, 954, 0, 0, 2699, 402, -320], [0, -1975, 0, 0, 365, 0, 0, -391, 215, 443], [0, 1728, 0, 0, -653, 0, 0, -1007, -5, -101], [-15638, 0, -364, 440, 0, 270, -112, 0, 0, 0], [-2853, 0, -381, 324, 0, 670, 598, 0, 0, 0], [0, 2163, 0, 0, 1131, 0, 0, -328, 580, -64], [0, 1590, 0, 0, 758, 0, 0, 1791, -172, 190], [-11507, 0, -511, 52, 0, 38, 126, 0, 0, 0], [-3810, 0, 1208, -192, 0, -226, -684, 0, 0, 0], [0, -1237, 0, 0, -1201, 0, 0, -3380, 88, -148], [4189, 0, -349, -76, 0, 662, 413, 0, 0, 0], [7100, 0, -1306, 63, 0, 214, 447, 0, 0, 0], [0, 2452, 0, 0, 611, 0, 0, -1151, -213, -149], [0, 2615, 0, 0, 420, 0, 0, -1955, -626, -78], [0, 1931, 0, 0, 233, 0, 0, -793, -39, 69], [4777, 0, 151, -936, 0, -586, 197, 0, 0, 0], [0, -1598, 0, 0, 29, 0, 0, 1918, 172, 143], [0, -993, 0, 0, 593, 0, 0, 2297, -113, 107], [3763, 0, 675, -492, 0, -586, 454, 0, 0, 0], [0, 821, 0, 0, -1603, 0, 0, 1362, -702, -499], [0, -2079, 0, 0, -119, 0, 0, -1209, -335, -363], [-17947, 0, 113, -444, 0, -82, 30, 0, 0, 0], [0, 813, 0, 0, 665, 0, 0, 721, 295, -653], [-5399, 0, 707, -970, 0, -1058, -293, 0, 0, 0], [0, -2679, 0, 0, 1618, 0, 0, -1687, 90, 336], [-13660, 0, 1335, -186, 0, -287, -263, 0, 0, 0], [0, -302, 0, 0, 713, 0, 0, -534, -170, 478], [-8826, 0, 256, 556, 0, 64, 148, 0, 0, 0], [0, 2863, 0, 0, 1321, 0, 0, -1924, 526, -243], [-25989, 0, -873, -148, 0, -70, -558, 0, 0, 0], [0, -2338, 0, 0, 1500, 0, 0, -1429, 348, -176], [-24430, 0, -391, -150, 0, 477, 147, 0, 0, 0], [-1605, 0, -814, -934, 0, -205, 175, 0, 0, 0], [-708, 0, 238, -149, 0, 950, 555, 0, 0, 0], [4335, 0, 217, -87, 0, -1384, -674, 0, 0, 0], [0, 862, 0, 0, 341, 0, 0, 2712, 400, 1048], [0, -753, 0, 0, 461, 0, 0, 4240, -650, 161], [-8427, 0, -353, 910, 0, -18, 39, 0, 0, 0], [-9667, 0, -1481, 784, 0, -344, 72, 0, 0, 0], [0, 2450, 0, 0, 622, 0, 0, 417, -1012, 542], [12008, 0, 450, -447, 0, 1350, -303, 0, 0, 0], [-15853, 0, 343, -324, 0, -14, -1054, 0, 0, 0], [-2236, 0, -84, -200, 0, -1160, 60, 0, 0, 0], [0, 158, 0, 0, -405, 0, 0, -2883, 471, 431], [-15176, 0, -710, -267, 0, 182, 549, 0, 0, 0], [0, 499, 0, 0, -261, 0, 0, 1578, -1070, 534], [0, 2237, 0, 0, 985, 0, 0, -3746, -358, -231], [0, -995, 0, 0, -5, 0, 0, 4316, 864, 76], [-25643, 0, 466, -274, 0, -1159, 469, 0, 0, 0], [17715, 0, -1103, -697, 0, -856, 472, 0, 0, 0], [794, 0, 219, -594, 0, 123, 257, 0, 0, 0], [15495, 0, -571, -41, 0, -1256, 116, 0, 0, 0], [0, -423, 0, 0, 244, 0, 0, 3231, -1078, -258], [0, -794, 0, 0, 1195, 0, 0, 3937, -1509, 11], [14973, 0, -743, 196, 0, 1502, -58, 0, 0, 0], [0, -638, 0, 0, 347, 0, 0, 691, -343, 129], [4312, 0, -308, -872, 0, 1532, -1052, 0, 0, 0], [0, -4300, 0, 0, 104, 0, 0, -3215, -632, -192], [0, -636, 0, 0, 1913, 0, 0, -2012, -756, 332], [-15845, 0, 369, -973, 0, -2248, -20, 0, 0, 0], [58, 0, 1223, -298, 0, 1375, -315, 0, 0, 0], [0, -127, 0, 0, 447, 0, 0, -3372, -888, -332], [0, -77, 0, 0, -1739, 0, 0, -5807, 767, 227], [0, 4002, 0, 0, 238, 0, 0, -1689, -380, 326], [-8935, 0, -353, -376, 0, -962, -1059, 0, 0, 0], [-20984, 0, 1026, -444, 0, 1406, 14, 0, 0, 0], [16418, 0, -988, 686, 0, 1008, -2, 0, 0, 0], [0, 3106, 0, 0, -84, 0, 0, -1845, -292, 464], [0, -691, 0, 0, -1781, 0, 0, 1038, 146, 783], [0, -2781, 0, 0, 1215, 0, 0, 3398, -106, -86], [0, -4730, 0, 0, -404, 0, 0, 497, -1532, 1240], [5663, 0, -195, 1927, 0, 1464, 332, 0, 0, 0], [8950, 0, -1640, 988, 0, -1736, 732, 0, 0, 0], [0, 2101, 0, 0, -2843, 0, 0, -3511, -1041, 91], [-27014, 0, 1016, 996, 0, 1784, -780, 0, 0, 0], [-6371, 0, -1916, 18, 0, -1401, 211, 0, 0, 0], [0, -1880, 0, 0, -181, 0, 0, -5769, 309, 525], [11130, 0, -320, -240, 0, 1090, -764, 0, 0, 0], [0, -1359, 0, 0, -2127, 0, 0, -3823, -849, 547], [2378, 0, 1684, 970, 0, -48, 50, 0, 0, 0], [0, 6330, 0, 0, -1901, 0, 0, -3600, 60, 873], [0, 3797, 0, 0, -541, 0, 0, -8568, 52, -48], [21840, 0, -176, 1248, 0, 2816, 272, 0, 0, 0], [0, -3588, 0, 0, -3330, 0, 0, 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