
# q-expansion of newform 531.4.a.f, downloaded from the LMFDB on 24 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 = 531
weight = 4
poly_data = [-384, 1940, -1476, -1023, 648, 89, -49, -2, 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], [[0, 1, 0, 0, 0, 0, 0, 0], 1], [[-13, 0, 1, 0, 0, 0, 0, 0], 1], [[-20832, 36118, 16089, -9408, -1483, 647, 34, -13], 2144], [[-33792, 28786, 12861, -12207, -734, 901, 11, -18], 2144], [[-6352, 15758, 3695, -5406, -345, 369, 8, -7], 536], [[-62496, 59042, 48267, -26080, -4449, 1941, 102, -39], 2144], [[29824, -89302, -20593, 36612, 1967, -2775, -46, 57], 2144]]

hecke_ring_character_values = None
aps_data = [[0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [2, -1, 0, 0, -1, 0, 0, -1], [7, -1, 1, 2, 0, 1, 0, -1], [7, -6, 1, -1, -2, 1, 3, -2], [9, 2, -2, 2, 3, 0, -3, 0], [-7, -6, 0, -1, 2, 2, -2, -2], [36, 7, -4, -1, 6, 0, 5, -1], [-28, 7, -2, -3, 8, -6, -3, 1], [18, 10, 2, -15, -5, 2, 1, 0], [68, -16, -4, -11, 1, -2, -3, 14], [57, -24, -1, -8, 9, -25, 4, -8], [-13, 15, -23, -4, 26, -5, -4, 29], [63, -56, -15, 3, 14, -25, -3, 18], [-126, 8, -5, 26, 0, -21, 11, 8], [102, 21, 6, 2, -1, -18, 16, -15], [59, 0, 0, 0, 0, 0, 0, 0], [-72, -32, -32, 19, 11, 22, -33, 16], [246, 3, 7, -21, -18, 23, -16, 11], [211, -6, -2, -8, 21, 22, -33, 10], [272, -52, 23, -6, -62, 25, -13, -22], [413, 76, -21, -11, 30, 13, 33, -8], [-279, 117, -35, -60, -40, -1, 24, 19], [-244, 98, -67, -18, 88, -85, 31, -4], [172, 27, 15, 67, 42, -101, -30, 45], [-243, 81, -92, 8, 68, -52, -49, 91], [882, -20, 11, -3, -59, 5, 10, 10], [128, 77, -40, 57, 104, -68, 11, 31], [460, 35, 3, -102, -33, 45, 23, -31], [144, 161, 97, -39, -60, -63, -72, 17], [564, 25, 31, -101, 34, -5, -48, 91], [-324, -32, 114, 132, 40, 8, -10, 56], [-67, 170, -29, 77, -36, 33, -97, -2], [620, 323, 65, 95, 8, 27, -22, 25], [739, -154, 36, -101, 76, -152, 50, 2], [366, 10, 14, 149, 17, -96, 93, -28], [-60, 297, -64, -184, -27, 28, 98, 69], [930, -188, -7, 86, 6, 93, -83, -58], [-806, 166, 38, 65, 69, 36, 47, 14], [-853, 5, -49, 236, 94, 109, -20, 13], [-387, 192, 31, -157, -74, 129, 139, -76], [-672, -256, -53, 2, 18, 3, -93, 156], [324, -113, -43, -5, 10, 197, -14, -27], [437, 421, 62, -200, -110, -68, 175, -145], [1442, -324, -17, -7, 137, -9, -86, 34], [-100, 206, 107, 116, -24, -231, -1, -56], [167, -178, 78, -41, -162, 248, -102, 56], [87, 231, -104, 74, 46, -106, 155, -211], [-971, 64, -20, 13, -4, -122, -246, 208], [-2277, 105, -30, 183, 41, -154, 6, -81], [536, -402, 6, 272, -274, 30, 90, -142], [1352, -17, 256, 348, 57, 318, 64, -23], [1359, -95, 266, -96, -388, 322, 45, -225], [-734, -37, -213, -91, 248, -47, -292, 359], [845, -219, 182, 144, -146, 140, -93, -157], [-289, 189, -156, 261, 527, -196, 108, 55], [1609, -162, -3, -133, -114, -269, 73, -76], [1229, -532, 9, -587, -54, 43, -57, 136], [1292, -611, 393, -73, -82, 47, 232, -379], [1331, 559, -60, -356, -242, -282, 131, -127], [-575, 572, -166, 21, 64, -412, 154, 74], [392, -355, 137, -14, -119, -11, 89, 1], [838, -364, 126, 390, 220, -4, -434, 184], [-2479, 1204, -67, 318, 127, -269, 206, 94], [742, 1343, -260, 61, -48, 90, -113, -5], [1250, 57, 338, 170, -153, -50, 244, -275], [1978, 178, -94, 212, 276, -644, 136, 190], [522, -513, 134, 551, -30, 232, -409, -81], [-1318, 578, -253, 90, 130, -163, -169, 74], [-443, 258, -128, -702, -381, -66, 215, -224], [1458, -381, -188, -245, -90, 330, -109, -295], [-2493, 51, 261, -285, -217, 79, 295, -173], [848, 59, -668, -122, 107, -224, -148, 153], [788, -131, -272, 9, 454, -58, -515, 85], [1282, 795, 33, 41, 284, 145, 38, 21], [-1239, 192, 166, -602, 55, 198, -165, 464], [3808, 1146, -469, -337, -233, 311, 62, 484], [4464, -117, 656, -260, -1129, 44, 546, -929], [118, -397, 478, 325, -652, 182, 523, -355], [-934, -796, -334, 486, 18, -222, -336, 664], [3657, -1149, -238, 244, -102, -96, -19, -95], [-2359, -1138, 6, -268, -707, 136, 371, -608], [-740, -606, -31, -622, -180, 13, -169, 486], [2953, -668, -438, 227, 14, -164, 264, -498], [8057, 70, -354, -855, 126, -214, 8, -256], [-2541, -470, -148, 997, 864, -310, -360, 642], [67, -1708, -339, 353, 292, 627, -613, 148], [4184, -1366, 123, 186, -76, 105, 79, 372], [6644, 881, -329, -64, 279, 507, -439, 157], [2562, 385, -201, 96, -457, -69, -281, 655], [313, 579, 462, 284, 290, 218, 69, -123], [-1000, 1125, -273, -702, 81, -909, 277, 545], [-1995, -296, 106, -609, -146, -414, -414, 186], [2462, -1072, 193, 94, -928, 571, -83, -214], [-3332, 2543, -373, 491, 128, 107, 112, 113], [-3504, -2426, -23, 150, -540, 453, -509, -238], [990, -778, -1121, -372, 792, -409, -39, 412], [164, -300, -503, 1042, 456, -299, -123, 308], [-3504, 601, 917, 637, -456, -375, 564, 203], [3917, -144, 712, 888, -633, 226, 269, 18], [-3346, -845, 468, -419, -118, 406, 85, -65], [-1426, -208, 1399, 651, -781, 219, -120, -830], [-5868, -209, -567, -955, -132, 363, 160, -303], [7718, -1738, -605, 282, 1322, -1373, 95, 608], [1294, -2225, 403, -399, 1056, -841, 162, -237], [3027, 908, -1323, -819, 1428, -1041, -241, 514], [-1651, 226, 449, -151, -1030, 557, 267, 62], [-2756, 1032, 563, -432, 1042, 199, -187, 264], [-301, -1417, -1287, -1587, 1013, -137, -55, 15], [2140, -1551, 116, 1023, -1062, 258, -329, 45], [6273, -1532, -421, 255, -112, 217, -37, 624], [21, 2425, 521, 493, -217, -21, 145, -535], [-2017, -1926, 946, -177, -912, 2084, -142, -246], [4896, -3559, 651, 13, -1792, 1311, -250, -569], [1261, -906, -919, 783, 802, 207, -295, 750], [4854, 2447, 1009, 495, -314, -703, 502, -107], [1684, 4031, -372, -1097, -1146, 554, -427, -485], [-11862, -1634, 941, -379, -1739, 417, 974, -1220], [4308, -484, 1102, 927, -1551, 570, -223, 134], [-5962, -147, -764, 339, -206, -404, 359, -217], [1612, 1818, -35, 72, -922, 1317, 251, 454], [6110, 596, 246, -1362, -70, -806, 572, -200], [562, -2410, 151, -1883, -385, -217, 646, 140], [2017, 3112, 467, -105, 166, -1017, -217, 464], [7299, -745, -598, 258, 1340, -826, 503, -743], [5901, 2629, -637, -342, 1398, -703, 1080, -159], [3306, 304, 638, 72, 172, -1238, 266, -576], [-1382, -2583, 220, 1437, -824, 1140, -75, -813], [-8069, -2879, 708, 570, -1192, 598, 675, -1749], [-5820, 4411, -585, 1697, 2126, -47, -376, 563], [-11077, -2850, -140, 193, 344, -786, -1122, 572], [-13256, 405, -643, -666, -435, -531, -505, -1063], [-2708, 2920, -259, -1909, -1991, 807, 48, -198], [-12232, -357, 964, 1255, -198, 338, 515, 103], [2566, -297, 927, -277, -1450, 979, -1432, 233], [3226, -4941, 196, -615, -1720, 1214, -641, -717], [-2886, 4562, -616, 1330, 2102, 326, 340, 762], [-666, 878, 1082, -112, -668, 208, -1200, 390], [-3915, -3615, -154, -2394, 238, 170, -505, 1263], [1566, 3459, 324, 785, 64, -948, -1103, 909], [-13707, -161, -740, 872, 1484, -1020, -503, 669], [-2937, 6405, -31, 1462, 88, -293, 772, 77], [-4796, -3599, -270, 86, -479, 198, -1598, 407], [-13128, 1083, -775, -1189, 680, 471, -922, 1867], [-810, -2186, -79, 790, 1322, -463, -269, 1870], [-1266, 1831, -447, -423, -1498, 2013, -526, 969], [6216, 157, 844, 875, -1692, 978, -1293, -819], [-892, -869, -1207, -1867, -766, -1809, -580, -1377], [-10280, -432, -184, -412, -630, 298, 146, -992], [-2974, 1, -1416, -1051, -408, -2168, -415, 795], [6100, -2314, 1962, -1042, 6, 1118, 174, -326], [6454, 828, 1558, 436, -2334, 1050, -248, -1260], [-26320, 1728, -104, 1964, 1348, -152, 188, 1188], [-25838, 3639, -211, -585, 924, -151, -880, 1333], [-12464, 2637, -551, -751, 1736, -1557, 36, 751], [-1567, 4421, 1266, 1251, -13, 1302, 770, -1297], [-4992, 172, 1279, -1603, -1941, 897, 1912, -2550], [-1182, -797, 1581, 35, -1446, -965, -200, -2093], [350, -2851, 493, -2179, -572, -2107, 240, -821], [9264, -795, 1347, 2899, -206, 2031, -180, -1587], [-2164, -298, -209, 1214, 980, 217, -291, 92], [14367, -5537, 2020, -946, -1888, 242, 1417, -2411], [5118, 4927, 575, 1962, -1505, 1231, -455, -1239], [-628, 3669, -2552, -627, 3136, -1332, -205, 1699], [-10642, -1591, 43, 1607, 520, 1127, -450, -1419], [1286, -1485, 669, 2797, -102, -427, -714, -635], [-638, 4109, -1133, 242, 265, 1843, -863, 1283], [3142, 3360, -421, -162, -704, 411, -491, 840], [-2760, -5092, -1288, -184, 1158, 552, -234, 728], [-13185, 1743, 154, -1000, 1084, -694, 1947, 493], [32718, -1889, -813, 1107, 86, 953, -758, 371], [4731, 6029, -3619, -745, 2231, -4485, -155, 1301], [-14109, -4948, -1127, 650, 793, -109, -884, 922], [7468, -281, -433, 1937, 1140, 299, -1340, 1697], [-12707, 1937, -2368, 1022, -438, 66, -171, 459], [-12111, 428, 659, 37, -2246, 1869, 967, -1424], [-18618, -5019, 301, -1973, -544, -505, 60, -61], [4473, 181, 2702, 1582, -1858, 3238, -677, -793], [10168, 4287, 952, 2333, -720, 742, -361, 491], [18042, 40, 576, -2688, -1472, 636, -1004, -500], [2056, 3322, 744, 1780, 1882, -668, 964, -186], [17566, 1094, -1002, -486, 1526, -1850, -1782, 2642], [31449, 2030, 2146, 960, 517, 2004, 803, -1704], [-9286, -1340, -746, -252, -1036, 1306, 1446, -1456], [6357, 1448, 974, -680, 65, 2014, -395, 1964], [-15152, -7595, 1422, -961, 978, 20, -1029, 831], [10584, 3262, 73, -2009, 543, -3747, 896, 1668], [12469, -4681, -660, -792, 216, -608, 1229, -819], [8370, 6253, -509, -733, 2246, -821, 2008, 1591], [-11944, 1849, 2924, 237, -2522, 968, 1199, -3877], [-10214, 1861, -727, 2661, 2116, 497, -1028, 371], [4885, -158, 879, 33, -262, -1873, 1261, -990], [-989, 1961, 1608, 1150, -3358, 1616, -955, -1325], [4990, 7408, 500, 1077, 865, 744, -2433, 2102], [21646, 1174, 1414, 810, -1310, 1910, 186, -778], [-11900, 1049, -306, -415, 2004, -1926, 1289, -2405], [-3974, -4177, 3328, -2475, -4230, -1050, 2967, -1733], [8655, 3593, -3772, 1389, 5387, -592, -1322, 2587], [4008, 5546, -1245, 670, -1192, 2861, 501, -760], [12050, 2981, -1476, -31, 1032, 1292, -907, 3403], [-10884, 6024, -3045, 1075, -655, 27, -946, 894], [-15584, -6079, -1989, 547, 3526, -2249, 1740, 1839], [5930, -1219, 796, 427, 1874, -318, -2363, 4277], [29894, -4241, -2554, -853, 418, 1644, 239, 1471], [2404, 10110, 878, 424, -814, -1022, 3614, -750], [10065, -4981, -273, 1980, -398, 2049, -514, -1881], [-7524, -1013, -1130, 206, 2411, 86, -2066, 3353], [16383, -2448, 2484, -1389, -2260, 2284, 290, -2492], [-16942, -4131, 810, -1529, -2132, -2946, 625, -2005], [-39471, -6209, 1454, 2470, -258, 2934, -535, -1927], [10048, 2455, -792, -1325, -840, -3062, 731, 2007], [13022, 1649, -172, 2378, 1183, 504, 350, -303], [6545, 1740, 154, 1067, -670, 14, 530, -398], [1624, -1806, -264, 6576, 2126, -248, -948, -382], [925, -1848, -60, -3868, -3211, -1992, 735, -284], [6094, 1187, -2872, 497, 5132, -2058, -2801, 4347], [11983, -4438, 1509, -4493, 180, 227, -933, 1462], [2217, 6953, -1417, 2232, -730, -1675, -1356, 2583], [-32056, 6361, -1360, -3211, -146, 80, -253, 1535], [-5554, -7053, -1304, 2652, 91, -1608, -3116, 3343], [13525, 1286, -1366, -156, 2949, -1412, 2851, 3904], [15990, 6063, -3768, 126, 5, 1456, 2314, -691], [17606, -1476, -4266, -6566, 1356, -424, 62, 3712], [-196, -7932, 2333, -146, -1384, -3053, 1075, -1390], [-1791, 4865, -722, -4606, -1230, 1352, -1893, 655], [3625, -9781, 644, -371, -2625, -2276, -1176, -1427], [-5156, -5849, 1914, -1503, -2698, 666, 1151, -2999], [-14392, -1358, 1640, 2610, 1480, 3828, -990, -234], [-6348, -414, 643, -2909, -833, 4057, 1874, -1460], [14462, 2125, 2752, -1133, -628, 3840, 395, -2789], [8785, 12015, -36, 1343, -2107, 5576, 282, -203], [-17059, -7196, -3791, -4109, 1144, 939, -2229, 1622], [-278, 7423, -3037, -3704, 2485, -1565, 3275, 2389], [25666, 2971, 2073, -3891, -3072, -2523, 2702, -3581], [-12144, -1052, 5118, 1374, -3672, 1300, -1498, 1364], [-23812, -2093, 1970, 30, 915, 2868, 1300, 921], [20070, 1371, 3015, 541, -4490, 3559, -322, -1099], 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