# q-expansion of newform 1160.4.a.c, downloaded from the LMFDB on 30 April 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 = 1160
weight = 4
poly_data = [-6912, -34400, 836, 4100, 38, -121, -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, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0], 1], [[-446310, -140258, 34126, 13747, -716, -182, 7], 26514], [[34668, -998696, -122864, 44266, 4825, -407, -41], 53028], [[546156, -288920, 56656, 28678, -5573, -425, 73], 53028], [[-182412, -160496, 68780, 9466, -2563, -107, 23], 17676], [[-783396, -178172, 86456, 9466, -2563, -107, 23], 17676]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-5, 0, 0, 0, 0, 0, 0], [9, 0, 0, 0, 0, -1, 0], [-1, -3, 0, 0, -1, 1, -1], [-20, -2, 2, -1, 0, 1, 0], [-24, 0, -1, 3, 1, 0, -1], [3, 1, -2, -2, 0, 4, -1], [41, -5, 2, -1, 5, 4, -1], [-29, 0, 0, 0, 0, 0, 0], [-7, -10, -4, 0, 1, -6, 0], [-52, -21, -3, 3, -4, 8, 4], [56, -26, 8, -8, -9, 1, 6], [-100, 8, 12, -3, -10, 4, -9], [-16, -7, 4, 9, -9, -18, -10], [-2, -35, -6, -6, 6, -5, 17], [-74, 20, -20, -7, 8, 5, 8], [94, -6, -10, 9, 6, -1, 8], [81, 7, -4, 16, 28, -16, 1], [22, 46, -14, -14, -5, -25, 12], [-110, -54, 31, 17, -16, -3, -17], [-85, -1, 0, 35, 6, 29, -19], [-253, 107, 2, -30, -9, -5, 13], [188, -82, -20, -2, 18, 12, 38], [-444, 1, -25, 8, 18, -17, -16], [72, -50, -6, 19, 29, -8, 8], [-593, 147, -50, 18, 22, 2, 23], [-213, 9, 40, -36, 17, -31, -3], [-682, -10, -36, 6, -17, -1, 8], [-718, 43, 15, 20, -28, -7, -6], [76, 75, -30, -33, -41, -2, 38], [-307, -27, 16, -6, -24, 16, 27], [304, 101, -25, -22, -13, 46, 38], [-794, 33, 2, -48, 39, 54, 21], [84, -48, 94, -56, 6, 4, -26], [80, -34, 44, -6, -61, 105, 18], [-564, -85, -37, -13, 43, 67, 16], [-1638, 19, 38, 61, -1, -24, -104], [295, 74, 36, 6, -67, -6, 20], [678, -2, 80, -1, 42, 77, -10], [-408, 70, -94, 61, 59, -42, 56], [-76, -2, 34, -64, -23, 17, 30], [-1451, 214, -80, 16, 26, -91, -48], [258, -69, 39, 80, -81, 0, -32], [430, 181, 124, -76, -61, -32, -1], [-762, 136, 62, -6, -21, 65, -30], [-2087, -273, 24, -88, -41, 5, 5], [-277, 107, 88, -95, -115, -52, 17], [-2541, 289, -32, 96, -46, -104, -83], [548, 113, 44, 90, 25, -82, 99], [-260, 300, -170, -10, -10, -94, 62], [44, 264, -108, -34, 133, 113, 84], [-232, -404, 136, -41, 53, -34, -80], [-291, -521, -18, 96, 159, 21, -81], [-1512, -410, 32, 72, 7, 77, -54], [144, 9, 256, 53, -25, -98, -226], [814, 261, 28, 34, 120, 1, -113], [-1353, -485, -66, -84, -55, 227, 5], [-1316, 146, -70, -26, 67, 39, 24], [1468, -193, -20, 8, 68, -59, -5], [-2449, 245, -20, -110, 10, -76, -5], [502, 41, 153, -71, -165, 197, 24], [-988, -89, -80, 111, 43, 180, 142], [529, -103, -176, 203, 37, -188, -99], [-750, 764, -90, 58, 39, -1, 62], [-416, 109, -179, 297, 166, 146, -118], [-2393, -539, 14, -30, -146, -36, -139], [-736, -202, -45, 149, 176, 51, 151], [-935, 681, -152, -82, 103, 89, 243], [1514, 74, 156, 42, -240, -60, -158], [60, 4, 82, -132, -77, -377, 174], [4459, -3, -68, -31, -149, -120, -109], [-1450, 25, 148, -15, 65, 184, -296], [-5580, 301, 150, -250, -9, -20, 9], [-845, -11, -6, 84, 221, -187, 75], [3051, 830, 146, -124, -134, -25, 10], [500, -80, 96, -172, -81, 207, -76], [4134, 687, -88, -312, -207, 48, 301], [1830, 128, -182, 343, -130, 285, -120], [1000, 24, 86, 192, -199, 125, -114], [-2232, 109, -48, -22, -148, 101, -397], [2858, 546, 44, 304, 189, -275, 146], [-2104, 148, -392, 234, 398, 100, 276], [44, 119, -91, -71, 418, 238, 306], [2910, -30, -152, -62, -245, -197, 22], [-1100, 537, 250, -34, -308, 68, -404], [3740, -704, 30, 10, -255, 367, -450], [1572, 194, 30, 86, -263, -199, -292], [176, 1483, 194, -192, -1, -68, -221], [-3857, 1171, -396, 366, 228, 34, 159], [-2628, -640, 250, -7, 88, 352, -137], [7119, 1264, -106, 132, -32, 281, -114], [-725, 1119, 382, -39, 174, -57, -473], [-903, -1261, 560, -124, -158, -186, 17], [2502, -150, -128, -97, 499, 318, 214], [3858, -451, -40, -301, 337, -170, -40], [9934, 1118, -60, 114, -235, 257, -316], [488, -1369, 440, -400, -318, 71, 107], [679, 577, 258, 374, -126, -534, -525], [2886, 391, 188, -142, -291, -442, -291], [-3619, 561, 270, -322, 147, 263, 151], [-2214, -818, 202, 285, -139, 396, -404], [106, 339, -448, -280, 284, 260, 274], [4894, 1368, -584, 122, 347, 343, 156], [-566, 1354, -184, 143, 233, -500, 254], [6278, -322, 155, -37, 589, 436, 293], [-7995, -1251, 750, -18, -810, -344, -81], [-2898, -78, -12, 394, -497, -291, -70], [10257, 944, -790, 216, 192, 455, -82], [1168, 1152, -192, -326, -123, -429, -48], [-1140, 497, 100, -427, 23, -138, 54], [-3972, 1107, -220, -196, -221, -32, 393], [-3796, -1206, 331, -359, -296, 23, 107], [7553, 339, -86, 246, -149, -479, 429], [-4252, -908, 394, -440, 436, -196, -34], [4416, 1982, 168, -372, -317, 509, 114], [2743, -1581, 308, -72, -454, 92, -391], [-1225, 1427, -258, -196, 735, -399, -315], [-2272, 113, 229, -25, -481, -451, 124], [-2357, -1109, 982, -140, -456, -284, -411], [-138, 444, 124, 316, -487, -939, -262], [-6232, -700, -728, 92, 412, 698, 380], [3170, -2063, 301, 43, 57, -247, -4], [-4005, -1027, 102, 202, 480, 120, -213], [-4310, 251, 334, -560, 312, 151, -369], [3198, 1244, 300, -316, -219, 805, 486], [-1254, 520, -340, 48, -189, 107, -42], [8854, 750, 318, 36, -683, -737, 196], [10980, 783, -246, 641, -427, -284, 100], [-13570, -2183, -195, 273, -92, -174, -46], [4575, -967, 874, -700, -224, -186, 7], [5956, -427, -154, -535, -213, 404, 182], [-273, 949, -1046, 44, 910, 580, 315], [-3590, -293, 191, 495, 195, -1043, -620], [-2068, 573, -80, 746, 169, -642, -549], [-794, -1998, 282, -512, 278, 864, 16], [3254, -867, -629, 285, 114, -620, 704], [1603, 825, -746, 312, 257, -321, -339], [5990, -2881, -881, 557, -83, -159, -164], [3480, -672, -312, 42, -632, -936, 972], [-5468, -1317, -40, -306, -327, -314, -175], [-5692, 1624, -650, 228, 552, -730, -258], [10242, -185, 178, -295, -527, -1332, 190], [-16436, 620, 1188, -321, -430, 392, -311], [-10622, 961, -180, 594, -132, -595, -837], [9707, -2093, -336, 952, 385, -435, 209], [-21410, -1831, -143, -289, -290, 502, 130], [5756, 652, 968, -254, -481, -627, 264], [15245, 2549, 74, -46, -35, 161, -25], [-2513, -2363, 930, 127, -622, 1065, -373], [-8878, 360, -890, -439, -254, -687, 922], [282, 3094, -178, 478, 221, -329, -760], [-10083, 1055, -1310, 1084, 1015, 45, 403], [2534, -1481, 404, -967, 201, 1096, 744], [-12776, -2788, 236, 87, -184, -1418, -351], [-11703, 479, 830, 149, 193, -676, -767], [11400, 190, 644, 0, 731, 263, -762], [-4746, -107, 486, 436, 55, -628, 93], [-14250, 276, 208, 284, 447, 1313, 604], [-2698, 2998, 124, -850, 592, -372, 422], [-4164, 1675, -1206, -342, 836, 844, 504], [-5946, 3116, 788, 64, -258, -326, 448], [-7266, -3963, 242, 25, 53, -580, 206], [-17555, 1165, 350, -340, -1142, -380, -23], [-4868, 492, 512, 358, -489, -1115, -172], [832, -1297, -2054, 803, 129, 662, 600], [-1020, 2240, -1588, 696, 871, -295, -240], [-6832, -1315, 337, 619, -429, -359, -260], [1324, 2535, -516, -86, 413, 928, -787], [-20580, 371, -1373, 423, -628, -1488, 322], [7634, -1396, -1308, 456, 295, -181, -28], [-13282, -2062, 664, -804, -290, 1276, 142], [-2830, 2706, 520, -48, -359, 619, -1698], [-28042, 3149, -59, -1442, -45, -88, 304], [-8034, -846, -218, 102, 449, -1387, 304], [4148, -3413, 1056, 836, -294, -825, -1333], [-6166, 1417, -1702, -872, 1028, 565, 1821], [-3038, -2446, -180, -457, 225, 710, 1250], [2678, -2339, 986, -353, -821, -1502, -544], [1434, 52, -1256, -512, 1030, 1038, 116], [4146, -935, 848, -323, 1181, 1310, -586], [-7513, -815, -1730, 462, -384, -122, 629], [-21184, -1277, 1638, 170, 439, 74, -249], [-7686, -1198, -290, -700, -367, -847, 484], [4911, 1038, -672, 404, -809, -1268, 1202], [19562, -2310, 676, 346, -252, -468, 786], [-29892, 319, 813, 211, -130, 542, -1396], [-8252, -245, 204, -1648, 326, -634, 308], [-16138, 1186, 552, 594, 303, -651, -98], [3366, 1468, 910, 102, 320, 372, -1434], [-25724, 544, 620, 318, 400, -1896, 464], [1978, 3804, 138, -757, -446, 176, -591], [-24555, -2459, 122, -640, 368, -526, 61], [-5412, 3473, -2184, 982, 800, 883, 1435], [-28094, -501, 150, -918, 1240, 40, 938], [-18068, 2268, -536, -424, -1272, -302, 80], [-4662, 1824, -490, -592, 853, -1565, 1822], [-23648, -3016, 1694, 411, -2119, 798, -314], [16216, 230, -1648, 1728, -922, -482, -658], [13653, -3935, -364, -277, -152, 837, -1263], [-310, 4176, -702, 223, 1189, -928, 746], [330, -2362, 1536, 488, -1894, 586, -82], [16394, -1501, -967, -23, 1596, -236, 1612], [8830, 180, 1370, -1082, -1732, -918, 142], [14141, 2747, 530, 384, 174, -1632, -319], [968, 513, 982, -510, 667, 3108, -491], [-5713, -415, 2608, -1191, -1019, 1150, -605], [-19369, -525, -1118, 520, 340, -812, -247], [9440, 224, 1600, 149, 214, -2091, -1374], [-27077, -717, 2702, -930, 395, 489, -589], [-7850, 3455, 1011, 779, 620, 1712, -680], [18724, -4936, -178, 42, -1204, 2164, -786], [8482, 2057, -56, 81, -253, 272, -950], [-4684, -707, 644, -187, -59, 48, 1070], [-1449, -6329, -664, -753, 1494, 921, 1577], [15230, 2484, 262, 1061, -61, -244, 234], [-14395, -6569, -1562, 1261, -576, -535, -511], [15140, -5770, -202, -686, 508, 1526, -212], [16095, -6709, 262, -836, 225, 1743, 453], [26286, -2211, -1193, -1531, -135, -965, 146], [588, -4425, -100, 1898, 356, -3083, -999], [6524, -600, 102, -370, 880, 1594, -246], [46474, 3420, 186, -1004, -658, 1358, 510], [-4304, 5011, -200, -445, 407, 1158, -1740], [11851, -3045, -1158, 2, -1253, -207, 71], [2270, -6972, -772, -437, 442, 963, -220], [18936, 3511, 685, -1803, 532, 1964, 1642], [-20369, -3551, -1122, 1769, 1298, 397, -1399], [20792, -3825, 28, 495, -711, 1088, -2032], [-43674, -3395, -626, 872, -454, -505, -875], [10188, 3555, 805, -921, 604, -1536, 2672], [-30714, 1742, -1466, 941, 1862, -1407, 2280], [-1293, 1937, -596, -1934, -2193, 1597, 421], [18486, -1116, 1062, 86, -1061, -953, -830], [12763, -5405, -308, -1034, -1185, 2697, -351], [-2432, -3057, 1692, 893, -1127, -800, -3242], [23688, -1528, -1134, 2240, -655, -2823, -786], [-9796, -7897, -893, 1221, 641, 51, 456], [1566, 3935, 2, -664, 1057, 2270, -545], [-1617, 1455, -3636, 1734, 2970, 44, 771], [-30654, 1796, 1500, 1321, -2168, -1448, -2393], [-9182, 7070, -730, 403, 936, 1533, 104], [47883, -8968, 934, 264, -322, 1401, -1568], [-8832, -5516, 1388, -427, -1905, -2522, 1296], [16912, 8186, -774, 2032, 1466, -1670, -1180], [-19798, -1776, 918, -1764, -172, 3546, 202], [-1515, -8320, 424, -2412, -192, -89, -568], [-10868, -2885, 1514, -872, -262, 1107, -413], [-41852, 2941, -1328, -334, 564, 849, -829], [24876, 6287, 648, 127, -1531, 436, -808], [29478, -1425, -1487, 187, 656, -470, -700], [32626, -8948, 740, 1248, 76, 16, -1268], [8688, -4105, -88, 371, 241, -818, -306], [-32630, -4534, -2184, 152, 209, 2593, 262], [8936, -6031, 1667, 905, 430, 2818, -1348], [-17172, -5019, 2760, 1146, 315, 1250, 747], [3246, -5678, 800, -2985, -115, -1562, -338], [6334, 3825, -216, -1984, 292, -1844, 230], [33410, -1238, -814, 39, 27, -1948, 2900], [4040, 304, 592, 2198, 1840, -1064, 304], [32614, -9355, 636, 931, -163, 1788, -966], [-17714, 237, -2896, 1068, -916, 508, 690], [21702, 6122, -2368, 632, 190, -1896, 190], [-10934, 1923, -1105, -3065, 50, 2820, 2154], [57238, 859, -1521, -1071, 1435, -1131, 1918], [-28499, 2037, -3952, 2390, 735, -917, -1019], [9110, -5131, 2166, -1120, -2401, -1440, -123], [47544, -2870, -2098, -658, 780, -202, -996], [9734, 6477, -388, -2622, -3284, 10, 1802], [-13734, 926, 2426, -1199, -2331, 1284, -2448], [20246, -130, 140, -592, 1484, 2250, -1946], [-16047, -10563, -302, 1514, -70, -980, -381], [-34486, 9704, -286, 424, -902, -1682, 1558], 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