
# q-expansion of newform 405.4.a.m, downloaded from the LMFDB on 19 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 = 405
weight = 4
poly_data = [288, -1200, -460, 479, 74, -44, -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], 1], [[0, 1, 0, 0, 0, 0, 0], 1], [[-13, 0, 1, 0, 0, 0, 0], 1], [[-1, -23, 0, 1, 0, 0, 0], 1], [[-240, -114, 310, 35, -39, -1, 1], 12], [[672, 756, -397, -140, 42, 4, -1], 24], [[-1104, 732, 595, -82, -48, 2, 1], 24]]

hecke_ring_character_values = None
aps_data = [[0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [-5, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, -1, 0], [-2, -3, 0, 0, -1, 0, 0], [15, -4, 1, 0, -1, -1, 1], [-22, 1, -2, 2, -1, -2, 2], [40, 3, 2, 2, -1, 0, 0], [-11, -6, -2, 2, 0, -1, 0], [46, -15, 5, 0, 0, 1, -3], [37, -8, -9, -4, -1, -1, 1], [61, -14, -11, 0, 5, -1, -1], [8, -4, 5, -2, -1, 9, 1], [82, -29, -12, -2, 5, 0, 0], [-30, 4, 9, -2, 5, -8, -1], [-74, 4, -24, -4, 4, -4, -10], [-1, 53, 1, 10, 4, -3, 13], [95, 101, -20, 2, -7, -8, 6], [171, -21, -12, 8, -7, 5, 0], [-89, 144, 3, 0, -1, -7, -7], [293, 29, -3, 10, 4, 3, -3], [322, -58, -6, -6, -8, 16, 16], [-223, 138, -14, 0, 4, 3, 8], [406, 133, 13, -2, -20, 19, 13], [307, -175, 37, 22, 8, 7, 3], [340, 126, 22, 4, 4, -22, 8], [195, 160, -19, 16, -15, -1, 33], [-400, 73, -17, -14, -10, -4, 13], [-21, -119, 6, -14, 11, -18, 2], [394, 50, -62, -28, -4, -10, -16], [346, -366, 1, 14, 3, 8, -17], [68, 204, 16, 22, -4, -30, 14], [-339, 55, 121, 18, -34, 15, 7], [-105, -257, -17, 26, -4, 37, -5], [263, -39, -74, 6, -9, -2, 2], [243, 28, 85, -10, 25, 51, -31], [-22, -26, 12, -48, 26, 8, -38], [-311, 396, 27, -4, 31, 53, -21], [-428, -158, 93, -20, 43, 72, -23], [372, 98, -72, 36, 38, -20, 36], [-139, -154, -59, 4, 3, 51, 3], [-52, -167, -15, -8, -60, -79, 61], [-281, -458, 59, -54, 13, -9, -21], [708, -369, 166, 14, 21, -22, -42], [-192, -40, 96, 44, 60, -20, -10], [659, 34, 101, 56, -1, -39, -5], [971, 474, 113, -70, 43, 23, -19], [1392, -46, 75, -80, 29, 8, -17], [-1280, 357, -128, 32, -61, -96, -6], [642, -237, 89, 32, 10, -107, 41], [-10, 59, 130, -50, 13, -2, 8], [672, 218, -46, -6, 64, 86, -56], [-272, 836, -87, -18, -5, 5, -49], [-2, -51, -298, -92, -15, -12, 22], [-2391, 495, 111, 62, 24, -123, -53], [771, 244, 13, -4, -79, 19, -63], [2165, 71, -24, -38, 111, -64, -48], [1765, 350, -165, -96, -43, 47, -15], [719, 108, -213, -52, -31, 1, -17], [2531, -687, -4, -80, -15, 70, -6], [2219, -58, -58, -78, -28, 115, -38], [-1563, -36, -55, -56, 15, -49, -19], [1058, 155, -313, -84, 40, 8, -89], [-1111, -700, 133, -78, -43, -71, 73], [824, 227, -186, 62, -119, 118, 36], [-1545, -696, 45, -180, 75, 67, -53], [4264, 176, 46, 112, 86, 20, -14], [771, 665, -325, 58, 106, 141, -55], [-1890, -851, 464, -18, 99, 108, -80], [408, -593, 27, -96, -32, 119, -11], [-4227, 133, -143, 54, 36, 15, -31], [1699, -274, -653, 54, -27, -33, 15], [3601, 56, 251, -32, -53, -51, 159], [594, 2200, 20, 32, -12, 4, -146], [2259, -823, -393, -112, 102, -77, -51], [1713, -216, -541, 0, -1, -55, 83], [5111, -243, 26, 134, -49, -122, 148], [1100, -18, 354, -60, 188, 14, -74], [-371, -649, -277, 14, -148, -47, 165], [-2645, -1109, -175, -94, -94, 327, -75], [1307, -498, 397, -66, -29, -5, 9], [-2436, -168, -202, 52, -206, 2, -38], [4002, -820, 14, 98, -30, -228, -16], [-5158, 285, -418, -282, -17, 118, -6], [3748, -910, 446, -22, -112, -74, 112], [2803, -1607, -220, -146, 119, 179, -118], [-757, -2029, 253, 126, -110, -141, 179], [-3070, 481, 58, -218, 199, 254, 4], [-2438, -183, 799, -60, 210, 347, -37], [-1873, 592, -781, 98, -125, -319, 257], [3820, -199, 18, -6, -131, 138, 74], [2208, 286, -248, -154, 186, -54, -342], [2591, 814, 851, 42, -71, 1, -99], [10631, 255, 117, -92, 34, -33, -85], [4183, 233, 421, 112, -80, -237, -77], [-7262, 670, 281, 24, 67, 102, 21], [-2708, 711, 23, -52, -96, -493, 41], [1645, 1134, -474, 0, -36, 254, -48], [5456, 1806, 523, -126, -11, 180, -309], [-4938, -339, 151, 328, -62, 99, 105], [-3460, -2139, -391, -92, 72, -2, 1], [-1532, 1074, 222, 20, 32, -398, 62], [-3727, 619, 274, -14, 151, 581, -36], [-6975, -21, -101, -102, -144, -299, -165], [-1895, -15, 1009, 118, -252, -153, 165], [-11599, 1251, -209, 34, 240, 277, -127], [7420, 857, 77, 44, 8, 250, 45], [-7007, 1624, -715, 116, -1, -381, 47], [2214, -1140, -666, 102, 278, -176, 180], [-5047, -1819, 699, -302, 198, 361, -405], [-2194, -1632, -99, -206, 17, 488, -141], [-1699, -892, 701, -156, -61, -61, 125], [6093, -1371, 229, -218, 68, -41, -221], [-1082, -717, -156, 2, -323, 278, 282], [-5500, 3338, -106, -90, -144, 2, 156], [324, 540, -19, -66, 99, 665, 3], [1545, 1453, -286, 74, 297, 213, 0], [4908, -278, -267, 52, 115, 316, -107], [2779, 3884, 347, 12, -351, 245, 193], [2490, 344, -252, 134, 132, -84, -38], [-3804, 1846, -30, 64, -240, -334, -94], [-10699, 2450, 439, 416, 183, 165, -171], [-1009, 1872, -191, 112, 319, -585, 7], [-2540, 1215, 782, 10, 391, -414, -70], [-1386, 3040, -238, -210, -18, 124, -60], [681, -1323, -102, 278, 83, 358, -68], [-2790, -6027, 561, 164, -36, -159, 293], [-6222, 460, 932, 180, -24, 130, -76], [6620, -244, -593, 378, -145, -46, -219], [-784, 1102, -82, -156, 388, 46, -282], [6573, -1623, 795, -66, -326, 315, 359], [-16249, -2190, -474, 344, 84, -245, -170], [4839, 2050, -161, 112, 293, 585, -23], [-2733, -3450, -379, 128, -95, -265, 189], [7352, -741, -775, -558, 722, 187, -393], [-5136, -1039, -65, 250, -134, 33, 705], [-8030, -474, 406, -252, -388, 170, -26], [7605, -1484, 867, -92, 11, 437, -101], [201, -696, -897, -500, 277, 289, -251], [-3731, 1523, 1003, -378, 492, 325, 37], [13207, -1645, -761, 76, 76, 263, 45], [5795, 1271, -828, 78, 47, -216, -266], [-2232, 5084, 1, 406, -287, -570, 399], [9234, -778, -863, 284, 371, 112, 27], [-5023, 613, -544, -154, 177, -496, -482], [-4963, -2560, 1751, 528, 115, 339, 349], [227, 920, -1723, -280, -113, -473, 271], [19797, 378, -25, 260, 155, 25, -421], [10756, -2393, 960, 598, 349, -276, 146], [-6228, -2438, 843, -32, 181, 912, -153], [-2929, 638, -1639, 204, 17, -257, -191], [10197, -727, 678, -264, -217, -324, 554], [4753, 1487, 122, -490, -173, 113, 196], [4845, 5078, -617, -20, -407, 785, -29], [11445, 1927, 1444, 50, 37, -43, -62], [13702, -4366, -180, 184, -534, -494, 376], [14320, -5926, -628, -118, 150, 190, 38], [-19380, 2396, -1730, -776, 166, 330, 158], [-4978, -1850, -1002, 456, -592, -378, 212], [10636, -2003, 1369, 184, -120, -31, 829], [7125, 169, 1014, -364, -239, 107, 334], [-15542, 951, -148, -66, -501, -40, 490], [11747, 1350, 1122, -66, 588, -399, -588], [10877, -1598, 787, -136, 677, 815, -123], [22660, 1699, -94, 710, 217, -638, 560], [-17723, -96, 158, 178, 738, -177, -308], [-1380, 2432, 934, 88, 26, 264, 250], [-15738, -2712, -884, -132, 112, 516, 478], [-5758, 4729, -195, -170, 100, 1019, -817], [-18595, -5908, -115, -28, -237, 399, 931], [9659, 1843, 2077, -516, 162, -493, 171], [-15708, 4181, -1207, -80, -252, 693, -557], [9607, 1772, 1305, -418, 165, -169, 25], [-4949, -3567, 499, -490, -446, 241, -191], [4818, -734, 1846, -508, 28, 1038, -598], [428, 984, -186, -316, 918, 626, -772], [25597, -2467, 291, -180, 118, 769, 109], [-4366, 2278, -944, -340, -810, 596, 70], [5369, 2528, -1742, 894, 190, -445, 740], [-13469, 2161, 1370, -222, 307, 1010, -96], [-3667, -844, -1289, -36, 655, 173, -85], [4608, 955, 1104, -192, -783, 246, 12], [-11061, -538, 1475, 340, 191, -243, -357], [30698, 2657, -620, 478, -111, -796, 272], [-6824, -3580, -515, -526, 733, 894, -1189], [-2900, -10095, 967, 896, 126, -509, 77], [-15330, 7070, 1814, -96, -16, 578, -6], [15737, 2782, 466, 340, 8, -1729, 368], [-9080, 5862, -897, -242, -721, -409, 971], [17670, -3770, -480, -36, 182, 58, 456], [-11918, -4339, 724, 342, -371, 912, -178], [-16437, -61, -1422, -20, 335, 945, 70], [11980, 1031, 1060, -332, 321, -1378, -892], [22829, -3969, 1454, -186, 473, 726, -574], [3281, -3233, 2036, 678, -431, 489, 118], [-14914, -4403, 244, -1182, 457, -612, 248], [4416, 2590, 853, 618, 285, 69, 555], [-15916, 1718, 1264, 20, -422, 768, -206], [8488, -197, -2492, -410, -9, -116, -402], [-36399, -4144, 362, -766, 738, 31, -216], [-22952, 2390, -2174, -768, -388, -402, 14], [-7275, -1058, 2001, 424, 199, 211, 431], [-8852, 3396, 908, 96, 72, 308, -64], [-10081, 286, 1950, -540, 1160, 370, 372], [31025, -2729, -1343, -186, -754, -181, -127], [-5011, -4732, 379, -128, 225, 781, -1025], [-12809, 6882, -243, -134, -901, -733, 1385], [-13344, -849, 671, 396, -856, -830, -17], [8941, 3266, -1012, 680, -1086, 472, 280], [13719, 8147, -2167, 524, -482, 523, 263], [9803, 3617, -1291, -830, -248, -585, -575], [23114, 1732, 1860, 68, -212, 512, -72], [14144, -9050, -1181, 108, 721, -392, 55], [-12249, 32, 2833, 56, -639, -737, 517], [6989, -1756, 1007, -318, -225, 53, 1355], [17084, -2956, -159, 1010, 455, -491, 19], [-3033, -2498, 1178, 1658, 604, -287, 308], [-22699, -961, -841, 1134, -368, -943, 363], [11820, 8078, 1999, -110, -431, 1362, -823], [118, -4098, -2318, 244, -1232, 14, 260], [-1185, -6451, -208, -66, -43, 0, -1166], [26851, 6748, 421, -758, 809, -577, -1319], [-6623, -6268, 780, -448, 1080, 1556, -1436], [12871, -6326, -2746, -536, -112, 215, -1122], [-12813, 4585, -3532, -520, -429, 209, 748], [19506, 13402, -424, 436, -810, -400, 990], [-4171, 9864, 641, -1308, -25, -1253, 327], [16435, -2932, 163, 724, -881, -803, 565], [-21294, 4558, -3133, -228, 153, -2054, 675], [14797, 7670, -3509, -996, 133, -297, -551], [-7254, 4980, 360, -852, 1876, 404, -194], [13851, 5090, -727, 10, -689, -1149, 1481], [19735, -9780, -3537, -80, 251, -511, 689], [23038, 3978, 79, -850, 659, -449, -1079], [7866, -6582, 903, -1272, 1529, 208, -947], [-16935, 4338, -2302, -332, 28, 749, 210], [24999, -3153, 2130, 852, -459, -628, 282], [-38123, 11512, 907, 472, 113, 953, -765], [24422, 2391, -2204, -14, 353, -1590, 518], [6671, -6494, 93, -294, 423, -65, 261], [-11072, 3557, 825, 230, -1580, 284, 1079], [27901, -5374, 2089, 696, 339, 2063, -753], [18166, -4454, -1979, -576, -1125, 142, 1205], [12114, 4517, -1399, 504, 1268, 1865, -507], [-13336, 899, -1300, -2, 939, 524, -1004], [48959, 3806, 203, -2022, 1149, -65, -1709], [17481, 11236, -3108, -722, -856, 735, 494], [-11486, 4333, -132, 738, 327, 1646, -426], [6613, 4098, -611, 1546, -149, 1041, 205], [41829, 870, -1200, 400, 746, 149, 516], [10501, -2612, 1833, -1492, -353, 1503, -485], [-2195, 9848, 2694, -836, 814, 1702, -124], [-32765, 4398, 143, 1306, 1285, -1587, -691], [25953, 3075, 3584, -332, 611, -626, -1076], [-1930, -7132, 3234, -568, 1150, 2206, -1462], [19133, 13139, -2063, -686, -762, -785, 661], [18704, -4008, -1082, -840, 306, 1542, 344], [19416, 1830, 5073, -770, 747, 1264, -383], [34023, -3698, -1709, -1692, 1855, 1197, -1467], [-5164, -5263, 740, -62, -487, 1536, 872], [10816, -6174, 2035, 2020, 33, 944, 239], [-5663, -9241, 3834, -168, -1101, -1389, -530], [-886, 9404, 620, -92, 180, 660, 246], [-2418, -1244, 796, 1196, 664, -940, 1786], [13667, 4722, 1725, 516, 1417, 143, -1771], [3277, 5855, -3763, -1420, 1230, 1141, -901], [-34750, -8000, 2300, 2184, 1228, -1360, -330], [41519, 8704, -92, 1216, -276, -360, 426], [21873, 4019, 486, 702, -1705, 425, 1004], [-20234, 5798, 36, 888, 50, 2236, 2050], [9260, -3368, 434, 424, 442, -530, -1456], [-21005, 4358, -3132, 2030, -222, -1873, -316], [-50574, 575, 2370, -474, -1011, -322, 90], [-12676, -3022, 550, -1052, 624, 1680, 118], [7185, 10593, 193, -158, 892, -597, -1731], [-52398, -2706, 4261, -592, -549, -526, 69], [-3146, -2850, 4683, -1462, 353, 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