# q-expansion of newform 1000.4.a.e, downloaded from the LMFDB on 21 June 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 = 1000
weight = 4
poly_data = [491, -5011, 12092, -1260, -8606, 1268, 1728, -17, -81, 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], [[483990018995724, -1296304299588845, -842003217059652, 591757916623593, 376295159418721, -3261100005150, -19796634603939, -537238497589, 254880482293, 8448418261], 13437186651284], [[-116891554368, 533838931380, -95070546510, -369795367065, 55673301455, 68640103940, -1202421465, -3102755360, 8280650, 37662805], 6447786301], [[324134012830974, -1442527568304079, 298489563148662, 997964816260497, -178154759871273, -205005577036204, 3177061584045, 9655201328695, -13510170967, -119408235789], 13437186651284], [[172213359178273, 536429741867342, -1227981080582185, -709285206047063, 593359418683964, 220479186917615, -27285450001829, -10142321149135, 337569845345, 121955932886], 13437186651284], [[127136826453869, 1110644944700915, -654079711026193, -852990619014132, 244827884253663, 223730439008789, -3244946477954, -11363090839802, -3865770664, 144960268225], 13437186651284], [[-41240082538025, -1184277212788558, 1122735454766241, 1016936343388819, -438644196450252, -268752832499355, 13210428711597, 13200591029555, -124798226137, -165549826026], 13437186651284], [[1193354356569341, -2620286943261028, 63814308155247, 1850778219316427, -203577357526330, -372943648087173, -1173853522419, 17790774353911, 69211818739, -221717266752], 13437186651284], [[-954198766579727, 3713693663149585, -1711882367759621, -2408159354012774, 874996964213617, 518662300254985, -35119242038924, -23345448604056, 409591047718, 280742230555], 13437186651284], [[1352064032382495, -7014374492853396, 2259013998356409, 5325134821473089, -1124087211857930, -1158937842005631, 21319804990887, 55405108838657, -104759826107, -689236045764], 13437186651284]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, -1], [1, 0, 0, 0, 0, 0, -1, 0, 0, 1], [-9, 0, -1, -1, 1, -1, 1, 0, 0, -1], [-16, 0, 1, -2, -1, -1, 1, -1, -1, 1], [3, 0, 1, -2, -2, -3, -1, 4, 1, -1], [-16, 0, 2, 2, 0, 0, 0, -1, 2, -2], [21, -1, -3, -15, 3, 2, 2, 1, -2, 2], [-31, 2, 0, -16, 2, 0, 1, -6, -2, 5], [-49, 0, -10, -11, 2, 0, 0, -3, 5, 2], [29, 5, 3, -19, -7, -6, -2, 7, 4, -4], [-76, 1, 20, 8, -2, 6, -2, -8, -8, 8], [-50, -2, -30, 15, 10, 10, 2, -2, -2, 4], [-20, -2, 24, -34, 0, 12, 0, 0, 0, 8], [-9, -10, 2, 2, -6, 10, -5, -2, -2, 9], [25, 2, -21, -34, 10, 14, -2, -10, -6, -10], [-227, -3, -16, 17, -16, -8, 8, 11, 6, -11], [-93, 4, 29, -6, -14, -3, -1, 0, 1, -1], [-137, -10, 37, -27, -5, -3, 7, 10, 6, 5], [-152, 10, -23, -42, 0, -11, 4, 14, 7, -32], [-328, 2, 20, 52, 18, 10, 2, -19, 2, 14], [73, -16, 33, -28, 0, -4, -10, -12, -8, 26], [-503, 18, -97, -49, 1, -9, 13, -2, 6, -25], [167, 22, -26, 35, -10, -1, 6, -2, 0, -3], [-263, 23, 90, 61, 10, 18, 2, -3, -14, 13], [-396, -7, -42, 52, -12, 4, -12, 2, 8, 8], [230, 8, 25, 158, 8, -18, 10, 12, 4, -36], [-455, -28, 65, 31, 19, 9, -5, -16, -14, 43], [-304, -34, -110, 49, -2, -2, 6, 4, 8, 24], [134, 12, 89, 40, 2, 33, -4, -16, 15, 80], [-870, 42, -65, -26, -15, 1, -5, -5, 3, 27], [-248, -2, -55, -80, -12, 13, 12, -14, 25, -2], [763, -12, 85, 134, -12, -6, 12, 20, 12, 26], [-890, 26, 75, 178, -4, -9, -18, -34, -31, -6], [-314, -22, -121, -84, 21, 9, 23, 27, -11, -7], [-199, 14, 130, -56, -10, 22, -2, 28, -6, -15], [-1212, 11, -148, -60, 6, 30, -10, 2, -14, 46], [-724, 22, -39, -78, 3, 3, 5, 35, -15, -29], [-657, 16, -82, -68, -52, -30, -7, -32, -6, 11], [275, 43, -95, -29, -41, -28, 16, -31, 22, -70], [-773, -42, 13, 160, 28, -27, -7, 106, 25, -55], [-373, -40, 214, 333, 34, 32, 32, -19, 1, 74], [-968, -68, -137, -98, -23, 1, -33, 25, 5, -53], [-1531, -56, -101, 116, 18, -29, 1, 44, 13, -85], [-802, -2, 255, -84, -16, -53, 22, 58, 39, -104], [-738, -17, -234, -106, 96, 72, 0, -122, -24, -10], [506, -38, -108, -96, 26, -30, 2, 41, -10, 16], [967, -18, 96, -233, 78, 95, -28, -78, -36, 103], [-1006, 40, -81, 226, -67, -59, -25, 75, 17, -3], [-918, -46, -71, -392, -8, 17, 2, 62, -71, 76], [1315, -52, -78, -9, -28, 9, -24, 36, -28, -11], [-718, 126, 252, -26, 22, -8, 4, -130, 12, 60], [700, -34, 302, 426, -46, -82, 14, 70, 50, -10], [780, -69, -144, -342, 10, 74, -46, 46, -66, 98], [1156, 38, 402, 154, -90, -174, -20, 150, 78, -164], [-1652, 34, 56, -40, -54, 40, 42, -70, 46, 76], [2734, -2, -131, 38, 43, -5, -15, 53, 29, -19], [123, 118, -442, -237, 46, 5, -6, -166, -86, 41], [97, -87, -122, 33, -78, -22, -62, 101, 48, -73], [-365, 60, 105, 227, 23, 141, -13, -130, -18, 77], [2413, 50, 316, -218, 72, 152, -48, -232, -78, 157], [-1317, -28, 130, -204, 128, 198, -19, -148, -118, 379], [2684, -10, -72, 732, -60, -12, 36, -12, 6, 142], [-1685, -76, 150, 31, 30, 32, 76, 41, 79, -24], [-730, 112, 625, -150, -2, 49, -56, -100, -63, -78], [-105, 170, 410, 261, 42, -24, -16, -23, 19, -36], [165, 82, -70, -400, 98, 42, 178, -226, -70, 45], [2261, -142, 98, 317, 58, 53, -38, -38, -108, 11], [1169, -158, 95, 57, 125, 55, 73, -18, -40, -135], [-3275, -54, -170, 454, 46, 10, -11, 58, 102, -57], [-1080, -36, 46, -250, 24, 80, -40, -75, -62, 62], [2445, -104, -988, 65, -88, -90, -50, 25, 83, 158], [464, -66, -626, 470, 114, -70, -66, -18, -78, 106], [-572, -54, -1046, 371, 46, 62, -10, -36, -16, 108], [370, -16, -125, 6, 168, 196, 130, -128, -98, 36], [3331, -54, -345, -569, -35, -125, 13, 272, 10, -143], [-359, 34, -101, 352, -30, 52, 62, -98, -126, 20], [3171, 28, 396, 515, -124, -191, -30, 160, 152, -333], [-1340, 28, -578, 578, -120, -170, 52, -28, -68, -262], [1852, 8, 511, 344, -80, 182, 52, -84, -134, 100], [-3174, -28, 95, 524, 134, 15, 68, -88, -3, -156], [835, -52, -932, 87, 96, 58, -62, -93, -15, 166], [-2139, -2, 879, 528, -224, -73, -113, 10, 119, -101], [-2064, -32, -58, -942, 156, 180, -20, -349, -304, 336], [2632, 74, -214, 774, 58, 154, -16, -30, 22, 72], [6592, -102, 860, 206, 124, 240, -12, -340, -28, 248], [471, -76, -472, 897, -68, -51, -68, -24, 78, -105], [3430, 8, -436, -152, 2, -326, 42, 67, 26, -424], [-4443, -116, 758, -196, -174, -118, 122, 66, 84, 83], [1344, 114, 261, 560, -64, 205, -16, 78, -43, -106], [-578, -281, 548, 42, 378, 130, 234, -224, -60, 226], [527, 42, 579, 598, -172, -165, -7, -106, -7, -85], [-223, 140, 51, 134, 14, 107, 101, -136, 23, -107], [2231, 46, 538, 470, -6, 114, 106, -178, 214, 135], [2488, 36, 362, 38, -36, -54, -54, 72, -204, -120], [-140, -142, 549, 214, 74, 24, -184, -34, 48, 64], [2574, 193, -1398, -480, -196, 60, 28, -144, 102, -224], [2065, 225, 247, 205, 165, -86, 48, -97, -48, 170], [2597, -152, -1436, 1022, 276, 204, -68, -100, -48, 47], [4344, -128, 1411, -164, 161, -67, 207, 213, 55, -187], [457, 73, -598, 297, -290, -282, -202, 145, 92, 19], [-1069, -61, 479, -393, 87, 8, -178, 149, -130, -92], [2236, -74, -726, -192, 202, 94, -2, -402, -60, 492], [8294, 384, -250, 432, -66, -226, 38, -396, 86, -20], [-5451, -75, 1606, 215, -86, 266, -158, 229, -110, 241], [-3873, -82, 316, 1217, 152, -98, 58, 535, 33, -354], [-4834, 212, 1085, 88, -54, -307, 0, 360, 59, -356], [-2333, -91, 121, 191, 249, -80, 60, 51, 116, -424], [6943, 129, 464, 405, -84, -308, 140, 417, 34, -505], [4456, 56, 156, 114, 200, 404, 80, -532, -322, 110], [-14016, -50, -19, 684, 155, 15, -179, -161, -21, 17], [4271, -164, -285, -1356, -274, -329, 11, 232, 197, -139], [-3901, 12, 4, 74, -356, -304, -215, 220, 82, 85], [-2665, 487, 867, -589, 67, 90, -102, -451, 60, -80], [896, 266, 1602, 98, 68, 36, -196, 95, 302, -182], [-6001, 21, -498, 83, -258, -226, -338, 107, 50, -113], [-2639, 210, 1124, -715, 44, -222, 26, 297, 149, -80], [4371, -258, -883, -1244, -164, 241, -203, 242, 65, 9], [-928, 340, -103, 4, -4, 80, -216, 124, -296, -440], [-1411, -24, -1171, 365, -285, -323, -253, 424, -36, -423], [3124, -82, 142, 54, -118, -34, -74, -362, 98, 550], [-12774, -60, -1600, -369, 124, -228, 92, 150, 70, -704], [4936, 698, -575, -704, -280, -215, -112, 110, 1, -338], [-5868, 36, 198, 46, 12, -190, 50, 384, 388, -896], [3769, -142, -820, -2397, 82, 199, 218, -62, 140, -411], [720, -296, -201, -1430, 450, 355, 94, -156, 287, -292], [8376, -71, -1150, -1556, -28, -100, 260, -76, 142, 94], [-2743, -266, 231, 431, 9, 503, -123, 62, -106, -97], [11274, -194, 729, 926, 212, 269, 222, -438, -285, 498], [-11301, 29, 1366, -1743, -42, -2, -2, 147, 96, 197], [-6842, -434, -1621, -882, -152, -285, 234, -222, -347, -74], [4257, 0, -579, 2103, 55, 53, -89, 372, 22, -681], [-2365, 34, -727, 1062, -126, 54, -276, -150, -14, 396], [196, 40, 227, -782, -496, 120, -122, 488, -154, 636], [-11604, -324, -2010, 1456, -198, -366, 326, 170, 86, -186], [-5524, -366, -276, 277, 184, 312, 16, -12, -132, 896], [-9939, 110, -487, -127, -533, 85, -333, 316, 2, -205], [6139, 52, 23, -598, -228, -470, -212, 708, 360, -90], [3295, 150, 1340, 754, -226, -652, 287, 134, 108, -83], [-5203, -108, 2569, -1068, 180, -88, -12, 364, 368, -332], [373, -85, -1132, -1881, -48, -144, 72, 383, -46, 177], [-2479, 189, -1050, -2297, 570, 482, 146, -511, -478, 413], [-9174, 6, -1241, 26, 446, 232, 196, -206, -360, 820], [4860, 284, 2865, 1454, 98, -527, 306, 296, 161, -920], [-2342, 428, 3396, 868, 296, 272, 164, -422, -16, -378], [-7577, 122, 1471, -671, 9, -765, -47, 428, 416, -685], [1356, 348, 134, -1314, -824, -110, -376, 548, -92, -326], [-8668, -74, -954, -786, 168, 648, 16, -609, 256, 300], [6281, -224, -1692, 1535, 404, 102, 66, -207, -235, -436], [-9655, -244, -988, 673, -100, -61, -450, 512, -216, -47], [-8060, 300, 1608, -1268, -614, -110, 146, -303, -152, 748], [-15770, 88, 1824, -2112, -366, 298, -238, -231, 146, 688], [2798, 258, -1190, -1522, -140, -796, -324, 643, 58, -552], [-931, -170, -260, 2868, -138, -460, -155, 414, 210, -55], [-4463, 762, 257, 134, 288, -23, 55, 126, -125, 165], [-18187, -120, -736, 1543, 72, 365, -180, -216, 276, 685], [-6218, -186, 2, -1828, -290, -546, -266, 1172, 500, -290], [-1588, -454, 1178, 218, 194, -210, 140, -70, -242, 232], [-15300, -346, 256, -875, -428, -788, 100, 432, 252, -344], [-13190, -704, 796, 1142, 668, 160, 144, -774, -390, 592], [-12172, 375, -1684, -2128, -266, -762, -106, 306, 194, -682], [13308, -116, -1549, -1394, 230, 147, 530, 144, -521, -256], [-2916, 106, 767, 822, -579, -523, 115, 565, 183, -699], [-12987, -201, -2320, -2097, 276, 380, -644, 187, 82, -151], [-19386, 38, -4630, 170, 138, 570, -12, -306, 86, -24], [-7024, -74, -3573, 556, 557, 241, 219, -1103, -351, -437], [6412, 16, -2489, 902, 592, -668, 494, 528, -130, -1376], [-11427, 130, -2212, 2415, 616, 90, 606, -375, -341, -302], [10081, 164, -2917, 578, 318, 275, 41, -896, -245, 189], [7091, -315, 3267, 1603, 369, 382, -246, 195, -6, 530], [-4035, 50, 294, -2862, -202, -310, 409, 530, 130, -649], [-2763, 774, 619, 159, -91, -645, -47, 246, 202, -425], [1235, 500, 5131, -2264, 134, 495, -113, -384, -175, 253], [1576, -144, -647, -166, -176, -590, 334, -276, -52, 44], [13514, -148, -384, 3236, -436, -436, -382, 76, 292, -82], [-7101, -91, -1807, -29, 257, -600, 678, 111, 256, 70], [8562, -204, 548, -31, 44, -1004, -116, 944, 182, -752], [946, 610, 1057, -1190, -70, 26, -340, -230, -70, -1088], [676, 402, 950, 557, -470, 74, -174, 948, 60, -112], [12133, -84, 462, -482, 48, -26, -221, -60, -464, -337], [3561, -268, 1991, 355, -371, -989, 121, 610, 352, -1667], [-1, 118, 584, 1507, 304, 1082, 426, -257, -77, 424], [-2750, 733, -138, -300, -476, -236, 244, 830, 28, -1170], [775, -159, 257, -2561, 413, 142, 454, -1169, -638, 1078], [2937, 620, -955, 943, -177, 357, -449, -388, 394, -573], [7824, -482, -530, 1705, -2, 238, -258, -642, -316, 620], [-3978, -44, 564, -2164, -500, 824, -36, -548, 40, 956], [5748, 310, -1545, -1260, -120, 67, -784, 818, 335, -258], [16078, -448, 1344, 632, 20, 516, -352, 770, 384, 474], [-15102, 77, -2910, -892, -276, -668, -596, 292, -314, -860], [-889, -730, 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