# q-expansion of newform 1035.4.a.p, downloaded from the LMFDB on 23 October 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 = 1035
weight = 4
poly_data = [-1152, -396, 548, 34, -46, -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], 1], [[0, 1, 0, 0, 0, 0], 1], [[-16, 0, 1, 0, 0, 0], 1], [[24, 212, 10, -34, -1, 1], 12], [[28, -22, -2, 1, 0, 0], 1], [[94, 56, -28, -3, 1, 0], 2]]

hecke_ring_character_values = None
aps_data = [[0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [5, 0, 0, 0, 0, 0], [10, 0, 2, -1, 0, 0], [-3, 0, -1, -2, 0, 1], [21, -4, -1, -2, 0, -1], [-23, 0, 1, -1, -2, 1], [45, 4, 3, -2, 2, 3], [-23, 0, 0, 0, 0, 0], [19, -8, 9, -7, 2, 3], [6, -8, -6, 1, -2, -4], [16, -12, -4, -1, -2, 10], [133, -16, -3, 9, -4, -3], [-4, -20, -14, -6, -6, -4], [-91, -24, -1, 0, -10, -7], [53, -4, -17, 9, 2, -3], [131, -92, 11, -3, 14, 11], [249, 72, -25, 20, -6, -13], [-22, 64, -48, -15, 4, -8], [-59, -68, 3, 13, -12, -11], [125, 72, -9, -2, -4, -1], [422, 108, -28, 18, -4, 0], [-315, 152, 21, -37, 0, -9], [-56, -128, -10, 10, 14, 22], [118, -16, -40, -44, -16, -22], [-315, 0, -11, -33, 0, 5], [-106, 0, 12, 14, 24, 6], [-85, 48, 9, 81, 6, -7], [191, 236, -23, 42, 18, -7], [-363, 192, -65, 45, -16, -11], [-97, 192, 23, -62, 48, 19], [1190, -28, 12, 18, 6, -28], [-932, 20, -96, -38, 24, -2], [-456, 52, 46, -39, -20, 46], [-397, 4, -49, 70, -10, -67], [1096, -84, -30, 82, 42, -4], [698, -192, -32, 51, -24, -40], [-300, -8, 60, 138, 32, -12], [423, 576, -123, -24, 10, -17], [468, -344, -176, 72, -34, -20], [630, -316, 46, 64, 32, 76], [1304, -452, -14, -98, -30, -22], [1707, -84, 129, 48, 38, 83], [-232, -124, 214, -130, -4, 22], [-50, 52, -118, -18, 26, -38], [-128, 60, -186, 92, -32, 30], [32, -92, -158, -37, 52, 30], [-56, 64, 228, -174, -20, 60], [-1444, 4, -128, 120, -46, -48], [1992, -320, -178, 194, 50, 22], [-588, -240, -132, 124, 22, -116], [-409, -836, 41, -77, 76, 115], [1081, -520, -105, 110, -68, 63], [-1608, 500, -180, -222, 2, -88], [1741, 356, 19, 224, -52, -55], [661, -612, 103, -161, -4, 35], [-141, -648, 277, 99, -18, -85], [1382, -996, -180, 93, 36, 48], [626, 444, 68, 360, 56, -12], [-1021, 176, 107, -64, 168, -39], [-2652, -172, -176, 189, -114, -186], [-435, -20, -95, -127, 64, -123], [57, -48, -267, -52, 48, -35], [-3906, 1036, 288, -2, 58, -68], [1756, -936, -88, -63, 28, -146], [997, 312, -85, 68, 12, 209], [2096, -104, 260, -51, -38, -10], [-1100, 172, -68, 42, 102, 292], [-1610, 284, 142, 16, -116, 34], [-1944, -640, -210, 373, -94, -144], [-811, -268, 131, -64, 256, 1], [-4259, 552, -365, 216, 10, -47], [1532, 460, -46, 11, -254, 62], [-1796, 188, 246, -36, -126, 92], [-1042, -76, -304, -394, 102, 4], [-5007, 764, 173, -15, -26, -13], [3116, 1044, 138, -44, 112, 162], [-1842, -808, -692, 342, 152, 116], [204, 220, -622, 252, -306, -62], [5200, -600, 618, 387, 70, 144], [-3429, 756, 53, 298, 120, -125], [-3397, 1292, 149, 94, 54, -71], [-3614, -656, -70, 300, -132, 386], [1894, -204, 132, -239, 158, 28], [5918, 692, 252, -126, -158, 82], [3679, 824, 393, -416, 170, -177], [6587, -576, 809, -175, 134, 71], [-4372, 2408, -236, -155, -52, -86], [1276, -264, -214, -64, 260, 310], [1123, -332, 355, 160, -220, 27], [2459, -488, 357, -687, 38, -143], [-2993, 1056, 133, -520, -102, -21], [-2053, 884, -737, 142, -340, -209], [4883, 828, 185, -27, 124, -197], [4720, 176, 152, 735, 162, 90], [-5447, -1100, -151, 601, 66, -109], [6232, 1160, 296, -218, -102, -190], [7757, -960, 453, 140, 44, 235], [-7512, -2168, 44, 212, -132, 80], [-3768, -44, -568, 1254, -98, -174], [-1308, -1524, 1076, 40, -112, -56], [7329, -1628, 115, 41, -306, -247], [-427, -2768, 595, -73, -70, 39], [8108, -1536, 650, -24, 274, -2], [3399, -4032, -423, 556, -30, 313], [5302, 2516, 140, 510, -376, -28], [4640, -828, 708, -214, -268, -272], [11115, -1424, -1371, 156, -108, -105], [1100, 668, -642, -546, 106, -162], [-2098, -1668, -90, -609, -112, -250], [-12399, 1568, 157, -82, 228, -75], [6080, 840, -584, -354, 216, 500], [249, 3216, 145, -71, -206, 33], [554, 1368, -28, 262, 322, -160], [6755, -3000, 145, 84, 200, -23], [3373, -532, -1259, 292, -268, -217], [1604, 1856, -370, 1257, 256, -46], [7667, 160, 305, 522, 494, 347], [5093, 232, 263, -146, -184, -7], [-5139, -2228, -101, -54, 168, 17], [-8580, -2328, -870, 496, 26, -426], [-507, -1360, 231, -512, 76, -85], [4635, -852, -751, 395, -762, -5], [1781, 732, 1075, -1056, 550, 377], [3452, -1404, 714, -34, -146, 112], [-239, 892, 725, -594, 514, 363], [8585, -256, 139, -724, 262, 99], [-1523, 1764, -1347, 33, -510, 217], [6604, -2016, 564, 213, 460, 678], [11498, -1596, 1058, 19, 478, 896], [17944, 1504, -1296, 373, 46, -618], [-3212, 2080, 484, -164, -196, -340], [-5619, 2460, -1525, 38, 332, 367], [-12264, 824, 910, -49, -40, -118], [8539, -432, -351, 419, -58, -69], [10317, 356, 235, 724, -688, -265], [-3680, 520, -338, 444, 48, 642], [-11404, 2616, 522, 549, 76, -422], [4903, -604, -351, 227, 658, 1019], [15491, 4024, -433, 471, 56, -265], [4500, 2900, -126, 743, -464, -994], [15600, -2024, 708, -238, 314, 338], [-1952, -3084, 1556, -1392, 320, 368], [7527, -1176, -851, -121, -298, -671], [20492, 4108, -24, -403, 176, 764], [-15336, -2984, -828, 334, 268, -260], [6168, 248, 198, -720, -452, 50], [8680, -2364, -788, -1206, -726, 84], [4484, -1636, -1022, -377, -664, -374], [-8382, 1916, 482, 246, -88, 510], [-5124, 5664, -710, -736, 528, -222], [15679, 3140, -1229, 214, 52, -1031], [-7173, -1812, -1569, -182, 328, 1019], [-13918, 3996, -854, -1014, 56, -186], [-13002, -4576, 1072, -287, -348, -264], [-3364, 1820, -816, 996, 132, 392], [-5162, 3432, -2912, 338, -852, -356], [9969, -712, -881, -603, -498, 77], [-2726, 680, -1072, 1756, 636, -438], [22049, 3176, 177, 386, -834, -181], [12034, 1840, -1042, 1340, 612, 122], [2258, 3596, 1856, -1804, 374, 236], [-4345, 1712, -73, -610, -332, -397], [12618, 600, 38, 572, 206, 126], [10355, 944, 2111, 1137, -350, -101], [-24507, -5500, 705, 1335, -78, -669], [30324, 92, -1814, 707, -396, -762], [-4744, -1132, 1458, 1934, 4, 730], [3614, 4140, -3768, -190, -364, -364], [8893, 2072, -2477, 372, -656, -479], [-1424, -868, 1524, -1208, 398, 1008], [-16466, -792, -500, -777, -938, -694], [-10420, 2844, 1044, -18, 324, 836], [-4568, 2580, 1058, 389, -518, 874], [4240, -1232, 1952, 936, 100, 1240], [6947, 5696, -1653, -1170, 368, 265], [3930, 676, 2316, -846, 130, 278], [-1831, 4848, 1891, -343, 134, -987], [-2746, -3092, -1416, 1255, 218, -448], [22445, -5684, 811, 214, 46, 111], [-4003, -300, -1371, 44, 412, 277], [-9843, 1240, -337, 72, 460, 809], [-3120, -856, 510, -1596, 296, 858], [21271, 1732, 625, 582, 400, 211], [-4767, -2812, 839, 1127, -56, 1007], [21903, 2272, 1299, -696, -154, 1025], [10429, -3092, -1721, -222, -712, -681], [-4822, 7152, -2628, 599, -676, 8], [25204, 1732, 456, -855, 700, -1032], [-18011, 1412, -993, 249, -584, 317], [12497, -1236, -153, 2454, -84, -177], [-4212, 6132, 776, -958, 116, 12], [-321, 10440, -23, -1454, 198, -1083], [1355, 1304, 403, 785, -672, -797], [-22136, 1196, -50, -218, -892, -784], [-4081, 6424, -1199, -1093, 328, -73], [3353, 5668, 2143, -1178, -1168, 439], [5606, 4772, -200, -748, -66, 218], [-4722, -3044, 596, -972, 298, -376], [-15460, 2504, -2292, 1122, -172, -104], [-13525, -8024, 2455, 538, -174, -695], [27260, -2428, 542, -1530, -318, -680], [2623, -9772, 1045, -586, 48, -755], [-6877, 484, -1755, -834, 712, -695], [-8733, -7572, 2109, -2300, 1304, 1271], [-6014, 7656, 1522, -1350, 1286, 734], [-24943, -7892, -17, 912, -884, -21], [-2889, -7588, 625, 1650, 1170, -733], [7873, -2032, 1925, 1169, -184, 1489], [12696, 4972, -310, -486, -702, 272], [-5984, -6976, -1494, -488, 848, 1870], [22848, 3304, 1234, 1780, -1124, -930], [-12782, 4696, -3692, 1358, -1072, -426], [-8360, 1700, 5026, -226, 604, -272], [-29187, 2564, -1187, 1005, 762, 821], [28389, 4584, 991, -644, 872, -1], [-18074, 4484, 3338, 1450, 232, 1162], [15469, 1960, 689, -3127, 868, 561], [-31339, 1956, -449, 1899, -584, -1045], [-1477, 3172, -1623, -38, 556, -1281], [12248, 4860, 2096, 545, -1048, 924], [-19660, -3928, -4254, 1924, 316, -506], [-24779, -10368, 531, 21, -542, 1877], [11126, 644, 3436, 1611, 1254, 272], [12647, -8968, 2729, 1070, -922, 515], [7984, -2008, 1454, 1593, 926, 44], [-28584, -2340, 1298, -1488, -222, -578], [-3465, 1964, 3629, -3657, -744, 967], [-10400, -6996, 1660, -144, -690, -1580], [-18021, 2012, 3397, 49, -396, -817], [-13795, 8900, 4339, -2242, 1128, 347], [16319, 6880, 2069, -830, -466, 1665], [-41456, 6296, 1186, -1416, -320, -934], [6003, 8344, -2235, 3701, 462, 99], [-28800, -624, 1810, -1327, 1020, 738], [-55268, -128, 498, -426, -694, 384], [-18657, 3320, -99, 2012, -1176, -2243], [16330, 6008, -1724, 744, 1712, -448], [-2491, 13532, -351, 65, -314, 585], [-19533, -4444, -941, -3897, 142, -225], [42399, -7192, -1525, -705, -900, -983], [-20174, -2272, 1874, 1261, 510, 588], [8142, -9192, -1014, 118, 1312, 650], [25912, -12676, 1616, 2, 274, 958], [-8348, -8504, -932, -1108, -1026, -324], [-13724, -14572, 2030, -470, 914, 874], [-373, 10016, 1279, -766, -1220, -969], [-2509, 200, 3237, -2226, 1868, 1231], [-42420, -1396, -3054, 416, -18, -494], [-7969, 6444, -2709, 2359, 1486, 1317], [-16682, -2052, 1100, 2656, 702, -1806], [-17250, -2492, 1964, 3168, -702, -728], [-16527, 11988, 2553, -1695, -758, -2237], [15456, -10064, 5574, -1280, -1308, -1298], [711, 2644, -1499, 3714, -164, 627], [33307, 4164, 3495, -603, 1234, 3551], [-7515, 4664, -581, 1616, 1326, -813], [-30296, -8168, -2496, -2144, 560, 928], [-10149, 400, -4903, 2350, -2716, -1653], [-30686, -340, -3116, 1276, 132, 1372], [10380, -10804, 1926, -657, -1550, -370], [31571, -3968, 2533, 1991, 302, -459], [24092, 1508, 4312, -3205, -452, -1500], [-14376, -4736, -1994, 1203, 92, -1618], [-34342, 4304, -14, -2572, 22, -106], [15415, -15488, -575, 48, -1874, -895], [-11443, -3632, -7527, 5814, -790, -1901], [-5842, -1672, -3884, 3052, 46, 1532], [-26664, -13436, -1872, 2565, -1294, -786], [1028, -10244, -48, -1128, -530, -1920], [-16978, -18644, 654, -1203, 716, 410], [-13304, -5728, 10994, -267, 616, 1210], [29160, -8112, 2600, 567, 44, 1918], [-3014, -4088, -1392, 3680, -1324, 244], [-32049, 7820, -4367, 1180, -2400, 1221], [39329, -4924, -1115, -2332, 280, -327], [7387, 1424, 405, -1200, -750, -845], [-45502, 8180, 1050, -979, -1856, -1766], [52171, 11672, 1413, -2996, -452, 2301], [18505, -12564, 5155, 184, 312, 873], [-9716, -884, 2368, 1402, 2060, -604], [6617, 5360, -8621, 4200, -1260, -1957], [778, -22672, 246, 664, 868, -454], [-2014, 6852, 264, -1278, 554, -1036], [25831, 22164, -861, 2472, -196, -365], [-20275, 876, -567, -729, -838, -755], [-24348, 5964, -3366, -1371, 194, 1662], [-8578, 1712, -4928, 108, 1976, 540], [-25214, -12508, -3368, 2634, 1098, -1514], [40684, -5016, -7322, -100, -1028, -2666], [29524, -1912, 1330, -3264, -276, 474], [-8137, 5608, 3671, -2959, 692, 1181], [36586, 2340, 7290, -3592, -1114, -90], [13553, 1840, 2023, 1204, -2052, -3047], [-36233, 14508, -3375, -1308, -816, 485], [17046, -16752, 3132, 1250, 2624, -460], [-15537, -1736, -6383, 3290, 1424, 497], [25341, -2408, -4237, -2634, 316, -19], [15803, 6204, 23, -1161, -138, -1941], [-18068, -9608, -1500, 2574, -1136, -3276], [90364, -13148, 3874, 1019, 1450, 1106], [-41703, -6252, 217, -3265, -2188, 1405], [-22376, -4704, -2794, -2242, -2836, -3394], [5665, 876, -1785, -3500, -1670, 605], [-11666, 6360, 7912, -94, 290, 2100], 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