
# q-expansion of newform 354.6.a.f, 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 = 354
weight = 6
poly_data = [-8433643, -7344954, -1919598, -192180, -6296, -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], 1], [[163231989243, 247814195181, 57421225892, 2534385203, 3994798, -405091], 4974382385], [[1019381753668, 595939730716, 84919990542, 3397439158, 4297543, -544836], 3553130275], [[9856228127, -10304481901, -2676131527, -126842723, -243763, 20471], 92979110], [[-21635683855423, -14910972914901, -2190963083437, -85051805363, -93564873, 13570971], 49743823850], [[-15137248321417, -9432995734379, -1475985997048, -60236074227, -80139592, 9653159], 24871911925]]

hecke_ring_character_values = None
aps_data = [[-4, 0, 0, 0, 0, 0], [-9, 0, 0, 0, 0, 0], [-8, 0, 0, 0, 0, 1], [-18, 0, -1, 1, 0, 1], [-108, 0, 0, 0, -3, 2], [106, -1, 4, -2, 7, -3], [115, 5, 1, -10, -7, -4], [-70, 1, -3, 1, 3, -22], [-573, -19, 1, 10, -4, -15], [261, 9, 23, 10, -16, -2], [995, -24, -37, 14, -26, -3], [6218, 21, -13, 7, -68, -25], [2319, 29, 33, -52, 103, -13], [1843, -80, -58, 39, 4, 87], [2027, 11, 81, 76, -26, 72], [3562, -90, 56, -70, 54, -97], [-3481, 0, 0, 0, 0, 0], [-4020, -226, -3, -23, 177, 88], [4397, 376, 78, 7, -255, -272], [-30863, -337, -208, 195, -104, -72], [-3742, 54, -279, -333, -83, -352], [-2780, 159, 78, 80, -149, -391], [-7325, 611, 151, -116, -263, -189], [-29870, 175, 67, -375, 1021, -657], [-36114, -201, 50, -218, 918, -228], [-70698, -282, 438, 694, -85, -135], [-20794, 400, 470, -62, -204, 752], [-93289, 262, -131, 140, -54, -882], [-15758, 209, -109, -563, 3, 922], [-98853, -15, -852, 363, -515, 765], [30162, -709, 394, -468, -158, -414], [-54940, 1186, -110, -1014, 1234, -200], [-89860, -838, 560, -174, 565, 1000], [-109550, -773, -676, 250, -452, 502], [-160127, 54, -93, -202, -1005, 1799], [62791, 1458, 1869, -1056, 1598, -7], [5580, 1017, -1054, -436, 422, -928], [-36664, -2060, -59, 475, 2741, 764], [-95928, -209, 1623, 1519, 2027, -2769], [-108314, -949, 241, -459, 2040, -1894], [-58346, 566, 442, 388, -3085, -4330], [27801, 90, -3473, 4, -762, 1301], [-155804, -1800, -262, 980, 4098, -2625], [191700, 1124, -224, 168, -4581, -615], [239522, -1408, -26, -114, -508, -5862], [-9270, 2552, 93, 241, 5525, -3514], [151936, 3820, 1275, 2503, -6672, -8534], [46536, -2452, -2068, -3050, 2553, -763], [66057, 1473, -4383, -5158, -4215, -4818], [140326, -4042, -3187, 2885, 1594, -1077], [-142714, 1882, 3482, 696, -3440, 9302], [-209571, -463, 5932, 485, 3121, -6873], [-191170, -5890, -6804, 1734, -3689, 9067], [-369593, 2179, 1294, -2751, 4261, 11697], [319602, 6710, 2172, -5398, -4641, -1029], [-314708, -847, 6099, 77, 2120, 11368], [-422637, 2229, -8768, 1737, -3184, 18272], [211060, -2891, 10300, -6722, 3877, -6501], [155997, -2844, -6992, 2633, 6157, -2782], [-160684, -8388, -3870, 2506, -7419, 3893], [-22874, -4838, 953, 3379, -11714, -10772], [466243, 12645, 661, -2884, 5810, 5713], [483050, -3150, 12500, -5210, -1940, -13578], [-858637, -5528, 723, 6432, 2219, -5311], [264064, -12323, -131, 9961, 13147, 5494], [28966, 718, 13064, 5954, 20782, -16729], [366772, 5204, 8682, -11638, 4232, 7680], [747398, -7407, -3747, 1531, 5833, 16544], [22373, -7523, -743, 9946, -1116, 26284], [227776, 4121, -8762, 25442, -7437, 16993], [-75257, -3342, 1589, -24500, 2772, -13880], [-790830, 7946, 7556, 8392, -3433, 5709], [-1013537, 16214, 7996, -1673, -14959, -15582], [480702, 1546, -10689, -2793, -38033, 10957], [-2672182, -10391, -10446, -404, -3160, 12390], [-1913619, 6421, 16642, 10261, -20596, 12956], [-394116, 7128, 4396, -10198, -2862, -34226], [552788, -4567, -2490, -17076, 15298, 4790], [-714923, 9844, -10583, -21898, -19554, -39086], [-2094518, 8152, 6112, 6684, -20632, 11864], [-1749396, -5660, 7612, -9682, 56291, 9801], [-1847186, 17647, -162, -14856, 36513, -12335], [-2639875, 7613, -16081, 16560, -36024, 12978], [804388, 8890, 14945, -4679, 20366, -2562], [-2835968, -1101, -1045, 12899, 37554, 14933], [718147, 4369, -4619, 5344, -25895, -19214], [580414, -10712, -4574, 11230, -35545, 55750], [826028, 7170, -19987, -14851, -6365, 21080], [-369907, -5141, 21901, 12710, -16988, -56011], [-1422650, 30390, 10857, 9297, -24687, 23027], [409818, -6314, -24728, 9528, 21493, 13619], [-136167, -15226, 17531, -38088, 69448, 22738], [1636058, 19137, -22011, -40997, -38322, 9611], [-1478304, -2343, -6731, 24793, -52261, 40441], [-1405894, 937, -4246, 15448, -5498, 37386], [419179, -8195, -437, -33170, 42746, 38504], [4652346, -16203, 7763, 1693, 19651, -25601], [665797, -21783, -22671, -4328, -46074, 14046], [782742, -4251, 31022, 19684, 12554, -62704], [601296, -24521, 15124, -2456, 34301, 36361], [175016, -35253, -16457, 15409, 9301, -20616], [4989020, 36782, -12516, 41962, 5524, 45126], [2141058, 54922, -33704, -9620, 17910, -12475], [-12168, -10651, -31411, 833, 23691, -73559], [-3296809, -32572, -23340, -15289, 22185, -338], [363401, -2564, 17660, 49223, 12226, 50053], [4033608, 59808, -8230, -15990, 37421, 60586], [4424297, -63343, 52645, 10036, 57702, -13802], [3075534, 81614, 13450, -7058, -68271, -51471], [-2991630, 46707, -23005, 26005, -23971, 176772], [-681500, 20289, -54776, -18486, -24667, 78551], [-1164928, -11066, -33566, 31394, 66893, 76187], [-3534097, -85127, -20217, 41124, 67293, 55368], [-1481486, -3425, -13088, -12098, 9232, 8214], [2667710, 29181, 22550, -48034, -87435, 31891], [-201040, 42562, 1978, -5730, 23776, -81383], [3393836, 31483, -66263, 23795, 1137, -1324], [-1562040, -17292, -64256, -1240, -112880, -48504], [3422455, 7947, 3261, 37714, 554, -122712], [-745131, -7605, 43301, -53700, 7240, -2945], [-2656465, -41832, 38455, 12250, 10836, 26959], [2179816, -27594, -18508, -18010, 76526, 65120], [4110026, -11868, -11774, 28422, 79292, -55688], [6565920, -3418, 11726, 14818, -4303, -131722], [-519084, 51500, 65498, -4930, -29003, -92421], [-416322, 72239, -38185, -62217, 81624, 114194], [-1607468, 4784, 17166, -22632, -39164, 175178], [689887, -17981, 41771, -59452, -47232, 32345], [-3281512, -31930, -136184, 18282, -54611, 168241], [-2578581, -68764, -66028, 92547, -31853, -2230], [-3555602, -37632, -22321, 39661, -73522, -191052], [2391789, 74740, 3493, -11930, 50234, 70532], [-827216, 76996, 36282, -26460, 46944, 101658], [-2034580, 13144, 14637, 47035, -100681, 142743], [5905098, -2060, -99240, 47422, 2980, 45197], [4019240, 15543, -62063, 961, -179937, 219728], [8603992, 20612, 15202, 78882, 37216, 115192], [1923242, -28870, 83094, 25384, 73154, -12176], [13195536, -35240, -29142, -14132, 78581, 144021], [4697755, -90626, -45037, -53614, 66694, -29510], [-2213882, -85592, 22258, -20216, 31513, 31869], [8239256, 135761, 27141, -27325, -78286, 5954], [-96389, 8262, -99034, 105993, 86255, 71752], [-4789473, 36035, 72934, -66775, -123513, 179325], [569741, -39204, -23147, -26290, 99830, 248169], [-14258946, 33738, -16060, 78956, 13320, 68807], [-9241428, 66806, -36931, -2881, -195225, 194639], [6501729, 190215, 3044, -84851, -24193, 7669], [-14369506, 51418, 56572, -90020, 83138, 16224], [11510415, -184177, -105321, 144920, 20000, 254305], [-415868, 88704, -83332, -134298, -235200, -44372], [-1975188, 173536, 111106, -40978, -9868, 49404], [-4831164, 88628, -56232, -16700, 68452, -114988], [6184036, 115178, -42326, 90240, -81916, -79159], [1000686, -73185, 140270, -20872, -67970, -208668], [-3834316, -112467, -135041, 104801, 29638, 220132], [1269050, 15124, 146750, 91286, -61906, -206268], [-6523315, -9731, 185730, 27379, 92379, -103109], [-13415086, -71289, 82506, 56954, 134780, -98966], [13783587, -113245, -32212, 163691, 97965, 248011], [1630528, 197381, -36925, -95169, -270917, -127709], [-7479334, -157858, -41700, -3290, 150197, 34149], [-21915114, 5954, 18559, 113447, -123557, -263939], [-10142135, 32224, 37277, 91074, 170976, 71538], [2200413, -27769, -56948, 122497, -139107, 111441], [-9313644, -142036, 142244, 49658, 277978, 334443], [-15120624, 100092, -91161, 135633, -145619, -313537], [-3382601, 198488, -31773, -23474, -176890, -241499], [-29220076, 153390, -153632, -47982, 96216, 162432], [7429650, 284310, 92680, -202556, -360761, 93107], [-7789535, 107431, -9924, -251763, 3937, -117621], [-2924370, -3278, -15336, 141882, 92963, -39759], [-3778635, -24530, -7321, 310498, 129183, 201569], [-1833732, -79169, 242840, -235258, 147250, 86150], [5090468, 62350, 36396, 150750, 209315, 452369], [-534774, -42726, 148740, 29964, -294669, -231896], [-20088718, -82671, -1820, -54498, -223142, 231294], [8486726, -149794, 273862, -9326, 135047, -212395], [-23369744, 20605, 49011, 46673, 352563, 370850], [-8974880, -80286, 120192, -31770, 101258, -626700], [643148, 259748, -233156, -243500, 53952, -40834], [-16617124, -54538, -93396, -59220, -432126, 286352], [5206342, -6303, 84754, -6912, -112783, -661377], [-10813632, -129336, -185270, 277036, -12540, 285918], [-20245251, 111389, 89220, -47125, 238470, -304800], [-14884749, -60330, -212573, -126410, 35460, 177218], [4988012, -235166, 62590, 213588, 128648, 431256], [18939482, -12640, 202954, -259336, 373277, 206721], [16299070, 183779, -54762, -152066, -280176, 204934], [10394451, 37331, 218735, -134980, 54278, -34703], [11264458, 124633, -51514, 36590, -254048, 241172], [29086582, 129316, 280280, -21418, -285319, -745020], [7487288, -177978, -95644, 240596, -94701, 390639], [-5088847, -65281, -16579, -350068, 182998, 54146], [13262860, 63026, -345268, 226236, -349374, -411672], [3032991, 146360, -123463, 112524, -45054, 276442], [24574098, 171124, 15947, -82733, -264387, -352543], [44267474, -123864, -20279, 153681, -3314, 78185], [9206800, 254283, 61761, 125589, -78743, 702597], [9023041, -57739, -8521, 95666, -183902, 425817], [963466, -41934, -32400, -252668, 19732, 274264], [8445976, 232051, -69098, -67110, -133696, 1156700], [30286266, -26630, -31087, -126571, 466187, -321695], [-4069556, 35149, -154247, 83093, -199709, 154542], [-2847884, -195238, -28210, 117818, 198870, -772840], [-9814198, 7081, -189055, 496105, 281130, -191486], [37524555, -128054, 25498, -129543, 124423, 22762], [18000313, -232611, -38153, -139756, 666885, -168652], [-7954993, 136308, 532727, -417218, 99214, -165466], [3896230, -3294, 291142, -100448, 246441, 503207], [20000316, -102223, -157833, -6357, 75755, 463594], [-6916228, 27918, -289678, -292894, 425562, 169777], [9698090, -84887, 53317, -290445, 407822, -206285], [-56104372, 191202, -55104, -63552, 366416, 179056], [-22046205, -200741, 65046, -120077, -62058, -996868], [-20055250, -84459, 9773, -52497, -28503, 338672], [20206048, -444755, 382194, 221224, -208351, -277595], [-22791183, -39153, -227121, 24948, -198583, -189375], [-6683197, -450143, -51881, 16920, -149130, -517801], [29614462, -196534, 149188, 22242, -65002, 276225], [52511714, -359743, 424854, 144944, 438413, -157829], [14767863, -737382, -251908, 254703, 478983, -523958], [-35332332, 221238, 100952, 94744, -374078, 570984], [54058072, -420768, 82809, -57925, 231339, -867534], [-45736336, -297490, -95350, -242836, 965533, 663901], [-9393902, 51142, -29167, -90763, 154656, 449553], [23020071, -29761, -492585, 528106, -433446, 111739], [-30058346, 520940, -108114, -597288, 1984, -499180], [7780116, 447656, 367692, -374466, -711810, -573008], [-1360919, 237830, 63843, -86024, -282436, -129876], [40457720, -199436, 111043, -444791, 800148, 45809], [-990767, -215226, -363162, 1565, 206056, 642887], [1368196, -349172, -857827, 374027, 125155, 1052753], [-67709015, 79917, -88500, -410103, 84593, -373303], [20630360, -53754, 102940, -104042, -697384, -119854], [-53915023, -444651, 102436, 641377, 132653, -457221], [-4916246, 366525, -447200, -344264, -603914, -500262], [66902028, 52421, 241667, 46819, 4213, -560269], [-16409738, -3924, -727608, 244708, -833922, -354999], [9410014, 90965, 106695, 238147, -290090, -508102], [3849293, 135253, -1678, 46427, 341687, -225981], [-19602342, -337160, 277780, 257250, -533495, -118829], [25834932, 88167, 772834, 23576, 479091, -766099], [-27469312, -57840, 505694, 5504, 96332, 750476], [-27692471, -170697, 111273, -219524, 882117, 580178], [-71406292, 17317, 287555, -154683, 725946, -333740], [5278141, 413774, -555032, -249763, -695149, -795702], [31001498, 239412, -271076, -54930, -836051, 1064284], [52083512, -208634, 33687, -319255, 604799, -848822], [25777255, 266643, -103686, 159931, -585486, -325428], [-25665606, -196176, 500595, 266643, -317947, 439443], [22688113, -153418, -303999, -263382, 266308, -583576], [-92253513, 449749, 570005, -771272, 212282, -776490], [-28755949, -41804, 411866, 542719, 769810, -417871], [15998702, -53872, -205002, -185960, -1337358, 177273], [57187583, 817901, 187666, 57193, 133217, 750975], 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