
# q-expansion of newform 900.3.f.f, downloaded from the LMFDB on 25 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 = 900
weight = 3
poly_data = [256, 0, 320, 0, 192, 0, 100, 0, 53, 0, 25, 0, 12, 0, 5, 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, 0, 0, 0, 0, 0, 0], 1], [[2240, 0, 1184, 0, 1512, 0, 35, 0, 467, 0, 96, 0, 115, 0, 11, 0], 2432], [[-6144, 0, -5584, 0, -2156, 0, -1179, 0, -807, 0, -196, 0, -139, 0, -47, 0], 1216], [[-28736, 0, -20192, 0, -11736, 0, -5959, 0, -2407, 0, -1136, 0, -471, 0, -127, 0], 2432], [[-2816, 0, -2040, 0, -1650, 0, -861, 0, -339, 0, -74, 0, -93, 0, -35, 0], 608], [[-36672, 0, -24960, 0, -12016, 0, -7603, 0, -3675, 0, -1576, 0, -835, 0, -179, 0], 2432], [[0, -19008, 0, -17152, 0, -10064, 0, -5541, 0, -2749, 0, -984, 0, -661, 0, -165], 9728], [[0, -17472, 0, -10816, 0, -7568, 0, -2021, 0, -1469, 0, -536, 0, -213, 0, -101], 4864], [[-7040, 0, -6088, 0, -2662, 0, -1497, 0, -667, 0, -318, 0, -185, 0, -59, 0], 608], [[0, 9152, 0, 7200, 0, 3168, 0, 1815, 0, 1135, 0, 440, 0, 231, 0, 71], 2432], [[24064, 0, 20832, 0, 10376, 0, 5259, 0, 3099, 0, 1344, 0, 603, 0, 211, 0], 1216], [[0, 36032, 0, 17152, 0, 6416, 0, 7631, 0, 3319, 0, 680, 0, 927, 0, 431], 9728], [[0, 1344, 0, -17408, 0, -9520, 0, -5679, 0, -3383, 0, -1736, 0, -767, 0, -495], 9728], [[0, -87104, 0, -63360, 0, -23440, 0, -14509, 0, -6853, 0, -3416, 0, -2333, 0, -621], 9728], [[0, -104768, 0, -100352, 0, -44848, 0, -24817, 0, -12681, 0, -5528, 0, -2849, 0, -977], 9728], [[0, 65344, 0, 50816, 0, 25168, 0, 14093, 0, 6757, 0, 3352, 0, 1661, 0, 525], 4864]]

hecke_ring_character_values = [[451, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [101, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [577, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0], [-1, 2, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 2, 0, 2, 0, -1, -2, 0, 0, -2], [0, 0, 0, 0, 0, 0, 3, 1, 0, 2, 0, 0, 1, 1, 3, 3], [-1, -2, 0, 0, -3, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, -2, 0, 0, 0, -1, -2, 2, 3, -3], [10, -5, -6, 2, -2, -4, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0], [-7, 4, -2, -2, -1, 2, 0, 0, 5, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 4, 0, 12, 0, -1, 0, 2, 8, 6], [0, 0, 0, 0, 2, 4, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 7, 0, 0, 0, -1, 7, 3, -7, 7], [0, 0, 0, 0, 0, 0, -4, 2, 0, 0, 0, -1, 2, -4, 5, -5], [0, 0, 0, 0, 0, 0, -1, -4, 0, -24, 0, 2, 4, 0, -1, 3], [5, -6, 0, 0, -1, -5, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [-20, -10, -12, 4, -2, -4, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 9, 0, 0, 0, 1, 9, -1, 1, -1], [2, 0, 0, 0, 2, 6, 0, 0, -2, 0, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 18, 8, 0, -5, 0, -2, 0, 4, 8, 4], [3, 14, 4, 4, 13, -6, 0, 0, 1, 0, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 5, 0, 0, 0, 5, 5, -5, 9, -9], [8, -22, 0, 0, 2, 4, 0, 0, 20, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -24, -2, 0, -37, 0, -2, -10, -6, -10, -14], [28, -9, -12, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -20, 4, 0, 0, 0, -7, 4, 1, 8, -8], [0, 0, 0, 0, 0, 0, -20, 7, 0, 0, 0, 1, 7, 13, 1, -1], [-42, -14, -24, 8, -4, -8, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 31, 15, 0, 36, 0, -4, -1, 7, 15, 7], [0, 0, 0, 0, 0, 0, 12, -10, 0, 0, 0, 1, -10, 5, -6, 6], [1, -8, 6, 6, -13, -9, 0, 0, 5, 0, -8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -19, 3, 0, -32, 0, -4, -13, -5, -7, -15], [23, -14, 28, 28, -19, -12, 0, 0, 5, 0, 11, 0, 0, 0, 0, 0], [-82, -3, -6, 2, -2, -4, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0], [3, 4, -6, -6, 13, 8, 0, 0, -9, 0, 11, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -57, -8, 0, -10, 0, 1, -4, -6, 20, 22], [0, 0, 0, 0, 0, 0, -20, 7, 0, 0, 0, 15, 7, 25, 3, -3], [0, 0, 0, 0, 0, 0, 12, -8, 0, 0, 0, -17, -8, -12, -9, 9], [0, 0, 0, 0, 0, 0, 67, 14, 0, 28, 0, -2, 6, 10, -5, -9], [11, 14, 8, 8, 17, -25, 0, 0, -3, 0, -14, 0, 0, 0, 0, 0], [-90, -18, 12, -4, 8, 16, 0, 0, 18, 0, -8, 0, 0, 0, 0, 0], [-12, 22, -10, -10, 20, -2, 0, 0, 2, 0, -14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 14, 6, 0, -77, 0, 0, 6, 6, 22, 22], [0, 0, 0, 0, 0, 0, 37, 6, 0, 80, 0, 0, 6, 6, -1, -1], [-45, 30, -20, -20, 5, 28, 0, 0, 25, 0, -17, 0, 0, 0, 0, 0], [31, -24, 26, 26, -19, 0, 0, 0, -5, 0, 31, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 28, -26, 0, 0, 0, 5, -26, -21, 24, -24], [0, 0, 0, 0, 0, 0, 40, -5, 0, 0, 0, 23, -5, 3, -15, 15], [-28, -12, -48, 16, 6, 12, 0, 0, -26, 0, -6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -59, -11, 0, -58, 0, 0, -11, -11, -7, -7], [2, 10, 10, 10, 2, 22, 0, 0, 8, 0, 24, 0, 0, 0, 0, 0], [30, 4, -48, 16, 4, 8, 0, 0, -40, 0, -4, 0, 0, 0, 0, 0], [31, -10, 28, 28, -7, -41, 0, 0, -3, 0, -10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 21, -23, 0, 24, 0, 12, 25, 1, -27, -3], [0, 0, 0, 0, 0, 0, -12, 52, 0, 0, 0, 47, 52, 20, -13, 13], [-150, 31, -6, 2, 6, 12, 0, 0, -41, 0, -6, 0, 0, 0, 0, 0], [-33, 48, -10, -10, 25, 12, 0, 0, 23, 0, -21, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -15, -16, 0, 12, 0, 5, 4, -6, -26, -16], [66, 4, -18, 6, 8, 16, 0, 0, -24, 0, -8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -48, 49, 0, 0, 0, 13, 49, 23, -11, 11], [0, 0, 0, 0, 0, 0, -5, 2, 0, 24, 0, -4, -14, -6, -27, -35], [0, 0, 0, 0, 0, 0, 60, 5, 0, 0, 0, 43, 5, -3, -19, 19], [-16, -20, 0, 0, -36, 34, 0, 0, 16, 0, 18, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 30, -22, 0, 91, 0, 16, 42, 10, 14, 46], [0, 0, 0, 0, 0, 0, 17, -30, 0, 60, 0, 16, 34, 2, -21, 11], [-75, 24, -26, -26, -25, 34, 0, 0, 49, 0, -41, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 18, 0, 0, -39, 0, 2, 8, 4, 4, 8], [0, 0, 0, 0, 0, 0, 24, -11, 0, 0, 0, 23, -11, 1, 9, -9], [-136, 82, 60, -20, 6, 12, 0, 0, -48, 0, -6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 65, -19, 0, 18, 0, 16, 45, 13, -3, 29], [122, -102, 90, 90, -70, -52, 0, 0, -32, 0, 70, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -24, 30, 0, 0, 0, 11, 30, 21, -16, 16], [0, 0, 0, 0, 0, 0, -43, 44, 0, 30, 0, -21, -40, 2, 76, 34], [33, -10, 12, 12, 11, 16, 0, 0, -21, 0, 49, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 52, -8, 0, 0, 0, 43, -8, 14, -15, 15], [-120, -5, 0, 0, 24, 48, 0, 0, -19, 0, -24, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 14, 0, 152, 0, -3, 2, 8, 54, 48], [112, 64, 42, -14, -10, -20, 0, 0, -26, 0, 10, 0, 0, 0, 0, 0], [90, 4, 60, -20, -4, -8, 0, 0, 40, 0, 4, 0, 0, 0, 0, 0], [-9, 16, -34, -34, 41, -39, 0, 0, -25, 0, -48, 0, 0, 0, 0, 0], [-24, 64, -12, 4, -10, -20, 0, 0, -62, 0, 10, 0, 0, 0, 0, 0], [28, 4, 24, 24, 8, 24, 0, 0, -4, 0, 52, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -106, 0, 0, 39, 0, -10, -40, -20, -4, -24], [-21, -10, -28, -28, -3, -48, 0, 0, -7, 0, -69, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 32, -53, 0, 0, 0, -33, -53, -77, 65, -65], [-70, 46, 96, -32, -20, -40, 0, 0, 38, 0, 20, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -52, 18, 0, -73, 0, -22, -70, -26, -82, -126], [10, 47, 18, -6, -6, -12, 0, 0, -29, 0, 6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -24, -10, 0, 0, 0, -77, -10, -83, 40, -40], [0, 0, 0, 0, 0, 0, -60, -1, 0, 0, 0, -35, -1, 1, 25, -25], [-40, 10, -18, -18, -12, 22, 0, 0, 22, 0, -18, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -60, -8, 0, 0, 0, -67, -8, -7, 8, -8], [51, -48, -38, -38, 41, -17, 0, 0, -89, 0, 34, 0, 0, 0, 0, 0], [45, -118, 40, 40, -113, 34, 0, 0, -5, 0, 79, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -44, -8, 0, 0, 0, -13, -8, 50, -11, 11], [132, 37, 72, -24, -16, -32, 0, 0, 27, 0, 16, 0, 0, 0, 0, 0], [50, -54, -36, 12, 8, 16, 0, 0, 22, 0, -8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 48, 51, 0, 0, 0, 49, 51, -23, -27, 27], [82, 34, 72, -24, 20, 40, 0, 0, -6, 0, -20, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 28, 11, 0, 0, 0, 39, 11, 11, -11, 11], [0, 0, 0, 0, 0, 0, 77, 18, 0, -4, 0, 6, 42, 30, 97, 109], [0, 0, 0, 0, 0, 0, -24, 37, 0, 0, 0, 23, 37, -11, 21, -21], [748, -12, -18, 6, -18, -36, 0, 0, 18, 0, 18, 0, 0, 0, 0, 0], [97, -96, -26, -26, 27, -78, 0, 0, -123, 0, 19, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -168, -66, 0, -309, 0, 26, 38, -14, 58, 110], [0, 0, 0, 0, 0, 0, 20, 67, 0, 0, 0, 17, 67, -15, -55, 55], [0, 0, 0, 0, 0, 0, -213, 11, 0, -136, 0, -28, -101, -45, -29, -85], [102, -158, 50, 50, -106, -64, 0, 0, -52, 0, 38, 0, 0, 0, 0, 0], [-66, 96, 84, -28, 52, 104, 0, 0, -92, 0, -52, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 50, 0, 0, 0, 95, 50, 99, -62, 62], [0, 0, 0, 0, 0, 0, -217, -70, 0, -76, 0, 23, 22, -24, 50, 96], [0, 0, 0, 0, 0, 0, -97, 15, 0, -258, 0, -12, -33, -9, 55, 31], [-25, -10, -80, -80, 45, 40, 0, 0, -55, 0, 15, 0, 0, 0, 0, 0], [-45, 12, -22, -22, -11, -14, 0, 0, 23, 0, -59, 0, 0, 0, 0, 0], [-98, -54, 54, -18, -8, -16, 0, 0, 98, 0, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -100, -13, 0, 0, 0, 31, -13, 101, 43, -43], [0, 0, 0, 0, 0, 0, -100, 132, 0, 0, 0, 55, 132, 66, -43, 43], [0, 0, 0, 0, 0, 0, -165, -20, 0, -80, 0, -8, -52, -36, -43, -59], [55, -90, -40, -40, 5, -47, 0, 0, -95, 0, 8, 0, 0, 0, 0, 0], [420, 58, 24, -8, 6, 12, 0, 0, -48, 0, -6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 130, -68, 0, 59, 0, 30, 52, -8, -208, -148], [0, 0, 0, 0, 0, 0, 157, 68, 0, 384, 0, -24, -28, 20, -13, -61], [0, 0, 0, 0, 0, 0, 12, -1, 0, 0, 0, -59, -1, -51, -19, 19], [-11, 100, 26, 26, 63, 66, 0, 0, 37, 0, 55, 0, 0, 0, 0, 0], [-232, -197, -168, 56, -36, -72, 0, 0, 121, 0, 36, 0, 0, 0, 0, 0], [220, -124, -36, 12, -42, -84, 0, 0, 142, 0, 42, 0, 0, 0, 0, 0], [-50, -12, -56, -56, -6, 84, 0, 0, -6, 0, 34, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 36, -80, 0, 0, 0, -7, -80, -19, 56, -56], [0, 0, 0, 0, 0, 0, -157, -4, 0, -160, 0, -1, -8, -6, 122, 120], [53, -34, 60, 60, -41, -126, 0, 0, 7, 0, -73, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 116, 66, 0, 0, 0, 151, 66, 84, -115, 115], [29, -80, -6, -6, -45, 98, 0, 0, -35, 0, 127, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 17, 16, 0, -28, 0, -23, -76, -30, -174, -220], [-26, 218, 144, -48, 4, 8, 0, 0, -126, 0, -4, 0, 0, 0, 0, 0], [374, 176, 24, -8, -8, -16, 0, 0, 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