
# q-expansion of newform 847.2.b.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 = 847
weight = 2
poly_data = [87025, 0, 113575, 0, 106925, 0, 42100, 0, 11605, 0, 2030, 0, 260, 0, 20, 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], [[487898960985, 0, 604192602745, 0, 352976217575, 0, 88532447865, 0, 15517814844, 0, 1368289173, 0, 79007122, 0, -1833998, 0], 141563707035], [[0, -70674046800, 0, -200532703805, 0, -74163373020, 0, -22443730260, 0, -4022687807, 0, -546435360, 0, -42954924, 0, -2307412], 141563707035], [[-712156894265, 0, -866089585460, 0, -521715656950, 0, -125469295005, 0, -21498650696, 0, -1270611026, 0, -22738192, 0, 14332206, 0], 141563707035], [[0, 8072787100, 0, 126280408730, 0, 8471357565, 0, 2563649095, 0, -656915387, 0, 62416920, 0, 4906553, 0, 2866677], 47187902345], [[-622504551990, 0, -703679870695, 0, -516762324905, 0, -117247734350, 0, -20102189386, 0, -832677109, 0, 15194037, 0, 17550090, 0], 141563707035], [[-8556925565, 0, -4868757880, 0, -4513152840, 0, -1648700690, 0, -423472000, 0, -64624056, 0, -6302671, 0, -312032, 0], 2399384865], [[0, 875123748100, 0, 1215421813590, 0, 918330446715, 0, 277910240545, 0, 58247425041, 0, 6766254120, 0, 531890783, 0, 11651382], 141563707035], [[-36566060, 0, 5713275915, 0, 5295988845, 0, 2038169710, 0, 496926000, 0, 75833523, 0, 7209311, 0, 366156, 0], 2399384865], [[0, -141348093600, 0, -353877505265, 0, -148326746040, 0, -44887460520, 0, -8045375614, 0, -1092870720, 0, -85909848, 0, -4614824], 47187902345], [[0, 571605367040, 0, 215720277125, 0, -15301965200, 0, -41205101320, 0, -13172985771, 0, -2577620454, 0, -259731426, 0, -16997173], 141563707035], [[0, -217803906170, 0, -15187573320, 0, 89465338220, 0, 63648831580, 0, 17195673578, 0, 3124055814, 0, 302686350, 0, 19304585], 141563707035], [[563850684635, 0, 564511445800, 0, 537175018240, 0, 109297120970, 0, 18037649624, 0, -357871758, 0, -157372815, 0, -34663122, 0], 141563707035], [[17923235, 0, 37136075, 0, 34423725, 0, 13503405, 0, 3230000, 0, 492915, 0, 45553, 0, 2380, 0], 6105305], [[0, 677017839760, 0, 556412873310, 0, 368268112995, 0, 153148476795, 0, 37010612536, 0, 6198172168, 0, 591428329, 0, 36161330], 141563707035], [[0, -149364120930, 0, 166033932275, 0, 294465298240, 0, 153943084810, 0, 41356996652, 0, 7316393468, 0, 729424284, 0, 45560235], 141563707035]]

hecke_ring_character_values = [[122, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [365, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 1, 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, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, -1, 1, 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, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, -4, 0, 1, 0, 0, 0, 0, 2, 0, 0], [0, 0, 1, 0, 2, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, -3], [0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, -4, 0, 3, 0, 0, 0, 0, 3, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, -4, 0, 0, 0, 0, -3, 0, 0], [0, 0, 2, 0, -1, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0], [0, -3, 0, 4, 0, -1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0], [0, 0, -1, 0, -3, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 1, 3], [0, 0, 4, 0, 2, 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, 2, 2], [0, 9, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0], [0, -3, 0, -4, 0, 6, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0], [-6, 0, 0, 0, 0, 0, -2, 0, -6, 0, 0, 0, 0, 1, 0, 0], [0, 5, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, -1, 0, 0, 2, 0, 3, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, -2, 0, -4, 0, 0, 0, 0, 10, 0, 0], [0, 0, -2, 0, -1, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 7, 0, -1, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 6, 0, 5, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -5], [4, 0, 0, 0, 0, 0, -9, 0, -5, 0, 0, 0, 0, 6, 0, 0], [0, 0, 2, 0, -3, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, -1, 0, -4, 0, 0, 0, 0, 0, 0], [14, 0, 0, 0, 0, 0, -2, 0, 4, 0, 0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -5, 0, 0, 3, 4], [6, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, -3, 0, 0], [0, 3, 0, -3, 0, -3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0], [0, 3, 0, 10, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 2, 0, 0, -2, -2], [0, -2, 0, -12, 0, 7, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, -4, 1], [0, 0, 6, 0, 6, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, -1, 5], [0, -2, 0, -3, 0, 1, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0], [0, 7, 0, 4, 0, -3, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0], [0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 6, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 3, 0, 0, -1, -2], [0, 10, 0, -4, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, -3, 0, 0, 1, 0], [0, 0, -2, 0, -2, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0], [0, -5, 0, -8, 0, -11, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0], [0, -15, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0], [0, 0, 8, 0, -3, 0, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, -3, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 5, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, 0, 0, 8, 5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 9, 2], [-13, 0, 0, 0, 0, 0, 10, 0, 5, 0, 0, 0, 0, -8, 0, 0], [-1, 0, 0, 0, 0, 0, -5, 0, -1, 0, 0, 0, 0, -1, 0, 0], [0, 2, 0, 18, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -13, 0, -19, 0, -6, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0], [0, 0, -5, 0, 0, 0, 0, -7, 0, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -6, 0, 0, 2, 8], [0, 1, 0, 18, 0, -10, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 1], [0, -12, 0, 3, 0, -11, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [10, 0, 0, 0, 0, 0, -3, 0, -2, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 9, 0, 0, -4, 0], [10, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 15, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 7, 0, 0, 0, 1], [7, 0, 0, 0, 0, 0, -6, 0, -10, 0, 0, 0, 0, 18, 0, 0], [0, 0, 1, 0, 3, 0, 0, -2, 0, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, -5, 3], [-1, 0, 0, 0, 0, 0, 8, 0, 11, 0, 0, 0, 0, -8, 0, 0], [0, 2, 0, 1, 0, 4, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 2, 0, 0, -4, 2], [0, 0, 2, 0, -2, 0, 0, -4, 0, -3, 0, 0, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, -17, 0, 0], [-7, 0, 0, 0, 0, 0, 3, 0, 10, 0, 0, 0, 0, -1, 0, 0], [0, -5, 0, -3, 0, 7, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0], [0, 0, 7, 0, 9, 0, 0, -4, 0, 5, 0, 0, 0, 0, 0, 0], [17, 0, 0, 0, 0, 0, -1, 0, -4, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 0, 0, -2, -8], [0, 0, -5, 0, -4, 0, 0, -11, 0, -3, 0, 0, 0, 0, 0, 0], [5, 0, 0, 0, 0, 0, -13, 0, -6, 0, 0, 0, 0, 16, 0, 0], [0, -12, 0, -14, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [-9, 0, 0, 0, 0, 0, 1, 0, -11, 0, 0, 0, 0, 20, 0, 0], [0, 0, 0, 0, -4, 0, 0, 5, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 4, 0, 0, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -9, 0, 0, 9, -2], [0, 0, 10, 0, 2, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, -14, 0, 3, 0, -5, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0], [0, -12, 0, -15, 0, -10, 0, 0, 0, 0, 0, 0, -21, 0, 0, 0], [0, -2, 0, -3, 0, -5, 0, 0, 0, 0, 0, 0, -18, 0, 0, 0], [0, 0, -4, 0, 1, 0, 0, 5, 0, 2, 0, 0, 0, 0, 0, 0], [0, -6, 0, -10, 0, 25, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0], [0, -18, 0, -16, 0, -6, 0, 0, 0, 0, 0, 0, -11, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -9, 0, 0, 2, -5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -2, 0, 0, 9, 1], [0, 0, -3, 0, -1, 0, 0, 6, 0, -4, 0, 0, 0, 0, 0, 0], [13, 0, 0, 0, 0, 0, -2, 0, 4, 0, 0, 0, 0, -8, 0, 0], [0, 0, 4, 0, 1, 0, 0, 8, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 5, 0, 0, 1, 0, -3, 0, 0, 0, 0, 0, 0], [0, 8, 0, 15, 0, 5, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0], [-10, 0, 0, 0, 0, 0, -19, 0, 0, 0, 0, 0, 0, 5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -4, 0, 0, 7, 6], [12, 0, 0, 0, 0, 0, -8, 0, 9, 0, 0, 0, 0, 11, 0, 0], [11, 0, 0, 0, 0, 0, 9, 0, 4, 0, 0, 0, 0, -15, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 1, -4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -5, 0, 0, -4, -2], [21, 0, 0, 0, 0, 0, -12, 0, -4, 0, 0, 0, 0, 23, 0, 0], [0, -15, 0, 6, 0, -3, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 1, -10], [0, -9, 0, -1, 0, -21, 0, 0, 0, 0, 0, 0, -11, 0, 0, 0], [0, 0, 0, 0, 8, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 0, 0, 0, 0, 13, 0, -5, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 10, -10], [0, -8, 0, -15, 0, -19, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0], [-31, 0, 0, 0, 0, 0, 8, 0, 7, 0, 0, 0, 0, -21, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, -13], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, -1, 0], [0, 0, 0, 0, 2, 0, 0, 9, 0, 3, 0, 0, 0, 0, 0, 0], [0, 18, 0, 13, 0, 12, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0], [0, 10, 0, 5, 0, 7, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 5, 0, -8, 0, 0, 0, 0, -5, 0, 0], [4, 0, 0, 0, 0, 0, -4, 0, -17, 0, 0, 0, 0, 9, 0, 0], [0, 0, 2, 0, -2, 0, 0, -2, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, -2, 0, 0, -4, 0, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 2, 0, 0, -7, -5], [0, 0, -2, 0, -7, 0, 0, 4, 0, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 11], [0, -4, 0, -12, 0, -4, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0], [0, 0, 1, 0, 10, 0, 0, -2, 0, 9, 0, 0, 0, 0, 0, 0], [0, 15, 0, 19, 0, 1, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0], [-5, 0, 0, 0, 0, 0, -9, 0, 8, 0, 0, 0, 0, -15, 0, 0], [0, 3, 0, 26, 0, -5, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -1, 0, 0, 4, 7], [0, 0, 1, 0, -6, 0, 0, -10, 0, -7, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 3, 0, 0, -4, 0, 10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, -12, 0, 0, 6, 0], [10, 0, 0, 0, 0, 0, 8, 0, -3, 0, 0, 0, 0, 27, 0, 0], [0, 24, 0, 19, 0, 12, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 5, 0, 0, 0, 0], [28, 0, 0, 0, 0, 0, -11, 0, 5, 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