
# q-expansion of newform 847.2.f.m, downloaded from the LMFDB on 17 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
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_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 = [1, -1, 1, -1, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = [[122, [1, 0, 0, 0]], [365, [0, 0, 0, -1]]]
aps_data = [[1, 1, -1, -1], [0, -2, 2, -2], [0, 2, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0], [2, 0, 0, -2], [-2, 0, -2, 0], [0, -4, 0, -4], [0, 0, 4, -4], [-4, 4, 0, 6], [4, -6, 6, -4], [-4, 4, 0, 6], [0, 2, 8, 2], [-8, 0, 0, 0], [0, -2, 6, -2], [-2, 6, -6, 2], [-2, 2, 0, 0], [-2, -4, -2, 0], [12, 0, 4, -4], [4, 4, 4, 0], [-2, 2, 0, -2], [-4, -4, 4, 4], [-12, 8, -12, 0], [2, 0, 0, 0], [-10, -2, 2, 10], [-6, 8, -8, 6], [6, -6, 0, -4], [0, 0, 4, 0], [2, 0, 4, -4], [0, 0, 2, 0], [-8, 16, -8, 0], [-12, 0, -16, 16], [-4, -10, -4, 0], [-4, 4, 0, -4], [0, 14, 0, 0], [0, -8, 4, -8], [0, 8, -2, 8], [-4, 16, -16, 4], [0, 8, -8, 0], [0, -2, 16, -2], [0, -8, 4, -8], [12, 6, 12, 0], [8, -8, 0, 8], [16, -14, 16, 0], [2, 0, 0, 0], [4, 0, -10, 10], [-4, -12, 12, 4], [0, -10, 6, -10], [-16, 16, 0, -4], [2, 2, -2, -2], [2, 10, -10, -2], [0, 16, -16, 16], [2, 0, 18, -18], [-8, 14, -14, 8], [0, 0, 0, 6], [0, 0, 0, 0], [-12, 6, -12, 0], [4, -4, 0, -8], [-10, -6, 6, 10], [4, -10, 4, 0], [0, -16, -4, -16], [18, -18, 0, -14], [4, 0, 8, -8], [0, 6, -18, 6], [-8, -2, -8, 0], [-14, 14, -14, 14], [-4, 0, -16, 16], [-4, 4, 0, -14], [8, 8, 8, 0], [0, 2, 4, 2], [10, 0, 16, -16], [8, -8, 0, -20], [-18, 18, 0, 12], [-6, 0, 0, 0], [-28, 8, -28, 0], [-4, 18, -18, 4], [16, -16, 0, -6], [-26, 0, -16, 16], [-8, -10, -8, 0], [-2, 20, -20, 2], [16, 0, 14, -14], [0, 20, -10, 20], [12, -12, 12, -12], [-4, 4, 0, 6], [-8, 0, -4, 4], [0, -8, -12, -8], [26, -22, 22, -26], [-24, 10, -24, 0], [6, 0, 10, -10], [-8, 0, 4, -4], [-26, 18, -26, 0], [-4, 20, -4, 0], [0, -20, -4, -20], [0, 0, 0, 0], [-4, 4, 0, -4], [0, -12, -4, -12], [0, 4, 34, 4], [20, -20, 0, -18], [0, 44, 0, 0], [6, 18, -18, -6], [0, 0, -28, 0], [8, -8, 0, -34], [20, -32, 32, -20], [0, 4, -14, 4], [20, 0, -8, 8], [-4, 26, -4, 0], [18, -18, 0, -16], [42, 0, 6, -6], [12, 4, 12, 0], [18, -18, 0, -26], [8, 0, 0, -8], [0, -28, 30, -28], [14, 0, -12, 12], [0, 18, -18, 18], [0, 0, 0, 24], [0, 4, -22, 4], [-18, 18, -18, 0], [-20, -6, 6, 20], [-8, 8, 0, -30], [8, 0, 16, -16], [-26, 0, -24, 24], [-14, 2, -2, 14], [-6, 0, -6, 0], [4, 0, -16, 16], [0, -30, 30, 0], [0, -4, 22, -4], [8, 2, 8, 0], [-38, 38, 0, 28], [4, 0, 34, -34], [2, -2, 0, 10], [20, -12, 12, -20], [16, -24, 16, 0], [0, -4, 36, -4], [-6, -10, 10, 6], [-18, -4, 4, 18], [-22, 0, 6, -6], [0, -20, -6, -20], [-12, -24, -12, 0], [8, 2, 8, 0], [-8, 34, -8, 0], [0, 24, -4, 24], [-24, 24, 0, 30], [24, -40, 40, -24], [-8, 0, 0, 8], [24, -24, 0, -22], [0, 26, 2, 26], [-14, -8, -14, 0], [-14, 0, -2, 2], [-8, 0, -30, 30], [-8, 16, -8, 0], [0, 12, -34, 12], [10, 0, -12, 12], [0, -8, 4, -8], [20, -20, 0, 8], [-4, -16, -4, 0], [-40, 36, -36, 40], [16, 16, 16, 0], [36, -6, 36, 0], [-2, -12, 12, 2], [22, -28, 22, 0], [-20, 0, 24, -24], [-8, 8, 0, 18], [4, 0, 16, -16], [0, 18, -18, 18], [-18, 10, -10, 18], [-10, 10, 0, -28], [32, 0, 36, -36], [14, -14, 0, 10]]
