
# q-expansion of newform 384.5.h.f, downloaded from the LMFDB on 24 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 = 384
weight = 5
poly_data = [9, 0, -4, 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], 1], [[0, -2, 0, 2], 1], [[0, 28, 0, -4], 1], [[-288, 0, 144, 0], 1]]

hecke_ring_character_values = [[127, [1, 0, 0, 0]], [133, [-1, 0, 0, 0]], [257, [-1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [3, -1, 0, 0], [0, 0, -1, 0], [0, 0, 1, 0], [-134, 0, 0, 0], [0, 0, 0, -1], [0, 28, 0, 0], [0, 58, 0, 0], [0, 0, 0, -2], [0, 0, -21, 0], [0, 0, 29, 0], [0, 0, 0, 5], [0, 152, 0, 0], [0, -50, 0, 0], [0, 0, 0, 8], [0, 0, 43, 0], [1382, 0, 0, 0], [0, 0, 0, -1], [0, 22, 0, 0], [0, 0, 0, 6], [4894, 0, 0, 0], [0, 0, -39, 0], [5914, 0, 0, 0], [0, 1068, 0, 0], [5858, 0, 0, 0], [0, 0, 391, 0], [0, 0, -75, 0], [-21274, 0, 0, 0], [0, 0, 0, 23], [0, 408, 0, 0], [0, 0, -563, 0], [-17338, 0, 0, 0], [0, 3248, 0, 0], [0, -2318, 0, 0], [0, 0, 247, 0], [0, 0, -619, 0], [0, 0, 0, -13], [0, 5298, 0, 0], [0, 0, 0, -126], [0, 0, -613, 0], [-6586, 0, 0, 0], [0, 0, 0, -95], [0, 0, 0, -36], [-24386, 0, 0, 0], [0, 0, -1693, 0], [0, 0, -899, 0], [0, -2082, 0, 0], [0, 0, 1137, 0], [-32678, 0, 0, 0], [0, 0, 0, 173], [0, -5620, 0, 0], [0, 0, 0, -232], [766, 0, 0, 0], [-2054, 0, 0, 0], [0, 2200, 0, 0], [0, 0, 0, 286], [0, 0, -201, 0], [0, 0, -1423, 0], [0, 0, 0, 73], [0, -13980, 0, 0], [0, 15882, 0, 0], [0, 0, -909, 0], [0, 2758, 0, 0], [0, 0, 0, -62], [-27458, 0, 0, 0], [0, 0, 4447, 0], [0, -9318, 0, 0], [63454, 0, 0, 0], [-28166, 0, 0, 0], [0, 0, 0, -209], [0, 3280, 0, 0], [0, 0, 0, -458], [0, 0, -5707, 0], [0, 0, 0, 509], [0, 30622, 0, 0], [0, 0, 0, 692], [0, 0, 1555, 0], [0, 0, 0, 735], [0, 22852, 0, 0], [-125086, 0, 0, 0], [72346, 0, 0, 0], [0, 0, 0, 789], [0, 0, 0, -872], [303266, 0, 0, 0], [0, 0, -6203, 0], [151354, 0, 0, 0], [0, 10172, 0, 0], [30626, 0, 0, 0], [0, 0, -8597, 0], [0, 0, 8045, 0], [-299558, 0, 0, 0], [0, 0, 0, -136], [0, 0, -4023, 0], [367142, 0, 0, 0], [0, -7090, 0, 0], [0, 0, 0, 818], [0, 0, -421, 0], [0, -24832, 0, 0], [0, -10938, 0, 0], [0, 0, 0, 775], [0, 30442, 0, 0], [0, 0, 3991, 0], [-238118, 0, 0, 0], [0, 19312, 0, 0], [0, 24410, 0, 0], [-306146, 0, 0, 0], [-460954, 0, 0, 0], [0, -30048, 0, 0], [0, 0, 0, -762], [74206, 0, 0, 0], [0, 0, 7893, 0], [0, 0, 0, 641], [0, -84460, 0, 0], [0, -18070, 0, 0], [0, 0, -11831, 0], [0, 8660, 0, 0], [0, 30730, 0, 0], [0, 0, 0, -702], [0, 0, -3805, 0], [383174, 0, 0, 0], [0, 0, 0, -179], [679778, 0, 0, 0], [0, 0, -177, 0], [552634, 0, 0, 0], [0, -38622, 0, 0], [0, 0, 22103, 0], [0, 0, 0, -11], [0, 0, 0, 3196], [0, 0, -21659, 0], [0, 0, 0, -369], [0, 41150, 0, 0], [0, 0, 0, -1654], [0, 0, -19855, 0], [0, 0, 0, 273], [0, 125208, 0, 0], [259006, 0, 0, 0], [0, 0, -15753, 0], [0, 107970, 0, 0], [0, 0, -889, 0], [0, 107720, 0, 0], [0, -142090, 0, 0], [0, 0, 11799, 0], [0, 0, -8623, 0], [44966, 0, 0, 0], [0, 0, 0, -21], [0, 0, 0, 1414], [0, 0, 0, -1955], [0, -13968, 0, 0], [0, 78718, 0, 0], [0, 0, 0, 724], [0, 0, 0, 2891], [0, 101640, 0, 0], [0, 47402, 0, 0], [0, 0, 0, -1446], [0, -123490, 0, 0], [0, 0, 0, 176], [0, 0, 23393, 0], [0, -30060, 0, 0], [952706, 0, 0, 0], [0, 0, -18221, 0], [-1241722, 0, 0, 0], [0, 11968, 0, 0], [0, 0, -17791, 0], [-125894, 0, 0, 0], [0, 208972, 0, 0], [0, 0, 0, -2606], [0, 0, 39241, 0], [0, 0, 0, -4947]]
