
# q-expansion of newform 384.4.f.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 = 4
poly_data = [169, 0, 0, 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, 0, 2, 0], 13], [[0, 13, -1, -1], 13], [[0, 26, 0, 2], 13]]

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], [0, 0, 1, 0], [0, 1, 2, 0], [0, 0, 0, 1], [0, -23, 0, 0], [0, -22, 0, 0], [0, 0, 0, -2], [0, -1, -2, 0], [88, 0, 0, 0], [0, 25, 50, 0], [0, 0, 0, 21], [0, 166, 0, 0], [0, 0, 0, -48], [0, -23, -46, 0], [-384, 0, 0, 0], [0, 45, 90, 0], [0, 315, 0, 0], [0, -118, 0, 0], [0, 5, 10, 0], [-680, 0, 0, 0], [422, 0, 0, 0], [0, 0, 0, 73], [0, 93, 0, 0], [0, 0, 0, -94], [-1062, 0, 0, 0], [0, 1, 2, 0], [0, 0, 0, -195], [0, -117, 0, 0], [0, 362, 0, 0], [0, 0, 0, 100], [0, 0, 0, 221], [0, 871, 0, 0], [0, 0, 0, 236], [0, 75, 150, 0], [0, -47, -94, 0], [0, 0, 0, -147], [0, -974, 0, 0], [0, -333, -666, 0], [-2008, 0, 0, 0], [0, -119, -238, 0], [0, 1191, 0, 0], [0, 1326, 0, 0], [-4784, 0, 0, 0], [-3074, 0, 0, 0], [0, -475, -950, 0], [0, 0, 0, 189], [0, 337, 674, 0], [0, 0, 0, 121], [0, -635, 0, 0], [0, 2278, 0, 0], [0, 0, 0, -86], [1920, 0, 0, 0], [-1618, 0, 0, 0], [0, 793, 0, 0], [0, 0, 0, -388], [4344, 0, 0, 0], [0, 269, 538, 0], [0, 0, 0, -319], [0, 2478, 0, 0], [0, 0, 0, -194], [0, 783, 1566, 0], [0, 181, 362, 0], [0, 813, 1626, 0], [-10104, 0, 0, 0], [1198, 0, 0, 0], [0, 493, 986, 0], [0, -601, -1202, 0], [390, 0, 0, 0], [0, -2183, 0, 0], [0, 2746, 0, 0], [0, 0, 0, 8], [-2216, 0, 0, 0], [0, 0, 0, -699], [0, 1014, 0, 0], [0, 129, 258, 0], [-2128, 0, 0, 0], [0, 189, 378, 0], [0, -3110, 0, 0], [0, 0, 0, -190], [-1270, 0, 0, 0], [0, -4563, 0, 0], [0, 1718, 0, 0], [1920, 0, 0, 0], [-15766, 0, 0, 0], [0, 0, 0, 717], [0, -4095, 0, 0], [0, 0, 0, 1350], [10682, 0, 0, 0], [0, -303, -606, 0], [0, 0, 0, 261], [0, -6123, 0, 0], [15008, 0, 0, 0], [0, 0, 0, 561], [0, -2509, 0, 0], [0, 1577, 3154, 0], [10632, 0, 0, 0], [0, -351, -702, 0], [0, 0, 0, -156], [0, -403, -806, 0], [0, -4742, 0, 0], [0, -537, -1074, 0], [0, -107, -214, 0], [0, 8949, 0, 0], [0, 0, 0, 2068], [0, 327, 654, 0], [-5066, 0, 0, 0], [0, -10021, 0, 0], [0, 0, 0, 2584], [4344, 0, 0, 0], [-18250, 0, 0, 0], [0, 0, 0, 517], [0, -5874, 0, 0], [0, 0, 0, 382], [0, -705, -1410, 0], [0, 0, 0, -2631], [0, 0, 0, -2318], [0, -1593, -3186, 0], [-7384, 0, 0, 0], [0, -1479, -2958, 0], [0, -10217, 0, 0], [0, -6074, 0, 0], [12730, 0, 0, 0], [0, 1929, 3858, 0], [0, -3279, 0, 0], [0, 2987, 5974, 0], [0, 3085, 6170, 0], [0, -12698, 0, 0], [-7888, 0, 0, 0], [0, 0, 0, -635], [0, 7386, 0, 0], [0, 2193, 4386, 0], [19080, 0, 0, 0], [0, 0, 0, 1993], [0, 2094, 0, 0], [0, 0, 0, -3016], [-10370, 0, 0, 0], [0, 1057, 2114, 0], [0, -2293, -4586, 0], [0, 317, 634, 0], [0, 0, 0, -3320], [0, 1029, 2058, 0], [0, 1481, 2962, 0], [0, 0, 0, 1761], [0, 3299, 0, 0], [0, 20514, 0, 0], [17816, 0, 0, 0], [0, -14154, 0, 0], [0, 0, 0, -3308], [0, 33, 66, 0], [-48208, 0, 0, 0], [0, -21422, 0, 0], [0, 0, 0, 684], [0, 471, 942, 0], [-37336, 0, 0, 0], [0, 1617, 3234, 0], [35360, 0, 0, 0], [0, 0, 0, -431], [0, 0, 0, 178], [34034, 0, 0, 0], [0, 3625, 7250, 0], [0, 12431, 0, 0], [0, 0, 0, 2236], [0, 0, 0, 3929], [0, 21761, 0, 0], [0, 0, 0, 1734], [-42360, 0, 0, 0], [0, 0, 0, -2471], [0, 2678, 0, 0]]
