
# q-expansion of newform 882.4.g.m, 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
    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 = 882
weight = 4
poly_data = [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 = [[785, [1, 0]], [199, [0, -1]]]
aps_data = [[0, -2], [0, 0], [0, 22], [0, 0], [26, -26], [-54, 0], [74, -74], [0, -116], [0, -58], [-208, 0], [252, -252], [0, -50], [-126, 0], [164, 0], [0, -444], [12, -12], [124, -124], [0, 162], [860, -860], [238, 0], [146, -146], [0, 984], [-656, 0], [0, -954], [526, 0], [1306, -1306], [0, 508], [0, 498], [614, -614], [1232, 0], [-2808, 0], [0, 520], [-2516, 2516], [-2672, 0], [0, -1164], [-1672, 1672], [-446, 446], [0, -428], [-4, 0], [0, -590], [3534, -3534], [1098, 0], [0, -4854], [1498, -1498], [-620, 0], [-32, 32], [4268, 0], [3464, 0], [-3252, 3252], [0, -418], [0, 2084], [-1662, 0], [-6182, 6182], [996, 0], [0, -5994], [6414, -6414], [2686, -2686], [0, -5100], [4426, -4426], [-7508, 0], [-3412, 3412], [4734, 0], [5836, 0], [5620, -5620], [0, -6082], [0, -7308], [0, 8020], [4590, 0], [-6546, 6546], [-7994, 0], [4650, -4650], [0, 346], [6784, -6784], [0, 6098], [-2660, 0], [0, -760], [-104, 104], [0, -4398], [0, 13236], [9490, -9490], [-4236, 0], [918, 0], [-11814, 11814], [-8374, 0], [0, -3840], [0, 10166], [-8200, 0], [0, 6074], [2006, 0], [-3728, 0], [0, -6380], [17180, -17180], [2728, -2728], [-2574, 0], [0, 7484], [-7920, 0], [0, 7254], [-17862, 17862], [0, 592], [0, 6402], [-8988, 0], [3244, -3244], [9812, -9812], [0, -12156], [-6876, 6876], [-20002, 20002], [-18404, 0], [0, -9846], [9234, -9234], [-1510, 0], [0, -17544], [-9246, 9246], [29212, 0], [-7096, 7096], [488, 0], [8756, -8756], [-3364, 0], [21804, -21804], [0, 13488], [-28946, 0], [20642, -20642], [-17602, 0], [0, 4266], [-26874, 26874], [0, 17128], [-11968, 0], [0, 5278], [0, 6720], [16804, 0], [0, -27522], [-21132, 21132], [30, 0], [0, 15480], [28770, 0], [0, -12418], [12346, 0], [38098, -38098], [-13824, 13824], [-22170, 0], [16288, -16288], [-8720, 0], [0, -31372], [17648, -17648], [2382, 0], [-26650, 26650], [-24092, 0], [-8194, 0], [-16962, 16962], [0, 48556], [0, -34274], [0, -7126], [9222, 0], [37652, 0], [0, -21996], [-14844, 14844], [19446, 0], [0, 39200], [0, -15954], [2546, 0], [430, -430], [0, 38266], [28216, 0], [-27712, 0], [0, 32976], [-8940, 8940], [-3288, 3288], [-33944, 33944], [54562, -54562]]
