
# q-expansion of newform 882.2.h.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, 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 = 882
weight = 2
poly_data = [9, -3, -2, -1, 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, 1, 0, 0], 1], [[-3, -2, 2, 1], 6], [[3, 2, 0, -1], 2]]

hecke_ring_character_values = [[785, [0, 0, -1, 0]], [199, [-1, 0, 1, 0]]]
aps_data = [[0, 0, 1, 0], [0, -1, 0, 0], [2, -1, -1, -1], [0, 0, 0, 0], [1, 1, 1, 1], [0, 0, -2, 0], [-1, 2, 2, -1], [-5, 0, 5, 0], [-4, -1, -1, -1], [2, 2, -4, -4], [-2, 0, 2, 0], [-2, 0, 2, 0], [1, -2, -8, 1], [-2, 3, -1, -6], [0, 0, 0, 0], [2, -4, 2, 2], [0, 3, -3, -6], [-3, 6, 7, -3], [-8, 3, 5, -6], [0, 3, 3, 3], [-3, 6, -2, -3], [3, -6, -5, 3], [-4, -4, 8, 8], [-8, -2, 10, 4], [-2, 3, -1, -6], [4, 1, 1, 1], [-10, 0, 0, 0], [4, 1, -5, -2], [0, 0, -14, 0], [1, -2, 1, 1], [-4, -3, -3, -3], [-4, -1, -1, -1], [13, 1, 1, 1], [-6, 12, 7, -6], [8, 2, 2, 2], [2, -3, -3, -3], [1, -3, 2, 6], [-2, 6, -4, -12], [4, -8, -8, 4], [0, 0, 6, 0], [2, -4, -10, 2], [8, 3, 3, 3], [3, -6, 9, 3], [7, 0, -7, 0], [-6, 0, 0, 0], [0, 0, 10, 0], [0, 0, 16, 0], [4, 0, -4, 0], [-17, -2, -2, -2], [-10, -3, -3, -3], [7, -2, -5, 4], [-5, 10, 7, -5], [11, -3, -3, -3], [-9, 0, 0, 0], [5, 5, 5, 5], [10, 1, 1, 1], [-1, 2, 5, -1], [4, -6, 2, 12], [2, 6, 6, 6], [-13, -1, 14, 2], [-17, -3, 20, 6], [1, -2, 7, 1], [-13, 0, 0, 0], [12, -6, -6, 12], [3, -6, 10, 3], [0, 0, 6, 0], [-6, 12, -2, -6], [3, -6, -2, 3], [0, 3, -3, -6], [22, 0, -22, 0], [-11, 1, 1, 1], [-5, -5, 10, 10], [2, 6, 6, 6], [-10, 0, 0, 0], [-1, 3, 3, 3], [-28, 2, 2, 2], [-8, -8, -8, -8], [22, 0, -22, 0], [-7, 2, 2, 2], [-8, 9, -1, -18], [7, -14, -11, 7], [-14, -6, 20, 12], [-4, 8, 20, -4], [-13, 3, 3, 3], [0, 0, -8, 0], [3, -6, 30, 3], [33, 0, 0, 0], [12, -24, -5, 12], [-19, 5, 14, -10], [3, -6, 13, 3], [4, -11, 7, 22], [28, -2, -2, -2], [-9, 18, -5, -9], [1, -2, 22, 1], [11, -3, -3, -3], [-18, 6, 6, 6], [-10, 8, 8, 8], [5, -10, 8, 5], [25, 3, -28, -6], [-14, 6, 8, -12], [-8, -3, 11, 6], [-4, 8, -16, -4], [-3, 0, 3, 0], [-3, 6, -6, -3], [-8, 9, -1, -18], [3, -6, -20, 3], [-1, -4, 5, 8], [-8, -2, 10, 4], [-24, 0, 24, 0], [-14, -3, 17, 6], [8, -6, -6, -6], [6, -12, 16, 6], [-5, 10, -2, -5], [17, -12, -12, -12], [-16, 9, 9, 9], [1, -14, -14, -14], [-9, 18, 4, -9], [-4, 8, 8, -4], [-22, 2, 2, 2], [14, 2, -16, -4], [-17, -3, 20, 6], [13, -3, -10, 6], [-12, 24, 6, -12], [-13, 26, 14, -13], [3, -6, 13, 3], [-24, 6, 6, 6], [0, 0, -44, 0], [4, -2, -2, 4], [16, 6, -22, -12], [-34, 3, 3, 3], [3, -6, -2, 3], [10, -20, -20, 10], [-10, 3, 3, 3], [32, -6, -6, -6], [26, -4, -4, -4], [0, 0, 10, 0], [5, -10, 23, 5], [4, 0, -4, 0], [-5, 10, 25, -5], [5, -10, -10, 5], [35, -3, -3, -3], [-6, 12, 24, -6], [-8, 6, 2, -12], [12, 0, 0, 0], [-6, 12, 34, -6], [8, -4, -4, 8], [25, 3, -28, -6], [-2, 16, 16, 16], [41, 3, 3, 3], [7, -5, -2, 10], [-16, 18, 18, 18], [-6, 6, 6, 6], [-19, 9, 9, 9], [2, -10, -10, -10], [-19, 3, 3, 3], [41, -1, -40, 2], [37, 3, -40, -6], [-6, 12, -24, -6], [2, -12, -12, -12], [-9, -3, 12, 6], [-3, 6, -24, -3], [21, -3, -3, -3], [3, -6, -41, 3], [-5, 1, 4, -2], [-30, -3, 33, 6], [-44, -2, -2, -2], [-12, 24, -8, -12], [2, 3, 3, 3]]
