# q-expansion of newform 880.2.bd.c, downloaded from the LMFDB on 31 May 2026.

# 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 = 880
weight = 2
poly_data = [1, 0, 0, 0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = [[111, [1, 0, 0, 0]], [661, [1, 0, 0, 0]], [177, [0, 0, 1, 0]], [321, [-1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [-1, 0, 1, -1], [0, -1, 0, -2], [0, 0, 0, -3], [0, -1, 3, 1], [3, 0, 3, 0], [-3, 0, 3, -3], [3, 0, 0, 0], [3, 0, -3, -4], [3, 3, 0, -3], [3, -3, 0, 3], [-2, -3, -2, 0], [0, -3, -6, -3], [3, 6, 3, 0], [-3, 2, -3, 0], [6, 0, -6, 5], [0, 1, 0, 1], [0, 3, -3, 3], [4, 0, 4, 0], [-3, -3, 0, 3], [6, 0, 6, 0], [-6, 3, 0, -3], [3, -6, 3, 0], [0, -6, -3, -6], [2, 6, 2, 0], [0, 6, 6, 6], [-2, 0, 2, -6], [-6, 0, 6, -6], [-6, -6, 0, 6], [-3, 0, 3, -2], [0, 0, 0, 6], [0, -12, -3, -12], [-6, 8, -6, 0], [6, -6, 0, 6], [-15, 3, 0, -3], [0, 0, -12, 0], [10, -9, 10, 0], [-1, 0, 1, 21], [0, 0, 0, 9], [12, -6, 12, 0], [0, 3, 18, 3], [12, 0, 0, 0], [0, 2, 0, -2], [3, 3, 3, 0], [3, 0, -3, -6], [0, -3, -9, -3], [0, -6, 9, -6], [8, 0, -8, -6], [-15, 0, 15, 0], [0, 15, 6, 15], [-9, -15, -9, 0], [-12, -9, 0, 9], [0, 9, 6, 9], [-12, -6, 0, 6], [-9, -10, -9, 0], [0, 9, 0, 0], [0, -14, 0, -14], [0, 9, 6, 9], [6, 0, -6, 6], [0, -12, 6, -12], [-9, 6, -9, 0], [-3, 18, -3, 0], [3, 0, -3, 12], [-3, -9, 0, 9], [8, 0, -8, -6], [-6, 5, -6, 0], [-12, 9, 0, -9], [-15, 0, 15, 3], [6, 0, -6, -18], [-6, -12, 0, 12], [-12, 0, 12, -4], [-18, 12, 0, -12], [2, 0, 2, 0], [3, 6, 3, 0], [0, -12, -12, -12], [0, 0, 0, 2], [0, -2, 0, -2], [-10, -12, -10, 0], [-9, -8, 0, 8], [-6, 12, 0, -12], [0, 0, -6, 0], [-18, -6, 0, 6], [0, -6, -12, -6], [-5, 0, 5, -24], [12, 9, 0, -9], [12, 0, -12, 10], [0, 6, -12, 6], [-3, 0, 3, 15], [0, -3, -3, -3], [-1, 0, 1, 24], [-9, 11, -9, 0], [24, -9, 0, 9], [-5, -24, -5, 0], [0, 6, 3, 6], [0, -12, 18, -12], [12, 18, 12, 0], [0, -10, 18, -10], [36, -4, 0, 4], [-6, 18, -6, 0], [0, 3, 15, 3], [18, 0, -18, 18], [15, 0, -15, -12], [-3, -12, -3, 0], [6, 6, 0, -6], [0, -6, 21, -6], [-7, 18, -7, 0], [3, -25, 3, 0], [-18, -6, -18, 0], [0, -9, -9, -9], [0, 24, 6, 24], [18, 0, -18, -3], [-18, 0, -18, 0], [24, -10, 24, 0], [0, 0, 10, 0], [11, 9, 0, -9], [-9, -2, 0, 2], [-5, 0, 5, -39], [3, 44, 3, 0], [-6, 0, 6, -5], [3, -18, 0, 18], [-4, 9, 0, -9], [9, 9, 9, 0], [-12, 0, 12, -30], [-3, 0, 3, -13], [-26, -9, 0, 9], [0, 9, -15, 9], [0, 0, 10, 0], [0, -21, -9, -21], [-10, -24, -10, 0], [9, -24, 9, 0], [0, -24, 0, 24], [-6, 3, -6, 0], [5, -27, 0, 27], [-8, -30, -8, 0], [0, -9, 6, -9], [18, -18, 0, 18], [0, 0, 0, 23], [-3, 0, 3, 12], [-6, -20, -6, 0], [12, -18, 0, 18], [0, 18, 21, 18], [0, -18, 24, -18], [1, 0, -1, -42], [3, 0, -3, 0], [0, -18, 20, -18], [0, 2, 36, 2], [15, -24, 15, 0], [-15, 0, 15, 15], [0, 0, -32, 0], [-9, 0, 9, 14], [18, 0, -18, -18], [0, -6, 0, 6], [19, 0, -19, -15], [-18, 0, 18, 6], [5, -9, 5, 0], [9, 25, 0, -25], [18, 3, 0, -3], [0, 2, 45, 2], [12, 0, -12, 18], [0, 33, 3, 33], [-9, 19, -9, 0], [-27, 3, -27, 0], [6, 0, -6, 3], [18, 22, 0, -22], [12, 22, 12, 0], [-12, 0, 12, 4], [0, -18, 0, 18], [33, 0, -33, 12]]