
# q-expansion of newform 4410.2.b.f, downloaded from the LMFDB on 25 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 = 4410
weight = 2
poly_data = [1, 0, 0, 0, 0, 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, 0, 0, 0, 0], 1], [[0, 0, 0, 0, 1, 0, 0, 0], 1], [[0, 0, 0, 1, 0, 1, 0, 0], 1], [[0, 0, 1, 0, 0, 0, 1, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 1], 1], [[0, 1, 0, 0, 0, 0, 0, -1], 1], [[0, 0, 1, 0, 0, 0, -1, 0], 1], [[0, 0, 0, 1, 0, -1, 0, 0], 1]]

hecke_ring_character_values = [[3431, [-1, 0, 0, 0, 0, 0, 0, 0]], [2647, [1, 0, 0, 0, 0, 0, 0, 0]], [1081, [-1, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -2, -1, 0, -2, 0, 0, 0], [0, 0, 1, -1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -1], [0, 0, 0, 3, -2, 0, 0, 0], [0, -2, 1, 1, -1, 0, 0, 0], [0, 2, 0, 2, 3, 0, 0, 0], [0, 0, 1, -2, 2, 0, 0, 0], [2, 0, 0, 0, 0, -2, 1, -5], [-2, 0, 0, 0, 0, -1, 4, -1], [-2, 0, 0, 0, 0, 4, 1, -1], [2, 0, 0, 0, 0, 3, 1, -2], [0, 4, -3, -2, -1, 0, 0, 0], [6, 0, 0, 0, 0, 2, 1, 0], [0, -2, 0, -1, 2, 0, 0, 0], [-4, 0, 0, 0, 0, 3, 3, -2], [0, -2, -4, -1, 0, 0, 0, 0], [0, -4, -5, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, -4, 0, -4], [-2, 0, 0, 0, 0, 2, 8, 2], [10, 0, 0, 0, 0, 0, 2, 4], [0, -2, 0, -4, 0, 0, 0, 0], [8, 0, 0, 0, 0, -2, -3, 0], [0, 2, 1, 2, 1, 0, 0, 0], [0, -6, 3, -2, -1, 0, 0, 0], [6, 0, 0, 0, 0, 3, 2, -7], [0, 0, -8, -4, 2, 0, 0, 0], [4, 0, 0, 0, 0, 6, 0, -4], [8, 0, 0, 0, 0, -4, 5, 2], [0, -8, -1, -1, -7, 0, 0, 0], [0, -4, 2, -3, -6, 0, 0, 0], [0, 10, 1, 2, 0, 0, 0, 0], [-4, 0, 0, 0, 0, 4, -5, -4], [0, -4, -4, -2, 6, 0, 0, 0], [0, 0, 0, 0, 0, 1, -1, -6], [-4, 0, 0, 0, 0, -8, 5, -1], [4, 0, 0, 0, 0, -4, 6, -8], [0, 6, 2, 4, -7, 0, 0, 0], [0, 2, -6, 1, 6, 0, 0, 0], [0, 8, -2, -5, -4, 0, 0, 0], [-4, 0, 0, 0, 0, 6, -8, -2], [0, 4, -9, 0, -3, 0, 0, 0], [0, 0, 13, 4, -4, 0, 0, 0], [-12, 0, 0, 0, 0, 5, -4, 1], [0, -6, -1, -2, 11, 0, 0, 0], [16, 0, 0, 0, 0, 4, -2, -4], [0, 0, -10, -1, -6, 0, 0, 0], [0, -4, 8, 6, -2, 0, 0, 0], [0, -6, -10, 9, -4, 0, 0, 0], [0, -8, 4, 8, -1, 0, 0, 0], [-2, 0, 0, 0, 0, -6, 2, 6], [14, 0, 0, 0, 0, 5, 3, 0], [0, -6, 13, -5, -9, 0, 0, 0], [6, 0, 0, 0, 0, -8, 4, 4], [0, 20, 2, 2, 3, 0, 0, 0], [0, 0, 0, 0, 0, 3, 5, -8], [0, -2, -2, -5, 0, 0, 0, 0], [0, -10, -3, 1, 1, 0, 0, 0], [4, 0, 0, 0, 0, 0, -2, 6], [0, 2, 4, 14, -6, 0, 0, 0], [4, 0, 0, 0, 0, 6, 4, 6], [0, 4, 9, 4, -1, 0, 0, 0], [0, -14, 0, -2, -6, 0, 0, 0], [-4, 0, 0, 0, 0, -11, -8, 3], [2, 0, 0, 0, 0, -10, 0, 0], [0, 0, 0, 4, 4, 0, 0, 0], [0, 6, 6, 3, -14, 0, 0, 0], [8, 0, 0, 0, 0, 4, 9, 9], [0, 12, 4, -5, -8, 0, 0, 0], [0, -4, -16, 0, 0, 0, 0, 0], [14, 0, 0, 0, 0, -8, -7, 3], [4, 0, 0, 0, 0, 2, 2, -12], [12, 0, 0, 0, 0, -10, -11, 1], [0, 26, -3, 2, 6, 0, 0, 0], [0, 4, 2, -6, 0, 0, 0, 0], [0, -6, 6, 3, 2, 0, 0, 0], [0, -20, 8, -8, -11, 0, 0, 0], [2, 0, 0, 0, 0, 20, 0, 0], [-14, 0, 0, 0, 0, -7, -2, -11], [0, -4, 2, -3, 2, 0, 0, 0], [0, 2, 12, 2, 14, 0, 0, 0], [0, 24, -14, -10, -7, 0, 0, 0], [0, 4, -8, 12, -6, 0, 0, 0], [0, -14, -8, 13, -6, 0, 0, 0], [-2, 0, 0, 0, 0, 8, 8, -2], [-6, 0, 0, 0, 0, -4, -2, 12], [-10, 0, 0, 0, 0, -18, 8, 4], [2, 0, 0, 0, 0, 14, -2, 2], [-16, 0, 0, 0, 0, 4, -12, -4], [-10, 0, 0, 0, 0, 10, 4, 6], [0, 22, -4, 4, 5, 0, 0, 0], [-8, 0, 0, 0, 0, -12, 12, -2], [-14, 0, 0, 0, 0, -3, -5, -4], [-10, 0, 0, 0, 0, 4, 7, -6], [-10, 0, 0, 0, 0, 8, 6, 10], [0, 10, 5, -3, 1, 0, 0, 0], [14, 0, 0, 0, 0, 17, -8, 9], [18, 0, 0, 0, 0, 6, -11, 7], [0, -14, 8, 2, 14, 0, 0, 0], [-4, 0, 0, 0, 0, -6, -4, 16], [0, 14, 0, 7, -4, 0, 0, 0], [-8, 0, 0, 0, 0, -2, -6, 20], [0, -8, 7, 18, 3, 0, 0, 0], [-14, 0, 0, 0, 0, -16, 0, 0], [6, 0, 0, 0, 0, -11, 3, -14], [0, 0, 0, 11, 6, 0, 0, 0], [0, -4, 11, 4, -2, 0, 0, 0], [0, -2, 7, 10, -1, 0, 0, 0], [-16, 0, 0, 0, 0, 3, -19, -8], [0, -4, 1, -15, 3, 0, 0, 0], [0, -2, 24, -5, -10, 0, 0, 0], [-6, 0, 0, 0, 0, -2, 1, -2], [0, -16, 10, 13, 2, 0, 0, 0], [0, 2, -1, 1, 23, 0, 0, 0], [-10, 0, 0, 0, 0, -5, 11, -6], [0, -22, -4, -10, 18, 0, 0, 0], [0, -18, 15, 8, 12, 0, 0, 0], [0, -22, -2, -13, 0, 0, 0, 0], [-16, 0, 0, 0, 0, 0, 14, -6], [-14, 0, 0, 0, 0, 0, -18, -6], [0, -2, -1, -16, 13, 0, 0, 0], [0, -2, 18, -3, -6, 0, 0, 0], [0, 14, -3, 4, 0, 0, 0, 0], [-10, 0, 0, 0, 0, 10, 10, 14], [-8, 0, 0, 0, 0, -3, -10, 15], [0, -6, -7, 6, -5, 0, 0, 0], [0, -18, -10, -16, 14, 0, 0, 0], [28, 0, 0, 0, 0, -8, 8, 8], [0, -12, 2, -14, 6, 0, 0, 0], [2, 0, 0, 0, 0, -14, -3, 2], [-14, 0, 0, 0, 0, -8, -1, -23], [-18, 0, 0, 0, 0, 10, 8, 4], [0, -12, 1, -6, -22, 0, 0, 0], [-22, 0, 0, 0, 0, 4, 6, 4], [0, 6, 0, -30, 2, 0, 0, 0], [26, 0, 0, 0, 0, -12, -2, 2], [0, 12, 2, 11, 16, 0, 0, 0], [0, -30, -8, 5, 4, 0, 0, 0], [0, 6, -16, 2, 1, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 0, 16], [0, 12, 10, -10, 0, 0, 0, 0], [0, 0, 18, 9, -12, 0, 0, 0], [-4, 0, 0, 0, 0, 5, 6, -17], [0, -10, 4, 10, 20, 0, 0, 0], [6, 0, 0, 0, 0, 5, -11, 6], [0, -6, -14, 11, 6, 0, 0, 0], [0, -28, -10, 4, -10, 0, 0, 0], [2, 0, 0, 0, 0, 0, -3, -3], [10, 0, 0, 0, 0, -7, -8, 1], [-12, 0, 0, 0, 0, 21, -5, -8], [-16, 0, 0, 0, 0, -6, 5, 3], [-8, 0, 0, 0, 0, -11, -11, 6], [0, 2, 22, -1, 0, 0, 0, 0], [-14, 0, 0, 0, 0, 4, -1, 12], [2, 0, 0, 0, 0, 20, -2, 22], [0, 6, -22, 0, 10, 0, 0, 0], [-8, 0, 0, 0, 0, 16, 13, 10], [0, -18, -19, 14, -11, 0, 0, 0], [0, -8, -5, -19, -11, 0, 0, 0], [-16, 0, 0, 0, 0, -2, -4, 10], [-26, 0, 0, 0, 0, 2, -10, -10], [0, -22, -18, 16, 2, 0, 0, 0], [0, 0, 0, 0, 0, -6, -11, -5], [-6, 0, 0, 0, 0, -4, 19, 8], [0, 2, -4, -6, 12, 0, 0, 0], [18, 0, 0, 0, 0, 4, -5, -8], [-10, 0, 0, 0, 0, 4, 20, -10], [0, 10, 17, 16, -14, 0, 0, 0], [0, 12, 8, -25, 8, 0, 0, 0], [0, 6, -14, 5, 4, 0, 0, 0], [-8, 0, 0, 0, 0, -26, 4, -14], [0, 28, 20, -14, 11, 0, 0, 0], [10, 0, 0, 0, 0, 14, -22, 10], [4, 0, 0, 0, 0, 23, -4, -5], [0, -6, -9, -10, 18, 0, 0, 0], [0, -10, 13, 20, 17, 0, 0, 0], [0, -34, 4, 1, -10, 0, 0, 0], [8, 0, 0, 0, 0, 10, 8, 20], [6, 0, 0, 0, 0, -14, 19, 12], [12, 0, 0, 0, 0, -19, 11, 18], [-4, 0, 0, 0, 0, -12, 9, 9], [0, -10, -5, -29, 5, 0, 0, 0], [8, 0, 0, 0, 0, -12, 10, -8], [0, 0, 0, 0, 0, -29, 1, -12], [0, 26, -15, -5, -3, 0, 0, 0], [-6, 0, 0, 0, 0, 0, 14, -12], [-12, 0, 0, 0, 0, 19, 2, 1], [0, -22, 10, -8, 8, 0, 0, 0], [0, -10, 7, -16, 19, 0, 0, 0], [32, 0, 0, 0, 0, 12, 2, 0], [-12, 0, 0, 0, 0, -18, 8, 10], [0, -6, 7, 8, -9, 0, 0, 0], [2, 0, 0, 0, 0, -21, -7, 14], [-22, 0, 0, 0, 0, -4, 9, 0], [4, 0, 0, 0, 0, 27, 3, -16], [30, 0, 0, 0, 0, -13, 3, -6], [-4, 0, 0, 0, 0, -10, -9, -29], [0, -6, -17, 18, 8, 0, 0, 0], [0, 8, 7, 12, 0, 0, 0, 0], [0, -32, 7, 3, -9, 0, 0, 0], [0, 20, 9, 2, -2, 0, 0, 0], [2, 0, 0, 0, 0, 2, -10, -18], [-12, 0, 0, 0, 0, 20, 4, -10], [0, 32, -17, -6, -6, 0, 0, 0], [0, -16, -10, 4, -2, 0, 0, 0], [0, 20, 8, 19, -4, 0, 0, 0], [0, 8, -6, -11, 2, 0, 0, 0], [8, 0, 0, 0, 0, 10, -16, 4], [24, 0, 0, 0, 0, -22, -16, -10], [-8, 0, 0, 0, 0, 8, 24, -4], [-28, 0, 0, 0, 0, 24, 6, 2], [0, 0, 0, 0, 0, 5, -6, -13], [0, -16, 6, -6, -13, 0, 0, 0], [10, 0, 0, 0, 0, 12, -4, 28], [-14, 0, 0, 0, 0, 13, -6, -5], [0, 2, 17, 5, -9, 0, 0, 0], [0, -10, 10, -6, -10, 0, 0, 0], [-6, 0, 0, 0, 0, -6, -16, -10], [0, 20, -12, -12, -1, 0, 0, 0], [0, -16, 12, 9, 2, 0, 0, 0], [6, 0, 0, 0, 0, -10, 12, 22], [-22, 0, 0, 0, 0, 14, 16, 18], [2, 0, 0, 0, 0, 2, -30, 2], [4, 0, 0, 0, 0, 2, -11, 1], [0, -2, -4, -1, 6, 0, 0, 0], [0, 20, -6, 10, 4, 0, 0, 0], [0, -6, -4, -20, 5, 0, 0, 0], [-14, 0, 0, 0, 0, 22, 5, 13], [0, 16, 2, 11, 0, 0, 0, 0], [-12, 0, 0, 0, 0, -18, 10, 18], [0, 24, -22, -5, -6, 0, 0, 0], [0, -2, 2, -12, 24, 0, 0, 0], [-2, 0, 0, 0, 0, 21, -3, 24], [0, -32, 1, 8, 8, 0, 0, 0], [0, 0, 15, 26, 3, 0, 0, 0], [0, -18, -7, -14, 12, 0, 0, 0], [-24, 0, 0, 0, 0, -9, -8, -13], [0, 2, -5, -12, -27, 0, 0, 0], [0, 24, -8, -3, 2, 0, 0, 0], [0, -12, -6, -12, -4, 0, 0, 0], [-10, 0, 0, 0, 0, 13, 4, -5], [-18, 0, 0, 0, 0, -9, -15, 4], [-16, 0, 0, 0, 0, 11, -6, -21], [0, 6, -1, 16, 3, 0, 0, 0], [16, 0, 0, 0, 0, 12, -19, 14], [8, 0, 0, 0, 0, 3, 24, 23], [0, -10, 0, -12, -18, 0, 0, 0], [-12, 0, 0, 0, 0, -13, -13, 30], [34, 0, 0, 0, 0, 7, 18, -7], [0, 26, 18, 12, 0, 0, 0, 0], [0, 24, 2, -10, -15, 0, 0, 0], [-8, 0, 0, 0, 0, -18, -12, 8], [0, -6, 14, -8, -17, 0, 0, 0], [-14, 0, 0, 0, 0, -11, -2, -21], [0, 38, 7, 9, -25, 0, 0, 0], [-38, 0, 0, 0, 0, 4, -16, -8], [0, 2, -24, 4, -10, 0, 0, 0], [-6, 0, 0, 0, 0, 8, -12, 14], [0, 24, 18, 6, 8, 0, 0, 0], [0, -42, -2, -11, -8, 0, 0, 0], [0, -24, 15, -9, -17, 0, 0, 0], [-12, 0, 0, 0, 0, 0, -1, 7], [0, 30, -14, 19, 10, 0, 0, 0], [0, -14, -16, -20, -5, 0, 0, 0], [-2, 0, 0, 0, 0, -2, 12, 20], [0, 0, 0, 0, 0, 5, 19, 16], [0, 4, -1, -2, -19, 0, 0, 0], [0, -22, 28, 9, 4, 0, 0, 0], [-10, 0, 0, 0, 0, -12, 7, -19], [0, 0, -9, 14, 25, 0, 0, 0], [2, 0, 0, 0, 0, -6, 20, 2], [0, 44, -11, -20, -3, 0, 0, 0], [0, 32, -4, 26, 18, 0, 0, 0], [0, 18, -7, -6, 3, 0, 0, 0], [2, 0, 0, 0, 0, 6, -6, -6], [14, 0, 0, 0, 0, 6, -21, 22], [14, 0, 0, 0, 0, -14, 9, 32], [14, 0, 0, 0, 0, 3, 33, -22], [2, 0, 0, 0, 0, 20, -8, 16], [-16, 0, 0, 0, 0, 0, -13, 3], [0, 46, -4, 1, -2, 0, 0, 0], [0, 6, 17, 13, -7, 0, 0, 0], [0, 18, 26, 23, -26, 0, 0, 0], [-12, 0, 0, 0, 0, -6, -20, -30], [0, -20, -21, 10, -21, 0, 0, 0], [0, -8, 11, 22, 12, 0, 0, 0], [-14, 0, 0, 0, 0, 0, -18, 18], [0, -22, -32, 4, -7, 0, 0, 0], [0, 0, 0, 0, 0, -22, -20, 22], [0, -32, 26, 8, -4, 0, 0, 0], [-8, 0, 0, 0, 0, -24, 3, -6], [20, 0, 0, 0, 0, -13, -13, 16], [-18, 0, 0, 0, 0, 2, -29, -16], [0, 8, 12, -2, 7, 0, 0, 0], [-16, 0, 0, 0, 0, 8, 26, -12], [-18, 0, 0, 0, 0, 12, -2, 12], [0, 16, 10, -8, 10, 0, 0, 0], [0, 6, 4, 28, 10, 0, 0, 0], [0, 22, 32, 12, -30, 0, 0, 0], [4, 0, 0, 0, 0, 14, 14, -6], [0, -34, 3, 14, -3, 0, 0, 0], [12, 0, 0, 0, 0, -18, 4, -16], [-10, 0, 0, 0, 0, 2, -8, 22], [0, 30, -13, -2, -25, 0, 0, 0], [0, 44, -20, -3, -10, 0, 0, 0], [0, -4, 28, -5, -24, 0, 0, 0], [-30, 0, 0, 0, 0, 24, 17, 3], [26, 0, 0, 0, 0, 3, 25, -26], [0, 2, -2, 14, -17, 0, 0, 0], [0, 16, 16, -25, 12, 0, 0, 0], [26, 0, 0, 0, 0, 8, 13, 21], [0, 4, -26, -6, -30, 0, 0, 0], [0, -12, 5, 18, -8, 0, 0, 0], [14, 0, 0, 0, 0, 0, 20, -20], [0, -16, 28, -5, -20, 0, 0, 0], [0, 34, -8, 6, 18, 0, 0, 0], [0, 26, -8, -23, -18, 0, 0, 0], [0, 32, 12, 5, -4, 0, 0, 0], [12, 0, 0, 0, 0, 16, -36, -6], [-14, 0, 0, 0, 0, -30, -17, 16], [2, 0, 0, 0, 0, 12, -4, 18], [0, -12, -17, -9, -19, 0, 0, 0], [0, 8, -14, -8, -23, 0, 0, 0], [12, 0, 0, 0, 0, 4, 14, -8], [0, 4, -28, 10, -4, 0, 0, 0], [0, -2, 20, 18, -22, 0, 0, 0], [0, 8, -15, -22, 3, 0, 0, 0], [-34, 0, 0, 0, 0, -4, 14, -6], [0, -30, 2, -18, -18, 0, 0, 0], [0, 4, 6, 48, -7, 0, 0, 0], [-12, 0, 0, 0, 0, 18, -24, -18], [4, 0, 0, 0, 0, -3, 6, -23], [24, 0, 0, 0, 0, 4, 20, 28], [10, 0, 0, 0, 0, -7, 14, -13], [-16, 0, 0, 0, 0, 0, -1, -7], [0, 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