# q-expansion of newform 1664.2.a.bb, downloaded from the LMFDB on 22 September 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, 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 = 1664
weight = 2
poly_data = [-2, 8, 9, -6, -2, 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], 1], [[-1, 2, 0, 0, 0], 1], [[-3, -1, 1, 0, 0], 1], [[1, -2, -5, 0, 1], 1], [[-4, -8, 5, 2, -1], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0, 0, 0], [0, 0, -1, 0, 0], [0, 0, 0, 1, 0], [-1, 0, -1, 1, 1], [1, -1, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, -2, 1, 0], [3, 0, 0, -1, 1], [0, 1, 0, 1, -1], [2, 1, 0, 1, -1], [-5, -1, -2, 0, 0], [2, 1, 2, 0, 1], [2, -1, 0, 1, 1], [2, 1, 1, 1, 1], [-1, -1, 3, -2, -2], [2, 1, 0, -1, -1], [1, -1, 0, 0, 0], [2, -1, -4, 1, 1], [3, 0, 0, 3, 1], [1, -2, 1, 1, 1], [0, 1, 0, -1, -3], [-6, 2, 0, 0, 0], [3, -2, -2, 1, -1], [0, 0, 0, 2, 2], [0, -1, 0, -3, -1], [6, 0, 4, -2, 0], [0, 2, 0, 2, 2], [-2, 1, 4, 1, -1], [6, -1, -2, -2, -1], [4, 0, 4, -2, -2], [-4, -1, 4, 3, 1], [0, 1, 1, -1, 3], [-2, 1, 0, 3, 3], [-4, -2, 1, -2, -2], [2, -2, 0, -2, -2], [-5, 1, 5, -4, 0], [6, 1, 0, 1, 3], [1, -1, 2, 4, 0], [-3, -1, -2, 2, -2], [-6, -2, 0, 0, -2], [-2, 2, 5, -4, 0], [2, 0, -4, -2, 0], [2, 3, 2, 1, -1], [-8, -2, 4, 2, 0], [6, 3, 6, 0, -1], [0, 0, -2, -4, 0], [12, 1, 5, -1, -1], [1, 1, 1, -2, 2], [-1, -1, 2, -6, -2], [-4, 2, 4, 1, -2], [-6, -1, 4, -6, -1], [9, 0, 3, -5, -1], [8, 0, -4, -2, 2], [8, -1, -4, 5, 3], [-2, 0, -2, -1, -2], [2, 3, -4, -3, -1], [-2, -1, 4, -3, 1], [-13, -1, -1, -2, 2], [-2, 3, 4, -3, 1], [0, -4, 0, -2, -2], [-12, -1, 0, 1, -1], [-10, -3, -2, 0, 1], [13, -1, 0, 0, 0], [0, 0, -6, 8, 0], [8, 3, 8, -2, -3], [-4, 0, -4, -2, -2], [11, 4, 10, -1, -3], [16, 1, 4, 0, -1], [-8, 2, -7, 6, -2], [0, -4, 0, -3, -4], [-10, -2, 0, -4, 2], [-15, 1, 6, 0, 0], [14, 0, 2, -2, 2], [2, -4, 0, -2, -4], [-1, 1, 2, 0, 0], [9, -1, 5, 0, 4], [-2, -1, -4, -5, -3], [-8, 0, -4, 6, 2], [-4, 1, 8, -3, -3], [-2, -1, -4, 1, 5], [-20, -2, -5, -2, -2], [8, -4, 0, 1, -4], [5, -2, -1, 1, -3], [-6, 3, -4, 0, -1], [-8, 5, -2, 5, -1], [0, -3, 3, 3, -1], [-12, -1, -8, 1, -5], [10, -2, 0, 6, 2], [-2, 5, -2, 4, 1], [1, 3, 6, 6, 2], [-8, 5, 0, -1, 1], [3, -3, -1, -4, -4], [-5, 3, 10, -8, 0], [-14, -3, 5, 1, -3], [-1, 3, 2, -2, 2], [-14, -2, -2, 0, 4], [8, -2, -4, -4, 4], [0, 1, 12, 0, -1], [-2, -3, 4, -3, -5], [10, -1, -2, -8, -1], [-8, -2, -5, 2, -6], [-6, -5, -6, 0, 3], [-10, -2, 5, -4, -4], [22, 0, -6, 5, 2], [2, 2, -1, -4, -4], [2, -7, 0, -3, 3], [11, -1, -2, -6, -2], [-2, -3, -4, 5, 3], [-2, -1, -2, -7, 3], [12, 4, 2, 5, 0], [2, 1, -10, 7, 1], [18, 0, 8, -4, 4], [-16, 2, 4, -2, -4], [5, -1, -6, 4, 0], [-1, 1, 5, -4, -4], [-4, 2, 4, -8, 0], [-9, 2, -8, 1, -5], [0, -5, -2, -5, 1], [-10, 0, 4, -6, -4], [6, -1, 4, 3, 5], [4, -4, 4, -6, -6], [-8, 4, -6, 9, 4], [18, -2, 4, 0, -2], [17, 2, -10, 7, 5], [-1, 0, 6, -1, 1], [-6, -3, -8, -5, -1], [8, 4, -12, 6, 6], [-16, -6, -6, 2, -6], [-8, -9, -2, -1, -3], [12, 4, -4, -1, 4], [17, 5, -14, 2, 2], [21, 1, 7, 2, -2], [-16, -3, -4, -3, 7], [10, -5, 4, -7, 5], [18, 7, 12, -1, 5], [12, 6, 0, -2, -4], [-12, 4, 4, 1, 0], [9, -1, 6, -12, -8], [2, 0, 16, -8, -4], [8, -7, -4, -8, -1], [11, 0, -6, 3, 1], [12, -2, 4, -9, -2], [-12, 5, 2, -7, -5], [-3, 0, -12, 1, -1], [-6, 0, -4, 4, 0], [5, -5, -10, -2, 2], [0, -4, 8, -5, 0], [10, 4, -8, 8, 4], [2, 5, 4, -7, -1], [5, 0, -13, -5, 3], [12, 0, -4, 7, 8], [22, 0, 2, -1, -2], [-2, -1, -1, 3, -1], [-22, 3, 0, 9, -1], [10, 3, -11, 3, 7], [-2, -3, 2, -1, -7], [-22, -2, 4, 4, 0], [-12, 2, -12, 14, 8], [22, -4, 0, 0, 4], [12, -4, 4, 3, 4], [9, -1, 8, -8, -12], [-26, -6, -10, 3, 0], [-3, 4, -3, 11, 7], [-4, 3, -1, 1, -3], [16, 4, 0, -6, 6], [-25, 0, -1, 1, -7], [14, 0, 14, -10, -6], [-2, 1, 4, 9, -1], [0, 3, 12, -11, -5], [-18, 6, 0, 6, -2], [23, 2, 8, -3, -9], [-26, 1, -10, 10, 9], [0, 1, 4, 5, -1], [8, 2, -8, 6, -4], [0, 0, -14, -4, 4], [14, -3, -4, -10, 1], [5, 2, -2, 3, 1], [16, -4, -4, -2, 6], [4, 4, -12, 8, 4], [2, 3, 0, 9, 9], [-3, -1, -10, 6, 2], [8, 5, -8, -1, 1], [-2, 3, -16, 11, 5], [-8, -1, -4, 9, -1], [5, -1, 2, -2, -6], [6, -5, 4, -5, 9], [14, 1, -4, 9, 3], [-1, -5, 8, -10, 6], [-6, 4, 0, -10, 8], [-23, -3, -10, 0, 8], [-4, -5, -4, -4, -3], [7, 0, 2, -5, -3], [8, 4, 15, -4, -4], [18, 8, 0, 0, 4], [-10, 3, -9, 7, 11], [-6, 6, 8, -6, -2], [18, -1, -4, 9, 1], [-14, -6, 12, 0, -2], [-2, -2, 4, -4, -6], [18, 3, 10, -3, 3], [12, 6, 12, -9, -10], [6, -4, 0, -2, 2], [10, 3, -6, 0, 3], [20, -1, 8, -8, -7], [13, 4, -6, 5, 7], [2, -3, 4, 5, -1], [9, 11, 9, -4, 0], [2, 3, 12, -5, -3], [-4, 7, -4, 5, -5], [2, 2, 3, 4, -4], [-6, 2, 0, 2, 10], [-2, -7, -8, 5, -1], [14, 4, 0, -2, -10], [29, 3, -12, 4, 0], [1, 0, -15, 15, -1], [-10, -6, 8, 4, -2], [14, -1, 20, -7, 3], [-18, 1, -12, 0, -7], [7, 5, -2, 6, 2], [-10, -3, -6, -2, 5], [22, 6, 8, 4, -6], [-25, -3, -15, 0, -8], [-12, -8, -4, 4, 2], [-3, -9, 1, -4, 8], [-26, -5, -5, -1, 3], [18, -3, 4, -3, 7], [6, -1, 8, 4, -1], [14, -1, -8, 5, 3], [4, -9, -6, 3, 1], [1, -2, 4, -13, -3], [18, 2, -4, 2, 2], [10, -1, -4, 7, -7], [-41, -1, 10, -4, -4], [-4, 2, -6, -11, 6], [-32, 3, 3, -7, -7], [-7, 2, 7, 9, -3], [0, -5, 12, 1, -5], [4, 0, 4, -5, -8], [-4, 1, -12, -5, 13], [10, -1, 10, 1, 7], [-1, -6, 4, -7, 3], [2, -7, -7, -7, 1], [26, -1, 8, 5, 3], [-8, -4, 16, -5, -8], [-30, -2, -8, 10, 10], [-24, 0, 8, -4, -4], [3, -10, -19, 3, -1], [17, 2, -10, -1, 1], [15, -3, 8, -16, 8], [28, -3, -2, 5, -1], [-8, 0, -4, 1, 0], [22, 5, 0, -3, 3], [-21, 5, 1, -8, -4], [-20, -4, -6, -5, -4], [-22, -2, 12, 8, 2], [-35, -2, -16, 3, 5], [30, 6, -4, -4, 6], [3, -9, -12, 2, 2], [-38, -2, 0, 4, 2], [-18, 7, 0, -9, 1], [0, 0, 4, 12, 0], [12, 7, 9, 1, 1], [10, -5, 14, -8, -5], [14, 6, -4, 10, 14], [4, -2, 8, 6, -12], [4, -5, -7, 5, 5], [38, 3, 6, -6, 11], [20, -1, 0, -9, -1], [-3, 2, -2, 3, -7], [-14, 9, 0, -1, -1], [30, -3, -4, 3, -1], [-21, 1, -4, 4, 4], [-32, 7, 4, 17, -1], [-24, -2, 12, -6, -6], [12, 8, 4, -6, -10], [7, 11, 2, 4, 4], [31, 8, 8, -5, 1], [-10, 2, 8, 2, -2], [8, 15, 4, 3, 3], [10, -7, 3, 1, 1], [30, 0, 4, -10, -2], [15, -11, -11, 0, 0], [-18, 4, -10, -2, 2], [2, 4, 0, 12, -8], [-27, 3, -6, 8, 4], [0, -6, -6, 10, -6], [-18, 3, -8, -14, -9], [2, 11, 2, -4, -5], [27, -2, -5, -9, -1], [18, 0, 0, -8, 0], [14, -6, 8, 0, 6], [8, 5, 5, -5, 7], [4, -10, 8, -10, -4], [21, -5, 0, 16, 8], [-6, -2, 4, 6, -2], [-42, -4, -8, 2, -8], [-6, -13, -2, -7, -9], [-14, -2, -20, -8, 2], [28, -1, -1, 9, -7], [-19, 4, 4, -7, -1], [14, -12, -2, -3, 6], [4, 6, -12, 3, 14], [-28, 4, 2, 0, -4], [-30, 7, 4, 7, 1], [-36, -5, -5, -7, -11], [-4, 1, 4, -3, 5], [22, 7, 1, 3, 7], [2, 6, -4, 8, -2], [-23, -9, 10, -2, -6], [14, 1, 4, -13, -1], [-22, 1, -8, 7, 13], [30, 5, -6, -6, -3], [40, -2, 6, 5, 6], [14, -3, -4, 17, -1], [11, -1, 1, -18, -6], [28, -2, -6, 15, -2], [-13, -6, 8, 9, -9], [-25, -9, -14, -4, -12], [-38, -1, 8, 9, 1], [-8, 2, 20, -24, -8], [24, 9, 8, -5, -3], [-14, 4, -4, 10, 0], [2, 7, 16, -7, -7], [7, 2, 1, 7, -1], [22, 8, -4, 2, 0], [6, 6, 8, 6, 10], [13, -2, 4, -9, -3], [19, -13, 2, -10, 2], [16, -4, 0, 8, 4], [10, 0, 0, 2, 8], [12, 8, 20, 2, -10], [2, 8, -4, 6, 16], [6, 7, 4, 1, 3], [-39, -1, -2, -4, 8], [19, 8, -2, -1, 9], [47, 10, 14, 1, 11], [24, -8, 4, -10, 2], [18, 6, 0, 0, -2], [12, 3, -4, -9, 3], [28, 3, -22, 7, 1], [8, -10, -24, 7, 2], [0, -1, -16, -10, 1], [34, -5, 6, 2, -1], [8, 8, 18, 8, 0], [-32, -10, -20, 5, 6], [22, -12, -7, -14, -2], [2, -1, -20, 11, 1], [25, 13, 14, -2, 6], [3, -2, 13, -1, 7], [14, -6, 8, -10, -10], [15, -7, -4, 4, -8], [22, -3, 16, -7, -13], [-38, -5, -6, 6, -1], [36, 0, 16, 0, -4], [18, -2, -12, 10, 2], [16, 4, 4, 2, -6], [7, 10, -7, -1, -1], [-13, 0, 0, -9, -3], [-22, 1, 20, -6, 1], [-11, 4, -3, -13, -1], [-12, -2, -4, 12, 16], [32, 0, 4, -6, -2], [-14, 2, 6, -12, -4], [35, 4, -14, -1, 5], [-20, 5, -16, 11, 13], [20, -6, -18, 3, 2], [34, -9, -10, 12, -1], [-35, -7, -2, -4, -4], [-4, 1, 4, 14, -9], [60, 6, -9, 2, -2], [46, 6, 19, -4, 4], [47, -3, 3, 8, 8], [-14, 2, -12, 2, -10], [-6, 7, 8, -3, 11], [-34, -4, -4, -8, 0], [-13, -1, 0, -10, -2], [-48, -4, 10, -8, 0], [-10, 1, 4, -13, -5], [-10, -10, -24, 14, 10], [-4, 4, 4, -2, -6], [26, -8, -8, 20, 4], [-22, 5, -8, 5, 7], [-27, -1, 2, 2, 6], [54, 1, 4, 1, 7], [-3, 14, 2, 3, -7], [-1, -4, -25, 17, 13], [1, -12, 7, -1, -13], [2, 7, -12, 9, -7], [7, 2, -12, -11, 7], [42, 0, 16, -10, -6], [26, -6, -8, 2, -2], [18, -3, -6, -10, 1], [9, -4, 12, 9, -9], [-8, 1, 21, 19, -5], [3, 4, -1, -15, 1], [-26, -4, -10, 3, 2], [-15, -9, 9, 4, -4], [-44, -3, 8, -14, -5], [-34, 5, -4, -11, -5], [-14, -9, -14, -2, 11], [-34, 5, 6, -12, 1], [36, -10, -12, -8, -4], [11, 14, 13, 3, -9], [22, 8, 0, 2, 0], [-12, -6, -12, 5, 6], [-10, -4, -2, -18, -6], [-48, -4, 16, 3, 0], [-8, -10, -16, -6, 0], [5, -5, -8, 16, -4], [-19, -2, -22, 15, 9], [14, 10, 10, -3, -4], [18, 7, 8, -5, -3], [20, 3, 27, 5, 5], [-18, 0, -3, -10, -14], [-22, 7, 20, -2, -1], [14, -12, -28, 2, 4], [13, -7, 1, -10, -2], [2, 2, 12, 8, 12], [-16, 4, -4, -14, 14], [12, 4, 14, 0, -8], [-6, -4, 28, -26, -16], [6, 10, 8, -2, 14], [15, 13, 2, -6, 6], [36, 11, 11, -7, -19], [-14, 6, 20, 0, -2], [14, 4, -8, 8, 4], [-6, -18, -15, 8, 8], [24, -2, 16, -14, 0], [5, -15, -10, 2, -2], [-2, 3, -2, -11, -1]]