
# q-expansion of newform 1638.2.bj.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 = 1638
weight = 2
poly_data = [81, -108, 45, 12, -20, 4, 5, -4, 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, 1, 0, 0, 0, 0, 0, 0], 1], [[-27, -45, 42, 10, 10, 2, -1, 1], 216], [[27, -27, 30, -10, -10, -2, 1, -1], 72], [[162, -153, 18, 52, -26, -4, 8, -5], 108], [[243, -225, 54, 26, -22, -2, 13, -7], 108], [[-891, 801, -150, -218, 130, 14, -49, 19], 216], [[270, -207, 24, 58, -32, -10, 14, -5], 36]]

hecke_ring_character_values = [[911, [1, 0, 0, 0, 0, 0, 0, 0]], [703, [1, 0, 0, 0, 0, 0, 0, 0]], [379, [0, 0, 0, 0, -1, 0, 0, 0]]]
aps_data = [[0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 1, -1, 0, -1, 2, 0], [0, 0, 0, 0, 0, 0, -1, 0], [-1, 1, 1, -1, 0, 0, 0, 0], [3, -1, 1, -1, 0, -2, 1, 1], [1, -1, -1, -2, -1, -2, 1, 1], [-2, 2, 3, 1, 0, -1, 2, 2], [-1, -1, 0, -2, -2, 1, 3, 1], [-1, 1, -1, 1, 0, 0, -4, 0], [-2, 0, -2, 0, -4, 1, -2, 1], [0, 2, -2, -1, -2, 1, -1, -1], [-1, -1, -3, 2, 2, 1, -1, -1], [0, 0, -5, -1, 1, -1, -2, 0], [3, -3, -2, -3, 0, 2, 1, 5], [3, 1, 4, -1, 0, -3, 3, 4], [-3, -3, -4, -1, -3, 1, -1, -3], [0, 0, 8, -2, 0, -2, 5, 0], [2, 1, -5, -3, -3, -2, 2, 2], [-1, 1, 2, 1, 0, -1, -2, 1], [2, -2, 6, -2, 0, -2, 8, 0], [0, -2, 1, 2, 0, -1, -1, -1], [4, 1, 6, 1, 10, 2, 5, 1], [3, -2, 10, 1, -1, -1, 1, 1], [-9, 1, -1, -2, -4, 2, 5, 1], [2, -2, 6, 2, 0, -4, 12, -4], [-8, 2, -10, -2, 0, 5, 2, -3], [5, -1, 3, -1, 4, 0, 8, 0], [-3, -3, -3, -3, -12, -1, 0, 2], [3, -3, -7, -2, 4, -2, -1, 3], [10, -2, -2, 0, 8, -2, -2, -2], [-1, 4, 5, -4, 0, 1, -3, 3], [-2, 2, 4, 2, 0, -2, 6, 2], [-6, 6, 1, -1, -2, -1, -2, -6], [-2, 2, 1, -1, 0, 1, 7, 2], [6, -6, -3, -6, 0, 3, 3, 9], [6, 2, 4, -2, 0, 0, -2, 2], [6, -4, -3, 1, 1, -1, 2, -4], [-4, 4, 0, -4, 0, 0, 0, 0], [3, -3, -11, -2, -2, -2, -3, 3], [6, 8, 8, 2, 14, 6, 6, 6], [-3, 5, -4, -5, 0, 4, 1, 1], [-7, 7, 2, -1, -8, -1, -2, -7], [-9, 1, 5, 1, 8, 2, -2, -2], [-4, -4, 1, 5, 8, 1, -1, -1], [5, -5, -8, 1, 6, 1, -2, 5], [2, -2, -2, -2, 0, 0, 0, 0], [-1, -3, -2, 8, 4, 5, -5, -5], [-16, -4, -4, 0, -10, 0, -8, -4], [-8, 8, 17, 8, 0, 4, 9, -4], [15, 1, 1, -1, 0, -1, 2, 2], [2, 2, -2, 2, 8, -1, -4, -3], [4, -4, -2, 2, 0, -2, -10, -4], [0, 0, 3, 3, -5, 3, 0, 0], [1, -6, 2, -3, -5, -3, 13, -3], [0, -2, 2, 5, -2, -7, 1, -7], [4, -4, -18, -2, 8, -2, -6, 4], [-5, 7, 6, -5, -2, 2, -2, -2], [-6, 6, -8, 10, 10, 10, -12, -6], [7, -3, -9, -3, 8, -3, -6, 0], [2, 6, 4, 8, 8, -2, -6, -2], [-6, -6, 2, 8, -6, -8, -4, -6], [-7, 9, 0, 9, 4, 2, -9, -7], [-11, 5, 0, -5, 0, 2, 1, 3], [15, 5, -3, -5, 0, 3, 2, 2], [-1, 5, -11, 5, 8, -2, -16, -7], [-6, 2, 2, 0, -2, 0, -14, 2], [2, -8, -5, 8, 0, -1, -1, -7], [7, -7, -4, -7, -6, -7, 5, 7], [-7, 1, 8, 4, 6, 5, -5, -5], [-4, -1, 1, -5, 5, -6, 6, 6], [5, 5, 7, 5, 20, 5, 2, 0], [5, 3, -6, 0, 8, 3, -15, 3], [-5, 5, -9, 4, 24, 4, -9, -5], [-12, 0, -10, 1, 12, 1, -1, -1], [-6, 6, 5, -1, 0, 1, 8, 6], [15, -3, 21, 3, 0, -10, -4, 7], [-4, 4, 8, 4, 0, -4, -5, 4], [-8, 6, -12, 2, 2, 8, -8, -8], [5, -5, 1, 6, 0, -6, 13, -5], [16, -5, -13, -10, 11, 5, -11, 5], [2, -8, -12, -8, -12, -4, -4, 4], [1, 3, 4, -1, -4, 2, -2, -2], [10, -10, 0, -2, 8, -2, 6, 10], [22, -6, -12, -10, 16, 4, -8, 4], [0, -4, 1, 4, 0, 7, -19, -11], [35, 1, -1, -2, 18, 2, -3, 1], [-8, 7, 7, 1, 1, 8, -8, -8], [4, 4, 4, 0, 4, 0, 20, 4], [0, -10, -11, -10, -20, -9, -1, 1], [17, 4, 3, -4, 0, 3, -5, 1], [19, -11, 13, 2, -8, -9, 9, 9], [-3, -9, 4, 13, -6, -13, 7, -9], [10, -2, -2, 0, 8, -2, -2, -2], [-4, 1, 14, 1, -6, 0, 13, -1], [-4, 4, 8, 12, 16, 12, -4, -4], [-12, 8, 0, -1, 4, 7, -7, -7], [-1, 1, 6, -1, 0, 2, -9, -1], [-15, -7, -7, -4, -22, -3, -3, -3], [-13, 5, -5, 5, -16, 5, -10, 0], [17, -7, -3, 7, 0, -3, 2, -4], [8, -4, -9, 3, -4, -1, 1, 1], [13, -13, -15, -8, 6, -8, 3, 13], [-4, 10, 14, 5, 6, 5, 13, 5], [-4, 7, -6, -7, 0, 0, 13, 7], [-13, 7, 1, 7, -12, -3, -6, -10], [-10, 7, -3, -4, 3, 3, -3, -3], [-10, 7, 7, 7, -6, -1, 0, -8], [7, 7, -5, -7, 0, 12, -12, -5], [-8, -6, 2, -11, -14, 5, 21, 5], [1, -1, 18, 3, 22, 3, 8, 1], [-4, -6, -18, 2, 10, -4, 4, 4], [-15, -5, -11, -6, -10, 6, 17, -5], [-17, 7, 2, 7, -20, 0, -5, -7], [17, -1, 4, 5, 8, -5, -11, -1], [8, -8, 6, -10, -8, -10, 12, 8], [-18, 10, 1, -9, -4, 9, -22, 10], [11, 3, 9, 0, 14, 3, 15, 3], [22, -2, -15, -3, 20, 1, -25, 1], [-1, 1, 2, 3, -4, 3, -1, -1], [5, -3, -8, -6, -2, -9, 9, 9], [-9, 6, -10, 2, -3, 4, -28, 4], [-9, 11, 13, -11, 0, 1, -4, 10], [0, -4, 4, 8, -2, -8, -24, -4], [-10, 0, 6, -2, 10, -2, 2, 2], [5, -5, -4, 5, 0, -5, 9, 0], [19, -7, 3, 10, 6, -10, 7, -7], [-10, 10, -13, 1, -20, 1, -12, -10], [9, 7, -5, -7, 0, 18, -24, -11], [-4, 0, -9, 0, -8, 10, -9, 10], [-10, 7, 9, -7, 3, 0, 0, 0], [9, -9, 10, 3, 0, -6, 6, 6], [9, 7, 15, -3, 16, 10, 26, 10], [6, 10, 4, 5, 16, 5, -7, 5], [-30, -2, -2, 0, -16, 0, 10, -2], [0, 2, 0, -11, -2, -9, 9, 9], [2, -2, -2, 0, 0, 0, 8, -2], [-16, 16, 13, -3, 0, 3, 2, 16], [-1, 1, 5, -2, 20, -2, 3, -1], [-22, 12, 2, 3, -10, 9, -11, 9], [0, -4, 34, -4, -8, -2, 38, 2], [4, 0, -5, -7, -4, -7, 7, 7], [12, -12, 6, -18, -6, -18, 18, 12], [14, -4, 6, -4, 20, 4, 10, 8], [13, -11, -11, -4, 2, -7, -7, -7], [12, 0, -8, -8, 6, 8, -20, 0], [20, -12, -15, -12, 16, -4, -3, 8], [-4, 6, -6, -6, 0, 18, -24, -12], [-3, -5, -16, 5, 0, 12, -13, -17], [6, 2, -32, 2, 16, 4, -34, 2], [-15, -5, -5, 0, -10, 0, -33, -5], [6, 9, 9, 11, 15, -2, -2, -2], [36, -7, 0, 7, 0, -6, 5, -1], [-3, -1, 15, 0, -4, -1, 31, -1], [24, 9, 17, 1, 33, 8, 24, 8], [7, -1, 7, 1, 0, -7, 6, 6], [5, -5, -44, -5, -2, -5, -17, 5], [21, 1, -3, -4, 11, 4, 9, 1], [-6, 8, 22, -8, 0, -6, -2, 14], [6, -20, 10, -20, -28, -12, 30, 8], [1, -7, -25, 7, 6, 0, 0, 0], [1, -1, -11, -8, 0, -8, -1, 1], [-2, 12, 10, 12, 20, 14, -2, 2], [-9, 9, -2, 3, -19, 3, -7, -9], [-12, 12, 20, -18, 0, -6, 6, 6], [-4, -8, -22, -8, -24, -12, -14, -4], [-27, 9, -9, -1, -18, 10, -26, 10], [2, -2, 4, -4, -8, -4, 5, 2], [3, 0, 9, 0, 6, -1, 9, -1], [-12, -6, 12, 6, 0, -10, 2, 4], [10, -6, -39, -6, 8, -1, -33, 5], [12, 8, 9, 1, 10, -1, -10, 8], [0, 0, 12, -8, 0, -8, 10, 0], [-13, 3, -10, -13, -5, 13, 1, 3], [25, -3, -3, 3, 0, -5, 10, 2], [9, -9, 9, 0, 29, 0, 9, 9], [-10, 19, 9, -5, -9, 14, -14, -14], [23, -3, 2, -3, 40, 9, 5, 12], [13, -7, 0, -9, 6, 2, 16, 2], [-12, -8, -7, -2, -20, -6, -4, -6], [4, 12, -8, 12, 32, 8, -20, -4], [-7, 8, -9, -8, 0, -1, 19, 9], [4, -8, -22, 8, 0, -8, 30, 0], [-18, 17, 11, 17, -2, -1, -6, -18], [-15, -1, 2, 5, 16, 4, -4, -4], [4, -4, -4, 8, -20, 8, -4, 4], [20, -12, -22, 12, 0, 0, 10, -12], [19, -13, 10, -13, 12, 4, 23, 17], [-2, -10, -2, 16, 12, 6, -6, -6], [4, 20, 17, -3, 12, 3, 26, 20], [-4, 4, 52, -4, 0, -4, 26, -4], [0, 0, 10, 10, 0, -10, 14, 0], [-5, 14, 7, -14, 0, -3, 13, 17], [-8, -10, 22, 6, 18, -4, 4, 4], [-6, 6, -30, 4, -10, 4, -20, -6], [-9, 17, 17, 14, 8, 3, 3, 3], [8, -20, -13, -20, -24, -7, 7, 13], [4, -4, 4, -12, -32, -12, 10, 4], [-25, 18, 11, 18, -14, 3, -7, -15], [15, 5, -18, -5, 0, 0, 23, 5], [11, 5, 13, 8, 8, -8, -12, 5], [10, -10, -45, -5, 4, -5, -15, 10], [2, -6, -4, 0, 4, -6, 6, 6], [-3, -5, -1, 5, 0, 1, -6, -6], [-16, 6, -14, 12, 10, 18, -18, -18], [-18, 6, 14, 8, -12, -2, 14, -2], [18, -10, -22, -10, 16, -6, -12, 4], [-2, 2, 38, 4, 28, 4, 16, -2], [-12, 8, 14, 0, 4, 8, -8, -8], [2, -2, 9, -9, -16, -9, 10, 2], [27, -10, 6, 4, 17, -14, 18, -14], [33, -5, 11, 5, 0, -1, -14, -4], [-3, 15, 25, 11, 12, 4, 24, 4], [7, -5, 8, 13, 1, -13, 1, -5], [16, -4, -5, -1, 6, 1, -36, -4], [1, -3, 33, -3, -4, -5, 36, -2], [8, 8, 1, -8, 0, 19, -31, -11], [3, -3, -25, -4, 35, -4, -9, 3], [-16, -2, -32, 7, 18, 5, -5, -5], [-4, -4, 7, -4, -16, -17, 11, -13], [-8, -4, 12, 10, -12, -14, 18, -14], [10, 6, -16, 6, 32, 12, -22, 6], [-11, -5, -11, -5, -32, 5, -6, 10], [-7, -1, 12, 13, -4, -13, 1, -1], [-10, -13, -5, -2, -23, -11, 5, -11], [37, 1, 11, -1, 0, -11, 12, 12], [35, -22, -26, -19, 13, -3, -11, -3], [-9, 9, -10, 1, 16, 1, -10, -9], [-2, 2, -30, 8, 12, 8, -20, -2], [-16, 18, 44, 18, 4, 10, 26, -8], [16, -12, -14, -10, 4, -2, -6, -2], [11, 13, 3, 26, 24, -13, -33, -13], [-7, 3, 11, -9, 4, -6, 6, 6], [-21, 7, -7, -7, 0, 13, -12, -6], [-9, -11, -8, 11, 0, 10, -23, -21], [-19, 23, 6, 23, 8, 22, -17, -1], [30, -6, -8, -2, 12, 2, -22, -6], [14, -22, 13, 9, 8, -13, 13, 13], [8, -8, -20, 0, 16, 0, -6, 8], [-25, -9, -5, -4, -34, -5, 3, -5], [-21, 11, 13, -3, 10, 8, -8, -8], [-17, 1, 7, 14, -16, -13, -1, -13], [-8, 8, 47, 13, 8, 13, 13, -8], [-14, 0, 2, 22, 14, 22, -22, -22], [39, 11, 7, -4, 25, 4, 23, 11], [-7, 1, 16, -1, 0, -4, -7, 5], [5, -17, -13, 4, -6, -4, 21, -17], [-24, 26, -10, -8, -2, 18, -18, -18], [6, 8, 5, -3, 7, 3, 42, 8], [24, -8, -18, -16, 16, 8, -12, 8], [-20, 10, -6, 4, 10, 14, -14, -14], [17, 1, 19, 0, -18, 1, -1, -1], [-1, -1, -15, -1, -4, -10, -14, -9], [-32, 18, 42, 20, -14, -2, 46, -2], [18, -8, 28, 8, 0, -28, 20, 20], [-13, -15, -6, 9, -14, -9, -31, -15], [-21, 21, 17, 14, 4, 14, -9, -21], [-12, 8, -22, -2, 4, 6, -6, -6], [-16, 16, 2, -16, 0, 8, -2, 8], [13, 7, -3, -10, 10, 10, 37, 7], [12, -12, 9, 12, 0, -3, -15, -9], [9, 9, 27, -1, 18, 10, 46, 10], [15, 3, -14, 3, 36, 4, -17, 1], [52, -4, -15, -6, 48, 2, -20, 2], [3, -9, -15, -6, -3, 6, 31, -9], [-6, 6, 30, 4, -18, 4, 10, -6], [1, -17, -21, -4, -8, 4, 0, -17], [-23, 10, 23, 10, -26, -5, 13, -15], [20, 14, 1, -13, 17, 13, 8, 14], [-5, 5, -1, -10, -20, -10, 2, -5], [11, 7, -25, -7, 0, -2, 36, 9], [5, 2, -7, 2, 14, -3, -9, -5], [-14, 14, -42, -5, 0, 9, -9, -9], [-3, 3, -44, 19, -4, 19, -33, -3], [17, -17, 1, 0, 36, 0, 9, 17], [-6, 0, -26, -5, 6, -5, 5, 5], [3, 5, 16, 11, 4, -11, 29, 5], [-6, 10, 2, 10, 8, 2, -8, -8], [-10, 4, -4, -8, -3, 8, -8, 4], [6, -6, -25, 1, -7, 1, -10, 6], [-30, 18, 34, 20, -12, -2, 30, -2], [20, -12, 6, 10, -8, -2, 2, 2], [-31, -3, 14, 3, 0, -8, -1, 5], [-9, -13, 35, 14, 22, 1, -1, -1], [-2, 0, 12, 0, -4, -16, 12, -16], [-29, 3, -9, -3, 0, 8, -4, -5], [23, 4, 16, -4, 0, 0, -12, 4], [-24, 8, -11, 8, -32, 9, -19, 1], [-48, -8, -9, -1, -28, 1, -9, -8], [11, -11, -4, 1, 21, 1, 3, 11], [-10, -14, -23, -13, -24, -1, -19, -1], [-27, 27, -35, 20, 30, 20, -41, -27], [-8, 8, 2, 15, 0, 23, -23, -23], [24, 10, 5, -5, 17, 5, 18, 10], [-1, 1, 31, 0, -14, 0, 15, -1], [-50, 16, 22, -16, 0, 0, -6, 16], [-35, 9, -3, -9, 0, 3, 6, 6], [-44, 8, 22, 23, -36, -15, 13, -15], [0, -3, 29, -2, -3, -1, 63, -1], [-9, 13, 11, -6, -4, 7, -7, -7], [7, -7, 13, -2, 10, -2, 11, 7], [-13, 9, 2, 9, -8, 5, -7, -4], [1, 11, 27, 13, 12, -2, 30, -2]]
