
# q-expansion of newform 1323.2.f.f, downloaded from the LMFDB on 23 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 = 1323
weight = 2
poly_data = [9, -9, 36, -3, 90, -36, 40, -8, 9, -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, 0, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-117, -873, -387, -204, -282, -72, -61, -3, -9, -1], 189], [[-1557, 36, -108, 303, -240, 204, -23, 48, -12, 7], 567], [[63, 270, 204, 123, 30, 68, 8, 12, -1, 2], 63], [[-180, 684, -351, 1680, -780, 663, -176, 156, -39, 16], 567], [[-63, 369, 1620, 1020, -57, 624, -4, 141, -24, 20], 567], [[-1368, -1071, -4401, -2757, -69, -1668, -11, -375, 60, -53], 567], [[720, -2898, -432, -7260, 2850, -3264, 632, -732, 165, -82], 567], [[801, -3222, 648, -8070, 3174, -3633, 704, -813, 174, -91], 567]]

hecke_ring_character_values = [[785, [0, 0, 0, 0, 0, 0, 1, 0, 0, 0]], [1081, [1, 0, 0, 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, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 0, -1, 1, 0, 0], [0, 0, 0, 0, 0, -1, -1, 0, 0, -1], [-3, 1, 0, -1, 0, -1, 0, 0, 0, 0], [1, -2, -1, 1, 1, 2, 0, 0, 1, 1], [0, 0, 0, -2, 0, 0, 0, 2, -2, 1], [-1, -1, 0, 0, 1, 0, -1, 2, 0, 0], [0, 0, 0, -1, 0, 0, -1, 1, -1, 1], [0, -2, 0, 0, 2, 2, 0, 0, 2, 0], [0, 0, 0, 1, 0, -1, 0, -1, 1, -2], [-3, 2, 3, 0, 1, 0, -3, 1, 0, 0], [4, 1, 1, 0, 1, 0, 4, -2, 0, 0], [-5, 1, 2, 0, 0, -1, 0, 0, 0, -2], [0, 0, 0, 1, 0, -1, -6, -1, -1, -1], [2, 1, -1, 0, 1, 0, 2, -1, 0, 0], [0, 0, 0, -2, 0, 4, -1, 2, -1, -1], [-2, 2, 2, -1, 3, -2, 0, 0, 3, -2], [4, 0, 0, -1, -3, 0, 0, 0, -3, 0], [-3, 4, 1, 0, 0, 0, -3, -2, 0, 0], [1, 0, -2, 0, 4, 0, 1, 2, 0, 0], [-7, 2, -1, 0, 2, -2, 0, 0, 2, 1], [2, 1, 2, 0, 1, 0, 2, 4, 0, 0], [1, 4, -1, 0, -3, 0, 1, -2, 0, 0], [0, 0, 0, 1, 0, -1, 2, -1, 3, 2], [-2, -2, 4, 0, 1, 2, 0, 0, 1, -4], [3, 4, 0, 0, -1, -4, 0, 0, -1, 0], [0, 0, 0, 1, 0, 3, 3, -1, -3, 0], [5, -2, 2, 1, 1, 2, 0, 0, 1, -2], [0, 0, 0, -2, 0, 5, -12, 2, 0, 0], [-1, 1, 2, 0, -1, 0, -1, -3, 0, 0], [0, 0, 0, -4, 0, -2, 5, 4, 1, 0], [0, 0, 0, 2, 0, 2, -8, -2, 4, 3], [-3, -2, 1, 0, -3, 0, -3, -5, 0, 0], [0, 0, 0, 1, 0, 5, 0, -1, -4, 0], [3, -3, -2, -2, -2, 3, 0, 0, -2, 2], [0, 0, 0, -3, 0, -4, -2, 3, 0, 2], [10, -6, -1, 0, -3, 0, 10, -1, 0, 0], [8, -1, -1, 0, -1, 1, 0, 0, -1, 1], [4, 2, 1, 7, 2, -2, 0, 0, 2, -1], [1, 5, 7, 0, -2, 0, 1, 1, 0, 0], [0, 0, 0, 6, 0, 3, -4, -6, 0, 3], [8, -7, -3, 0, 1, 7, 0, 0, 1, 3], [-6, 5, -6, 0, 1, -5, 0, 0, 1, 6], [0, 0, 0, 3, 0, 7, -2, -3, 3, 1], [-5, -1, -3, 0, 1, 0, -5, 0, 0, 0], [5, -5, 0, 0, -2, 0, 5, -7, 0, 0], [0, 0, 0, 2, 0, 4, -12, -2, -6, 1], [-8, 0, -4, -5, 0, 0, 0, 0, 0, 4], [0, 0, 0, -2, 0, -2, 4, 2, 4, 3], [3, -1, 2, 0, 3, 0, 3, -4, 0, 0], [-5, -2, 0, 3, 2, 2, 0, 0, 2, 0], [0, 0, 0, -5, 0, -3, 1, 5, 1, 0], [-6, 8, 2, 0, 4, 0, -6, -6, 0, 0], [2, -4, 1, 0, -6, 4, 0, 0, -6, -1], [-7, 7, 3, -6, 2, -7, 0, 0, 2, -3], [2, 1, -3, 0, 5, 0, 2, 0, 0, 0], [1, 8, 2, 0, 3, 0, 1, -2, 0, 0], [0, 0, 0, 4, 0, 0, 0, -4, 7, -1], [0, 0, 0, 2, 0, 4, -2, -2, 1, 0], [-9, 7, 1, -4, 0, -7, 0, 0, 0, -1], [0, 0, 0, -1, 0, 7, -17, 1, -7, 7], [1, -4, 3, 0, 1, 0, 1, 7, 0, 0], [5, -1, -1, 0, 5, 0, 5, -4, 0, 0], [-4, 4, 9, 0, -1, 0, -4, -4, 0, 0], [0, 0, 0, 5, 0, 0, 3, -5, -1, -2], [0, 0, 0, -1, 0, 2, 7, 1, -1, 7], [4, -11, -2, 0, -6, 0, 4, -6, 0, 0], [-6, 4, 7, 0, 5, 0, -6, 3, 0, 0], [-5, 8, -3, -4, 2, -8, 0, 0, 2, 3], [-5, -8, -5, 0, 3, 0, -5, 5, 0, 0], [0, 0, 0, -2, 0, 3, -9, 2, -7, 4], [2, 7, -4, -5, 1, -7, 0, 0, 1, 4], [0, 0, 0, -1, 0, -12, -6, 1, -1, -4], [4, -10, 0, 0, -7, 0, 4, 1, 0, 0], [11, 0, -2, 7, 1, 0, 0, 0, 1, 2], [0, 0, 0, -3, 0, 4, -10, 3, 1, 2], [0, 0, 0, 0, 0, 12, -8, 0, 3, 6], [0, 0, 0, 3, 0, 1, -18, -3, -4, -7], [-1, -4, -4, 0, -6, 0, -1, 2, 0, 0], [2, -3, 7, -4, -3, 3, 0, 0, -3, -7], [-3, 7, -5, -4, -3, -7, 0, 0, -3, 5], [-10, -6, 3, 0, 0, 0, -10, -2, 0, 0], [-15, 1, -2, 0, -1, 0, -15, -5, 0, 0], [18, -12, -3, 6, -6, 12, 0, 0, -6, 3], [0, -9, -7, 0, -4, 0, 0, 2, 0, 0], [19, -4, -7, 0, 3, 0, 19, 3, 0, 0], [0, 0, 0, -8, 0, 1, -10, 8, -10, 5], [6, 11, 0, 1, 0, -11, 0, 0, 0, 0], [20, 0, -2, 0, 6, 0, 20, 9, 0, 0], [2, 2, 1, 2, -4, -2, 0, 0, -4, -1], [0, 0, 0, -4, 0, 7, -12, 4, 2, -7], [0, 0, 0, 3, 0, -7, 4, -3, 2, -3], [-9, 0, 3, -6, -6, 0, 0, 0, -6, -3], [0, 0, 0, 11, 0, 14, -7, -11, -5, 5], [-3, 10, 6, -10, -3, -10, 0, 0, -3, -6], [6, 0, -13, -7, -7, 0, 0, 0, -7, 13], [2, 4, 5, -3, -8, -4, 0, 0, -8, -5], [-6, 4, 4, 0, -3, 0, -6, -4, 0, 0], [6, 9, -1, 6, 4, -9, 0, 0, 4, 1], [0, 0, 0, 11, 0, 11, -4, -11, 0, -5], [10, -1, -5, 0, -5, 0, 10, -5, 0, 0], [0, 0, 0, -2, 0, -3, -15, 2, -1, 1], [-16, 4, 9, 3, 5, -4, 0, 0, 5, -9], [1, 1, 9, 0, 5, 0, 1, 4, 0, 0], [7, -11, -4, -6, 0, 11, 0, 0, 0, 4], [0, 0, 0, 6, 0, -6, 4, -6, 1, -12], [-5, -4, -3, 0, 7, 0, -5, 2, 0, 0], [0, 0, 0, 7, 0, 1, -13, -7, -2, -6], [-8, 2, 9, -9, -8, -2, 0, 0, -8, -9], [0, 0, 0, 10, 0, -8, -8, -10, 4, 0], [24, -12, -5, 0, -5, 0, 24, -2, 0, 0], [-14, 6, 11, -8, -7, -6, 0, 0, -7, -11], [-25, 10, -1, 0, -1, 0, -25, 3, 0, 0], [0, 0, 0, -1, 0, 2, -3, 1, -7, 12], [-1, -15, -9, 8, 1, 15, 0, 0, 1, 9], [0, 0, 0, 7, 0, -12, 3, -7, 4, -5], [5, 18, -1, 0, 2, 0, 5, -2, 0, 0], [0, 0, 0, -9, 0, -3, -22, 9, -7, 4], [-4, -5, -9, 0, 11, 0, -4, 0, 0, 0], [6, 5, -7, 0, -11, 0, 6, -10, 0, 0], [-2, 4, 5, 3, -3, -4, 0, 0, -3, -5], [-2, 7, 4, 0, -3, 0, -2, -5, 0, 0], [-13, 10, 4, 10, 6, -10, 0, 0, 6, -4], [3, 8, -12, 0, 4, 0, 3, 3, 0, 0], [0, -4, 0, -11, -6, 4, 0, 0, -6, 0], [2, 14, 4, 0, 2, 0, 2, -9, 0, 0], [0, 0, 0, 0, 0, -7, 9, 0, -4, -3], [-10, 14, 7, 1, -1, -14, 0, 0, -1, -7], [0, 0, 0, -15, 0, -7, 3, 15, -5, -5], [0, 0, 0, -7, 0, 10, -27, 7, 1, 3], [0, -5, 9, 5, -1, 5, 0, 0, -1, -9], [0, 0, 0, 3, 0, -19, 10, -3, 11, -2], [0, 0, 0, 2, 0, -2, -6, -2, 6, 1], [9, -1, 8, 4, 13, 1, 0, 0, 13, -8], [0, 0, 0, -1, 0, -7, -2, 1, -8, -5], [0, 0, 0, -8, 0, 1, -1, 8, 3, -2], [14, -15, -2, -1, 6, 15, 0, 0, 6, 2], [4, -9, 0, 0, -15, 9, 0, 0, -15, 0], [17, -6, -2, 0, 0, 0, 17, 13, 0, 0], [0, 0, 0, -6, 0, -2, -6, 6, 1, 0], [-29, 5, 8, -1, 3, -5, 0, 0, 3, -8], [-6, 15, 5, -8, 5, -15, 0, 0, 5, -5], [2, 12, 8, 0, -10, 0, 2, -6, 0, 0], [-10, 12, 3, 0, -3, 0, -10, 3, 0, 0], [-24, -5, -11, 0, 2, 0, -24, -6, 0, 0], [0, 0, 0, -5, 0, -7, 20, 5, -3, 2], [10, -3, -1, -5, -3, 3, 0, 0, -3, 1], [0, 0, 0, 3, 0, -16, 25, -3, 5, -9], [5, -14, -5, 4, 9, 14, 0, 0, 9, 5], [9, -7, -1, -10, -9, 7, 0, 0, -9, 1], [0, 0, 0, -4, 0, 14, 5, 4, 4, 2], [-4, -4, -4, 0, 12, 0, -4, 8, 0, 0], [8, -6, 11, 0, 2, 0, 8, 4, 0, 0], [-5, -11, 4, 4, 3, 11, 0, 0, 3, -4], [-1, -13, -5, 0, 6, 0, -1, 13, 0, 0], [30, -21, -7, -6, -10, 21, 0, 0, -10, 7], [0, 0, 0, -9, 0, -4, 22, 9, -11, -9], [1, -8, -10, 0, 9, 0, 1, 8, 0, 0], [14, -9, -16, 5, -4, 9, 0, 0, -4, 16], [0, 0, 0, 3, 0, 1, 1, -3, 6, 4], [-12, -1, 9, 1, 0, 1, 0, 0, 0, -9], [0, 0, 0, 3, 0, 3, 13, -3, -2, 15], [17, -5, -6, 0, -11, 0, 17, -7, 0, 0], [14, -3, 3, -3, 0, 3, 0, 0, 0, -3], [27, 0, 7, 0, 7, 0, 27, -5, 0, 0], [2, -15, -9, 14, 0, 15, 0, 0, 0, 9], [0, 0, 0, 7, 0, 9, -8, -7, -11, -3], [14, 1, 2, 0, -1, 0, 14, 3, 0, 0], [0, 0, 0, 7, 0, -14, 31, -7, 2, -7], [0, 0, 0, -14, 0, -18, 6, 14, -2, -2], [15, -24, -10, 0, -1, 0, 15, 4, 0, 0], [0, 0, 0, -7, 0, -2, -6, 7, -9, -11], [0, 0, 0, 5, 0, 19, -8, -5, 1, 6], [20, 8, 14, 0, -6, 0, 20, -3, 0, 0], [27, 6, 2, 0, -8, -6, 0, 0, -8, -2], [-8, -13, 7, 8, 2, 13, 0, 0, 2, -7], [10, -7, -6, 0, 4, 0, 10, 9, 0, 0], [-14, 5, -2, 0, 14, 0, -14, 3, 0, 0], [15, 16, 2, 0, 4, 0, 15, -9, 0, 0], [0, 0, 0, -7, 0, -3, 10, 7, -5, -7], [0, -9, 2, 21, 7, 9, 0, 0, 7, -2], [0, 0, 0, 10, 0, 2, -22, -10, 4, -1], [-18, 17, 0, 0, -3, 0, -18, 2, 0, 0], [-15, 3, 1, 10, 19, -3, 0, 0, 19, -1], [3, 7, -12, 0, -4, 0, 3, 4, 0, 0], [0, 0, 0, -1, 0, -6, 3, 1, 5, -11], [-14, 2, -9, -1, -7, -2, 0, 0, -7, 9], [-25, 0, 7, -12, -11, 0, 0, 0, -11, -7], [22, -5, 3, 0, -7, 0, 22, 2, 0, 0], [17, 9, 3, -12, 0, -9, 0, 0, 0, -3], [8, -6, 18, 0, 7, 0, 8, 6, 0, 0], [10, -7, -17, -1, 1, 7, 0, 0, 1, 17], [0, 0, 0, 4, 0, 5, -9, -4, -1, 11], [0, 0, 0, 8, 0, -7, 4, -8, 14, 3], [-6, -16, -3, 0, -5, 0, -6, 12, 0, 0], [-7, 8, 9, 0, -2, 0, -7, -5, 0, 0], [-30, 20, 0, -2, 6, -20, 0, 0, 6, 0], [0, 0, 0, 11, 0, 8, 13, -11, 8, -11], [-19, 14, -8, 0, 11, 0, -19, -9, 0, 0], [0, 0, 0, 11, 0, -3, 17, -11, 8, -5], [-20, -4, -12, 0, -2, 0, -20, 3, 0, 0], [-24, -11, -1, -4, 1, 11, 0, 0, 1, 1], [28, 10, 4, -6, -11, -10, 0, 0, -11, -4], [16, -5, 11, 11, 13, 5, 0, 0, 13, -11], [0, 0, 0, -1, 0, -13, 37, 1, 7, -12], [6, -7, 7, 0, 2, 0, 6, -12, 0, 0], [0, 0, 0, 13, 0, -6, 12, -13, 1, 8], [-1, -4, -2, 16, -1, 4, 0, 0, -1, 2], [0, 0, 0, 8, 0, -16, 14, -8, 8, -3], [-10, 26, 2, 0, 6, 0, -10, -6, 0, 0], [-11, 2, 3, 0, -16, 0, -11, -12, 0, 0], [0, 0, 0, 15, 0, -4, -14, -15, 7, -6], [-11, 17, 3, 0, 4, 0, -11, 12, 0, 0], [0, 0, 0, -5, 0, 3, 27, 5, 2, -2], [9, -14, 7, 0, -7, 0, 9, 9, 0, 0], [5, -12, -3, 14, -2, 12, 0, 0, -2, 3], [0, 0, 0, -6, 0, -1, 18, 6, -2, 4], [0, 0, 0, 8, 0, 7, 1, -8, 12, -5], [0, 0, 0, 5, 0, 15, -14, -5, -12, -3], [0, 0, 0, 17, 0, 9, -22, -17, 8, 3], [-27, 24, 10, -1, 4, -24, 0, 0, 4, -10], [0, 0, 0, -11, 0, 14, -39, 11, -14, 10], [-2, -5, -7, 0, 4, 0, -2, -3, 0, 0], [-9, 3, 1, 0, 23, 0, -9, 13, 0, 0], [-28, 2, -1, 3, 14, -2, 0, 0, 14, 1], [-4, 1, -13, 0, 1, 0, -4, -9, 0, 0], [-39, 1, 13, 0, -1, 0, -39, -2, 0, 0], [0, 0, 0, -4, 0, -26, 24, 4, 7, -12], [0, 11, 12, 5, 10, -11, 0, 0, 10, -12], [0, 0, 0, -5, 0, -5, 6, 5, 10, 9], [0, 0, 0, 9, 0, -10, 9, -9, 9, 6], [27, 6, 7, 0, -5, 0, 27, 0, 0, 0], [25, 2, 4, 0, 1, 0, 25, -6, 0, 0], [0, 0, 0, -7, 0, -14, 19, 7, 8, -16], [23, 4, 2, 13, 2, -4, 0, 0, 2, -2], [0, 0, 0, 6, 0, -1, -15, -6, 1, -2], [-26, 6, -2, -5, -8, -6, 0, 0, -8, 2], [20, -11, -1, 0, -16, 0, 20, 0, 0, 0], [6, -11, 7, -3, 0, 11, 0, 0, 0, -7], [0, 0, 0, -2, 0, -27, -5, 2, -9, -12], [29, -2, -1, 2, -2, 2, 0, 0, -2, 1], [0, 0, 0, 8, 0, -4, 14, -8, -16, 6], [30, -6, -2, 0, -7, 0, 30, 1, 0, 0], [18, 1, -2, -4, 9, -1, 0, 0, 9, 2], [0, 0, 0, 3, 0, -10, -26, -3, 4, 3], [0, 0, 0, 15, 0, 6, -32, -15, 1, -10], [15, -9, 9, 0, -9, 9, 0, 0, -9, -9], [0, 0, 0, -6, 0, -7, -4, 6, 4, -5], [3, 5, 9, -5, -24, -5, 0, 0, -24, -9], [0, 0, 0, -3, 0, 2, -9, 3, -15, -3], [-8, -17, -2, 0, -6, 0, -8, -9, 0, 0], [6, 7, 1, 0, -1, 0, 6, 6, 0, 0], [12, 12, -8, 9, 8, -12, 0, 0, 8, 8], [-3, 17, 11, -13, -3, -17, 0, 0, -3, -11], [-23, 0, -1, 0, 14, 0, -23, 11, 0, 0], [10, -7, 11, 14, 6, 7, 0, 0, 6, -11], [-8, 20, 7, -6, -7, -20, 0, 0, -7, -7], [16, 8, 1, 0, 2, 0, 16, 12, 0, 0], [-8, 2, 11, 0, 18, 0, -8, 7, 0, 0], [0, 0, 0, -3, 0, 6, 31, 3, -12, -6], [-11, 16, 8, -21, -5, -16, 0, 0, -5, -8], [0, 0, 0, -6, 0, 26, 4, 6, -8, 0], [29, -15, 3, 0, 3, 0, 29, -12, 0, 0], [17, 4, -8, 4, 3, -4, 0, 0, 3, 8], [2, -3, -4, 0, -16, 0, 2, -15, 0, 0], [0, 0, 0, -11, 0, -20, 17, 11, 14, -7], [0, 0, 0, -20, 0, -11, -5, 20, 0, -10], [0, 0, 0, 7, 0, 5, -27, -7, -1, 15], [5, -15, 12, 12, 12, 15, 0, 0, 12, -12], [25, 17, 0, 0, 1, 0, 25, -6, 0, 0], [0, 0, 0, 11, 0, -8, 1, 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