# q-expansion of newform 2352.2.bl.q, downloaded from the LMFDB on 19 June 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 = 2352
weight = 2
poly_data = [4, 0, -8, 0, 14, 0, -4, 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, 1, 0, 0, 0, 0, 0, 0], 1], [[20, 0, 0, 0, 0, 0, 1, 0], 14], [[0, 34, 0, 0, 0, 0, 0, 1], 14], [[2, 0, -28, 0, 7, 0, -2, 0], 14], [[0, 16, 0, -28, 0, 7, 0, -2], 14], [[2, 0, -21, 0, 7, 0, -2, 0], 7], [[0, 6, 0, -70, 0, 21, 0, -6], 14]]

hecke_ring_character_values = [[1471, [-1, 0, 0, 0, 0, 0, 0, 0]], [1765, [1, 0, 0, 0, 0, 0, 0, 0]], [785, [1, 0, 0, 0, 0, 0, 0, 0]], [2257, [1, 0, 0, 0, 1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, -1, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, -1, 0, -1, 1, -1], [0, -2, 0, 0, 0, 2, 0, 1], [0, 0, 2, -1, 0, 2, 1, -1], [0, 1, -2, 1, 0, -2, -2, 2], [2, -1, 1, 1, -2, 0, -1, 0], [2, 1, -1, -2, 0, 1, 0, -1], [0, -2, 0, -1, -4, 1, 2, 1], [0, -2, -1, -2, 0, 4, -1, -4], [2, -1, 1, 0, 4, 1, 2, 0], [2, -2, 0, 0, 4, 2, 0, -6], [2, -1, -2, 1, 2, 2, -2, 2], [0, -4, 0, -2, 2, 2, 2, 2], [0, 6, 0, -1, 6, -3, 2, 1], [4, 0, -2, -1, -4, 0, 2, 0], [-8, 0, 0, -2, -4, 2, 0, -2], [-4, -3, 1, 0, -8, 3, 2, 3], [8, 0, 4, 1, 4, -2, 2, 1], [-2, 4, 0, 4, 2, 0, 0, 0], [0, 2, -2, -8, 0, 2, 0, -4], [4, -2, -3, -7, -4, 0, 3, 0], [4, 6, 2, 0, 8, -6, 4, 1], [4, 0, 6, 0, 2, -3, 3, 0], [0, -1, -6, -3, 0, 2, -6, -6], [-2, -5, 1, 5, 2, 0, -1, 0], [0, -8, 0, 0, 2, 4, 3, 0], [0, 4, 0, -4, 0, 4, 0, -2], [0, -2, 0, 0, 0, 2, 0, 2], [8, -4, 2, 2, 8, 8, 2, 4], [0, -6, 0, -1, 6, 3, 1, 1], [0, -2, -4, 0, 0, -2, 0, 0], [2, 0, 0, -4, 2, 0, 0, -8], [-4, 0, -8, 2, -2, -6, -4, 2], [0, 0, 4, -2, 0, -3, 2, -2], [0, -6, -4, -2, 0, 0, 4, 0], [-10, 1, 4, -10, 0, 1, 0, -5], [0, -10, -1, 1, 0, 0, 1, 0], [8, 0, 2, -5, 4, 7, 1, -5], [0, -1, 0, 0, 0, 1, 0, 6], [0, 7, 5, 3, 0, 0, -5, 0], [0, 0, 0, -4, -2, 0, -8, 4], [4, -2, -10, 0, 0, -2, 0, 0], [0, -4, 0, 4, 0, 2, -4, -4], [4, -4, -4, 0, 8, 4, -8, 0], [4, -4, 8, 0, 0, -4, 0, 0], [0, 6, 0, 1, 2, -3, 8, -1], [0, 1, -4, 10, 0, 0, 4, 0], [-2, -3, 3, 5, -2, 6, 3, 10], [2, 7, 7, 0, 4, -7, 14, 3], [0, 0, 4, 4, 0, 1, 2, 4], [10, -1, -12, -6, 0, -1, 0, -3], [-2, -1, -1, 0, 2, 0, 1, 0], [0, 0, 10, 7, 0, -1, 5, 7], [4, 0, -2, -2, 2, 5, -1, -2], [4, 7, 2, -1, 4, -14, 2, -2], [0, 8, 0, 4, -4, -4, 2, -4], [14, -1, -5, 2, 0, -1, 0, 1], [0, -10, 0, -5, 12, 5, -6, 5], [-6, -1, 1, 0, -12, 1, 2, -6], [16, -1, -2, 6, 0, -1, 0, 3], [0, 14, 0, 1, -2, -7, 0, -1], [-8, 0, 4, 3, 8, 0, -4, 0], [0, 2, 10, 0, 0, -4, 10, 0], [-4, -12, 4, -8, 4, 0, -4, 0], [-4, -6, 9, -4, 0, -6, 0, -2], [12, 0, 2, 5, 6, 3, 1, 5], [0, -1, -6, 0, 0, 1, -12, 6], [4, 0, -6, -10, 2, -3, -3, -10], [2, 9, -1, 15, -2, 0, 1, 0], [0, 12, 0, 0, -4, -6, -8, 0], [12, 0, -2, 4, 12, 0, -2, 8], [-4, 4, -4, 0, -8, -4, -8, 0], [8, 0, 4, 0, 8, 0, 4, 0], [0, 6, 0, 5, 14, -3, -5, -5], [4, 2, 6, -9, -4, 0, -6, 0], [14, 1, 11, 1, 14, -2, 11, 2], [-8, 0, -12, -3, -4, -10, -6, -3], [-6, 9, 2, 2, 0, 9, 0, 1], [0, 0, 16, 0, 0, 0, 0, 0], [0, 0, 6, 1, 0, 13, 3, 1], [0, 7, 0, 0, 0, -7, 0, 2], [-12, 4, -4, 2, -12, -8, -4, 4], [0, -7, 3, -11, 0, 0, -3, 0], [0, -6, 10, 8, 0, -6, 0, 4], [8, 4, 10, 0, 8, -8, 10, 0], [-8, 10, -1, 0, -16, -10, -2, -7], [12, 2, 2, 0, 24, -2, 4, 6], [-10, -7, 10, 7, -10, 14, 10, 14], [0, 14, 0, -5, -10, -7, -2, 5], [-12, 0, 0, 14, -6, 2, 0, 14], [-2, 13, 1, 0, -4, -13, 2, -11], [6, 2, -6, -10, -6, 0, 6, 0], [16, 4, -2, -12, 0, 4, 0, -6], [-10, -9, 3, -8, 10, 0, -3, 0], [-4, 0, -2, -10, -2, 7, -1, -10], [0, 0, 8, -2, 0, 0, 8, -4], [-6, 0, 6, 8, -6, 0, 6, 16], [-10, -8, 0, 0, -20, 8, 0, -4], [0, 8, 0, 2, 4, -4, -20, -2], [0, -6, 0, 3, 14, 3, 12, -3], [14, 5, 5, 1, 14, -10, 5, 2], [4, 0, -4, 12, 2, 12, -2, 12], [16, 0, 0, 0, 8, -7, 0, 0], [10, -5, -6, -10, 0, -5, 0, -5], [4, 0, 5, 9, -4, 0, -5, 0], [0, 0, -2, 13, 0, -15, -1, 13], [-16, 2, 0, 0, -32, -2, 0, -1], [-4, 0, -8, -2, -4, 0, -8, -4], [0, 12, 0, -2, -6, -6, -9, 2], [18, -1, 5, -6, 0, -1, 0, -3], [0, 0, 0, 6, -12, 0, -8, -6], [4, 6, 2, 0, 8, -6, 4, -10], [0, 10, 0, -1, -6, -5, 7, 1], [8, 9, -2, -10, 0, 9, 0, -5], [0, -2, 0, 3, 2, 1, -18, -3], [-10, -5, 11, -1, -10, 10, 11, -2], [8, 1, -3, 0, 16, -1, -6, 11], [16, 0, -4, -6, 8, 1, -2, -6], [6, -4, -13, 0, 0, -4, 0, 0], [-4, -12, -3, 17, 4, 0, 3, 0], [8, 0, 14, -7, 4, 9, 7, -7], [8, 2, -4, 14, 8, -4, -4, 28], [18, -3, -7, -10, 0, -3, 0, -5], [-18, 4, 17, -8, -18, -8, 17, -16], [-4, -2, -4, 2, -4, 4, -4, 4], [-16, -7, -6, 10, 0, -7, 0, 5], [0, -7, 4, 6, 0, 0, -4, 0], [-16, 0, -12, 2, -8, 10, -6, 2], [4, 19, 3, 0, 8, -19, 6, -7], [0, 12, -6, -8, 0, 0, 6, 0], [2, 0, 7, 8, 0, 0, 0, 4], [-4, 10, -3, 1, 4, 0, 3, 0], [8, -1, -12, 0, 16, 1, -24, 0], [16, 0, 14, 9, 8, -14, 7, 9], [0, -12, 0, -4, -12, 6, 20, 4], [-14, 15, 1, 0, -28, -15, 2, -2], [0, -20, 0, 2, -10, 10, 6, -2], [-8, -4, -8, -8, 0, -4, 0, -4], [-16, -2, -14, 4, -16, 4, -14, 8], [28, 0, 12, 4, 14, 0, 6, 4], [6, -3, 1, 0, 12, 3, 2, -7], [0, 0, 0, -11, 0, 4, 0, -11], [-10, -1, 0, 18, 0, -1, 0, 9], [-8, -7, 10, 0, -16, 7, 20, 8], [24, 0, 6, -5, 12, 6, 3, -5], [-4, -3, -14, -7, -4, 6, -14, -14], [6, -3, -9, 11, -6, 0, 9, 0], [34, 2, -9, -2, 34, -4, -9, -4], [4, -4, 3, 0, 8, 4, 6, 15], [10, -14, 2, 0, 20, 14, 4, -6], [6, 5, -2, -9, 6, -10, -2, -18], [-20, 0, -8, 6, -10, -2, -4, 6], [-8, -13, -5, 0, -16, 13, -10, 9], [-6, -6, 12, 10, 6, 0, -12, 0], [-4, 22, -7, -3, 4, 0, 7, 0], [0, 6, 4, 0, 0, -6, 8, -5], [20, 0, 2, 24, 10, -7, 1, 24], [-2, 7, -7, 1, 2, 0, 7, 0], [2, 2, 18, -20, 0, 2, 0, -10], [-4, 2, 6, 0, -8, -2, 12, -6], [-16, 10, -6, 0, -16, -20, -6, 0], [0, 6, 0, -7, 2, -3, 3, 7], [0, -16, 0, -4, 20, 8, -4, 4], [-12, 0, -12, -14, -6, -6, -6, -14], [0, 0, 4, 10, 0, 1, 2, 10], [12, 0, 14, -8, 0, 0, 0, -4], [22, -1, 3, 6, -22, 0, -3, 0], [-28, 0, -6, 13, -14, -13, -3, 13], [20, -10, 6, 0, 40, 10, 12, -7], [-12, -9, 5, -5, 12, 0, -5, 0], [0, 12, 0, -2, 18, -6, 11, 2], [0, -12, 0, -8, -8, 6, 8, 8], [14, 1, -3, 0, 28, -1, -6, 20], [-14, 4, -2, 0, -28, -4, -4, -8], [0, -14, 0, 3, 22, 7, 3, -3], [-4, 1, -26, 2, 0, 1, 0, 1], [-8, 7, -6, -16, 8, 0, 6, 0], [2, -12, 12, 8, -2, 0, -12, 0], [-8, 4, 2, 20, 0, 4, 0, 10], [-12, 8, -13, -16, 0, 8, 0, -8], [2, 5, -1, -20, -2, 0, 1, 0], [-12, 0, 10, -15, -6, 11, 5, -15], [-40, 0, 2, 1, -20, -8, 1, 1], [0, -8, 0, -8, -4, 4, 10, 8], [0, 28, 0, 2, 4, -14, 12, -2], [6, 6, -13, -14, 6, -12, -13, -28], [0, -12, 0, 2, 8, 6, 28, -2], [16, -1, -2, 2, -16, 0, 2, 0], [4, 7, -13, 0, 8, -7, -26, -11], [16, -8, 12, 0, -16, 0, -12, 0], [14, 7, -9, 2, -14, 0, 9, 0], [-28, 0, -14, -15, -14, 3, -7, -15], [8, 0, 22, -13, 4, 8, 11, -13], [0, -8, 0, 0, 18, 4, 5, 0], [0, 0, -13, 8, 0, 0, -13, 16], [8, -20, 5, 0, 16, 20, 10, 11], [-22, 9, -4, -3, -22, -18, -4, -6], [0, -36, 0, 2, -6, 18, -10, -2], [32, -4, 4, 4, 0, -4, 0, 2], [-8, 16, -8, 11, 8, 0, 8, 0], [0, 0, 12, 14, 0, 27, 6, 14], [-28, 4, 12, 16, 0, 4, 0, 8], [0, 0, 10, 23, 0, -10, 5, 23], [36, -4, -16, 4, 36, 8, -16, 8], [-4, 17, -9, 5, 4, 0, 9, 0], [4, -8, 12, 36, 0, -8, 0, 18], [0, -22, 0, -3, 16, 11, -10, 3], [-28, 0, 17, 0, -28, 0, 17, 0], [-6, -13, -1, 0, -12, 13, -2, 10], [-18, -2, 2, 0, -36, 2, 4, -18], [-8, 0, -18, -2, -8, 0, -18, -4], [0, -6, 0, -9, -10, 3, -12, 9], [-8, -1, -16, 6, 8, 0, 16, 0], [12, 0, 16, -2, 6, 6, 8, -2], [36, 0, -10, 0, 18, 5, -5, 0], [-10, -15, -5, -5, 10, 0, 5, 0], [10, 7, -15, 2, 0, 7, 0, 1], [18, 0, -1, -4, 18, 0, -1, -8], [4, 3, -2, -38, 0, 3, 0, -19], [10, 14, -14, -10, 10, -28, -14, -20], [4, -16, 2, -24, -4, 0, -2, 0], [12, -20, 14, 4, 0, -20, 0, 2], [36, -2, 3, 20, 0, -2, 0, 10], [12, 22, 9, -11, -12, 0, -9, 0], [-48, 0, 14, -23, -24, -3, 7, -23], [12, 7, 2, 15, 12, -14, 2, 30], [12, -11, -1, 5, -12, 0, 1, 0], [0, 8, 0, -8, 18, -4, 21, 8], [0, -10, 0, -1, -16, 5, -10, 1], [6, -26, 4, 0, 12, 26, 8, 6], [0, 4, 0, 0, -40, -2, -6, 0], [16, -11, -10, -14, 0, -11, 0, -7], [0, -16, 0, 18, -4, 8, 10, -18], [-16, -9, -14, 10, 16, 0, 14, 0], [10, -12, -16, 4, 10, 24, -16, 8], [-6, 15, -9, 0, -12, -15, -18, 19], [0, -6, -12, 12, 0, -6, 0, 6], [12, 0, 6, -17, 6, 5, 3, -17], [-20, -14, -4, 4, -20, 28, -4, 8], [0, -20, 0, -14, 36, 10, -16, 14], [10, 16, -2, 0, 10, -32, -2, 0], [10, -1, 13, 0, 20, 1, 26, 8], [-10, 13, -8, -7, -10, -26, -8, -14], [0, 7, 6, -22, 0, 7, 0, -11], [0, -6, 0, 1, -14, 3, -6, -1], [20, 0, 16, 20, 10, -24, 8, 20], [8, -7, -3, 0, 16, 7, -6, -9], [-20, -8, 6, -8, 0, -8, 0, -4], [10, 5, 7, 0, -10, 0, -7, 0], [8, 0, -14, -5, 4, 11, -7, -5], [16, 13, -10, 0, 32, -13, -20, -10], [-12, 0, 2, 30, -6, -3, 1, 30], [-2, 29, 3, -1, 2, 0, -3, 0], [0, 16, 0, 0, -4, -8, 17, 0], [0, 26, 0, 11, -4, -13, -2, -11], [-14, 23, -1, 0, -28, -23, -2, -16], [-8, 1, 12, -14, 8, 0, -12, 0], [-4, 0, -24, -16, -2, 20, -12, -16], [-14, -19, 9, 0, -28, 19, 18, 25], [-16, 0, -32, 13, -8, -16, -16, 13], [0, 1, 6, 0, 0, -1, 12, -20], [-48, 0, 10, 3, -24, -2, 5, 3], [16, -6, 4, 2, 16, 12, 4, 4], [-30, 0, 16, 16, 0, 0, 0, 8], [-4, 28, 7, 0, -8, -28, 14, 7], [-2, 4, 12, 0, -4, -4, 24, -12], [0, 6, 0, -3, -2, -3, -15, 3], [8, -15, 18, 6, -8, 0, -18, 0], [24, 0, -20, -4, 12, 4, -10, -4], [48, 0, -12, 16, 24, 3, -6, 16], [12, 8, -16, 0, -12, 0, 16, 0], [-16, -11, -2, 0, -32, 11, -4, -22], [0, 9, 10, -1, 0, -18, 10, -2], [-2, -5, -15, 13, 2, 0, 15, 0], [0, -24, 0, 8, -36, 12, -8, -8], [10, 0, -24, 4, 10, 0, -24, 8], [-18, 5, 0, -3, -18, -10, 0, -6], [0, 16, 0, -4, 36, -8, 8, 4], [-16, 0, 24, 18, -8, 30, 12, 18], [-8, -12, -14, -20, 0, -12, 0, -10], [-4, 24, 6, 0, -8, -24, 12, -1], [0, -10, -24, -4, 0, 20, -24, -8], [0, 15, -7, -41, 0, 0, 7, 0], [0, 16, 0, -8, -22, -8, -18, 8], [-48, -6, -2, 8, 0, -6, 0, 4], [0, 26, 0, -1, -28, -13, -10, 1], [0, -24, -11, 0, 0, 24, -22, 7], [0, 36, 0, 10, 10, -18, -14, -10], [0, -16, 0, 8, -28, 8, 4, -8], [-6, 7, -1, 27, -6, -14, -1, 54], [18, 1, -24, 14, 0, 1, 0, 7], [-6, -8, 12, -16, 0, -8, 0, -8], [20, 0, -26, -8, 10, 13, -13, -8], [12, 4, -12, 20, 12, -8, -12, 40], [-16, -4, 11, 0, -32, 4, 22, -11], [-34, 1, 8, -7, -34, -2, 8, -14], [-24, 2, 20, 40, 0, 2, 0, 20], [0, 0, 12, 29, 0, 0, -12, 0], [-38, 5, -9, 5, -38, -10, -9, 10], [32, 0, -8, 18, 16, 30, -4, 18]]