
# q-expansion of newform 5040.2.k.f, downloaded from the LMFDB on 24 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 = 5040
weight = 2
poly_data = [81, 0, 0, 0, -10, 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, 1, 0, 0, 0, 0, 0, 0], 1], [[0, 0, -1, 0, 0, 0, 1, 0], 18], [[0, -7, 0, 0, 0, 1, 0, 0], 6], [[0, 0, 19, 0, 0, 0, -1, 0], 18], [[-45, -3, -1, 1, 9, 3, 1, -1], 36], [[-45, 3, 1, -1, 9, -3, -1, 1], 36], [[0, 0, 0, -10, 0, 0, 0, 1], 27]]

hecke_ring_character_values = [[3151, [1, 0, 0, 0, 0, 0, 0, 0]], [3781, [1, 0, 0, 0, 0, 0, 0, 0]], [2801, [-1, 0, 0, 0, 0, 0, 0, 0]], [2017, [-1, 0, 0, 0, 0, 0, 0, 0]], [3601, [-1, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 1, 0, -1, 1, -1], [0, 0, 1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1, 1, 0], [0, 1, 0, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, 0, 0, 1], [0, 3, 1, 2, 0, -1, 1, 1], [5, 0, 0, 0, -1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 3, 2, 4, 0, -2, 2, -1], [0, 0, -1, 0, 0, -1, -1, 0], [0, 1, -2, 0, 0, 2, -2, -1], [0, 0, -5, 0, 0, 1, 1, 0], [0, -4, -1, -2, 0, 1, -1, -2], [7, 0, 0, 0, 1, 0, 0, 0], [0, -4, -1, 0, 0, 1, -1, 4], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 2, 2, 0], [0, 0, 0, 0, 0, 1, 1, 0], [0, -1, -2, 0, 0, 2, -2, 1], [-4, 0, 0, 0, 4, 0, 0, 0], [0, -4, -3, -6, 0, 3, -3, 2], [0, -1, -1, 0, 0, 1, -1, 1], [0, 5, 4, 0, 0, -4, 4, -5], [0, -1, 2, 0, 0, -2, 2, 1], [0, -4, 0, 0, 0, 0, 0, 4], [9, 0, 0, 0, 3, 0, 0, 0], [2, 0, 0, 0, 6, 0, 0, 0], [-3, 0, 0, 0, 3, 0, 0, 0], [0, 0, 2, 0, 0, 2, 2, 0], [0, -2, 1, 0, 0, -1, 1, 2], [1, 0, 0, 0, -5, 0, 0, 0], [0, 3, -2, -4, 0, 2, -2, 7], [0, 0, -7, 0, 0, -2, -2, 0], [0, 0, 0, 0, 2, 0, 0, 0], [0, -5, 3, 0, 0, -3, 3, 5], [0, 0, 0, 0, 0, 0, 0, 0], [0, -10, -4, -8, 0, 4, -4, -2], [0, -11, -3, -6, 0, 3, -3, -5], [0, 0, 2, 0, 0, -3, -3, 0], [0, 6, 7, 14, 0, -7, 7, -8], [0, 0, -4, 0, 0, -3, -3, 0], [0, 0, 3, 0, 0, 3, 3, 0], [-3, 0, 0, 0, -9, 0, 0, 0], [0, -3, 3, 6, 0, -3, 3, -9], [0, 0, 0, 0, 2, 0, 0, 0], [0, -6, -5, 0, 0, 5, -5, 6], [0, 2, 5, 10, 0, -5, 5, -8], [0, 12, 5, 10, 0, -5, 5, 2], [19, 0, 0, 0, 1, 0, 0, 0], [0, 0, -8, 0, 0, -1, -1, 0], [0, 6, 1, 2, 0, -1, 1, 4], [0, 6, 3, 0, 0, -3, 3, -6], [0, 7, 0, 0, 0, 0, 0, 7], [7, 0, 0, 0, 1, 0, 0, 0], [0, -7, 2, 0, 0, -2, 2, 7], [0, -3, -4, -8, 0, 4, -4, 5], [0, 0, -18, 0, 0, 0, 0, 0], [0, 0, -7, 0, 0, 4, 4, 0], [0, 10, 5, 0, 0, -5, 5, -10], [0, 1, 6, 12, 0, -6, 6, -11], [0, -8, -7, 0, 0, 7, -7, 8], [0, -8, -5, 0, 0, 5, -5, 8], [0, -1, 1, 0, 0, -1, 1, 1], [9, 0, 0, 0, -9, 0, 0, 0], [4, 0, 0, 0, -6, 0, 0, 0], [0, 0, -9, 0, 0, -3, -3, 0], [21, 0, 0, 0, 3, 0, 0, 0], [0, 6, 8, 16, 0, -8, 8, -10], [0, -11, -3, -6, 0, 3, -3, -5], [0, 0, 18, 0, 0, 1, 1, 0], [0, 10, 5, 0, 0, -5, 5, -10], [0, 0, 6, 0, 0, 6, 6, 0], [8, 0, 0, 0, -2, 0, 0, 0], [0, -10, -4, -8, 0, 4, -4, -2], [0, 0, -7, 0, 0, 4, 4, 0], [0, -3, 5, 0, 0, -5, 5, 3], [0, 0, 7, 0, 0, -6, -6, 0], [0, -6, -3, -6, 0, 3, -3, 0], [0, 0, 6, 0, 0, -6, 6, 0], [8, 0, 0, 0, 6, 0, 0, 0], [0, 0, 2, 0, 0, -3, -3, 0], [0, 11, 1, 0, 0, -1, 1, -11], [0, 9, -1, -2, 0, 1, -1, 11], [7, 0, 0, 0, 1, 0, 0, 0], [0, 0, 15, 0, 0, 2, 2, 0], [0, 0, 15, 0, 0, -3, -3, 0], [0, -11, -5, 0, 0, 5, -5, 11], [0, 0, 12, 0, 0, 0, 0, 0], [0, -4, -5, -10, 0, 5, -5, 6], [0, -8, -5, 0, 0, 5, -5, 8], [0, 0, -3, 0, 0, 3, 3, 0], [0, 0, 0, 0, 0, 7, 7, 0], [-24, 0, 0, 0, 2, 0, 0, 0], [0, 2, 2, 4, 0, -2, 2, -2], [0, 9, 0, 0, 0, 0, 0, -9], [0, 3, -6, 0, 0, 6, -6, -3], [0, -4, 0, 0, 0, 0, 0, 4], [-4, 0, 0, 0, -6, 0, 0, 0], [0, 0, -18, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -7, 0, 0, 0], [0, -10, 2, 4, 0, -2, 2, -14], [0, 0, -21, 0, 0, 2, 2, 0], [-12, 0, 0, 0, -4, 0, 0, 0], [0, 7, -3, 0, 0, 3, -3, -7], [0, -10, -8, -16, 0, 8, -8, 6], [0, 7, 7, 14, 0, -7, 7, -7], [0, 0, 0, 0, 0, 7, 7, 0], [0, -18, -4, -8, 0, 4, -4, -10], [0, -12, -5, 0, 0, 5, -5, 12], [0, 0, 14, 0, 0, 2, 2, 0], [-11, 0, 0, 0, 7, 0, 0, 0], [0, 3, 6, 12, 0, -6, 6, -9], [0, 0, 0, 0, 8, 0, 0, 0], [0, 0, 9, 0, 0, -4, -4, 0], [0, -10, -3, 0, 0, 3, -3, 10], [0, 8, 6, 12, 0, -6, 6, -4], [-7, 0, 0, 0, 11, 0, 0, 0], [0, 0, -14, 0, 0, 5, 5, 0], [0, 0, 2, 4, 0, -2, 2, -4], [0, 0, 19, 0, 0, -5, -5, 0], [0, -11, -8, -16, 0, 8, -8, 5], [1, 0, 0, 0, 7, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 3], [0, 0, -15, 0, 0, 2, 2, 0], [28, 0, 0, 0, 8, 0, 0, 0], [0, 16, 13, 0, 0, -13, 13, -16], [0, 4, 8, 0, 0, -8, 8, -4], [0, -7, -8, 0, 0, 8, -8, 7], [-12, 0, 0, 0, 8, 0, 0, 0], [21, 0, 0, 0, 3, 0, 0, 0], [24, 0, 0, 0, 2, 0, 0, 0], [0, 0, -7, 0, 0, -1, -1, 0], [0, 17, 5, 0, 0, -5, 5, -17], [0, -18, -4, -8, 0, 4, -4, -10], [0, -5, 5, 10, 0, -5, 5, -15], [0, 10, 8, 0, 0, -8, 8, -10], [0, 13, 5, 10, 0, -5, 5, 3], [0, 0, -9, 0, 0, -10, -10, 0], [0, -3, -4, -8, 0, 4, -4, 5], [0, 0, 7, 0, 0, 0, 0, 0], [0, 0, -14, 0, 0, -2, -2, 0], [-45, 0, 0, 0, -3, 0, 0, 0], [0, 0, 5, 10, 0, -5, 5, -10], [0, 2, 2, 0, 0, -2, 2, -2], [0, 1, 3, 0, 0, -3, 3, -1], [0, 1, 5, 10, 0, -5, 5, -9], [0, -21, -7, -14, 0, 7, -7, -7], [9, 0, 0, 0, 15, 0, 0, 0], [0, 0, -12, 0, 0, 0, 0, 0], [0, 7, 7, 0, 0, -7, 7, -7], [0, 0, -9, 0, 0, 3, 3, 0], [0, 8, -1, -2, 0, 1, -1, 10], [0, 0, 3, 0, 0, -9, -9, 0], [0, 0, -6, 0, 0, -11, -11, 0], [-32, 0, 0, 0, -6, 0, 0, 0], [0, 5, 11, 0, 0, -11, 11, -5], [0, 5, -5, 0, 0, 5, -5, -5], [0, -3, -3, 0, 0, 3, -3, 3], [-21, 0, 0, 0, -3, 0, 0, 0], [-21, 0, 0, 0, 9, 0, 0, 0], [0, 0, 12, 0, 0, -6, -6, 0], [0, 16, 10, 0, 0, -10, 10, -16], [21, 0, 0, 0, 15, 0, 0, 0], [0, -4, 5, 10, 0, -5, 5, -14], [12, 0, 0, 0, -10, 0, 0, 0], [0, 5, 7, 0, 0, -7, 7, -5], [-22, 0, 0, 0, 0, 0, 0, 0], [0, -5, -4, -8, 0, 4, -4, 3], [0, 0, 14, 0, 0, 3, 3, 0], [0, 24, 4, 8, 0, -4, 4, 16], [0, 0, -18, 0, 0, -5, -5, 0], [0, 0, -24, 0, 0, -6, -6, 0], [0, -21, -5, -10, 0, 5, -5, -11], [0, 1, -2, 0, 0, 2, -2, -1], [8, 0, 0, 0, -8, 0, 0, 0], [0, 0, 9, 0, 0, 2, 2, 0], [0, 14, 0, 0, 0, 0, 0, -14], [0, 6, -2, -4, 0, 2, -2, 10], [0, 0, -6, 0, 0, -12, -12, 0], [0, -6, 6, 0, 0, -6, 6, 6], [0, 0, 2, 0, 0, 8, 8, 0], [0, 1, 6, 12, 0, -6, 6, -11], [-29, 0, 0, 0, 13, 0, 0, 0], [0, -17, -8, 0, 0, 8, -8, 17], [0, 0, 21, 0, 0, 3, 3, 0], [0, 10, -10, 0, 0, 10, -10, -10], [38, 0, 0, 0, 0, 0, 0, 0], [0, 12, -3, 0, 0, 3, -3, -12], [0, -7, -8, 0, 0, 8, -8, 7], [7, 0, 0, 0, 13, 0, 0, 0], [8, 0, 0, 0, -14, 0, 0, 0], [0, 5, 11, 0, 0, -11, 11, -5], [-13, 0, 0, 0, -7, 0, 0, 0], [0, 13, 7, 14, 0, -7, 7, -1], [12, 0, 0, 0, 4, 0, 0, 0], [0, 0, 0, 0, 0, -6, -6, 0], [0, -23, -2, -4, 0, 2, -2, -19], [0, -16, -14, -28, 0, 14, -14, 12], [0, 0, -9, 0, 0, 2, 2, 0], [0, -21, -1, -2, 0, 1, -1, -19], [0, -13, -2, 0, 0, 2, -2, 13], [0, 0, 7, 14, 0, -7, 7, -14], [0, 6, -12, 0, 0, 12, -12, -6], [0, 7, 13, 26, 0, -13, 13, -19], [0, -3, -11, -22, 0, 11, -11, 19], [-5, 0, 0, 0, 13, 0, 0, 0], [0, 0, 35, 0, 0, -2, -2, 0], [0, -15, -3, -6, 0, 3, -3, -9], [0, 0, -8, 0, 0, 4, 4, 0], [0, 1, -8, 0, 0, 8, -8, -1], [0, 0, 21, 0, 0, 9, 9, 0], [0, 14, 7, 14, 0, -7, 7, 0], [0, 16, -2, 0, 0, 2, -2, -16], [0, 12, -2, -4, 0, 2, -2, 16], [0, 0, 7, 0, 0, -11, -11, 0], [0, -7, -7, 0, 0, 7, -7, 7], [-21, 0, 0, 0, -15, 0, 0, 0], [45, 0, 0, 0, 3, 0, 0, 0], [-24, 0, 0, 0, 4, 0, 0, 0], [0, 3, 10, 20, 0, -10, 10, -17], [0, 0, -35, 0, 0, 0, 0, 0], [0, 0, 7, 0, 0, -5, -5, 0], [0, 26, 6, 12, 0, -6, 6, 14], [-28, 0, 0, 0, 0, 0, 0, 0], [0, -17, -18, -36, 0, 18, -18, 19], [0, 0, -2, 0, 0, -7, -7, 0], [0, -12, 4, 0, 0, -4, 4, 12], [0, 0, -14, 0, 0, 5, 5, 0], [0, 0, 3, 0, 0, 9, 9, 0], [0, 21, 7, 14, 0, -7, 7, 7], [8, 0, 0, 0, 10, 0, 0, 0], [0, 0, 21, 0, 0, 8, 8, 0], [0, 14, -9, 0, 0, 9, -9, -14], [0, 8, -1, -2, 0, 1, -1, 10], [0, -6, -5, -10, 0, 5, -5, 4], [-21, 0, 0, 0, 9, 0, 0, 0], [0, 0, -14, 0, 0, -7, -7, 0], [0, -22, -19, 0, 0, 19, -19, 22], [63, 0, 0, 0, -3, 0, 0, 0], [0, -3, 6, 12, 0, -6, 6, -15], [0, -10, 3, 0, 0, -3, 3, 10], [28, 0, 0, 0, -16, 0, 0, 0], [0, 7, 7, 14, 0, -7, 7, -7], [0, -22, 2, 0, 0, -2, 2, 22], [0, 22, 14, 0, 0, -14, 14, -22], [0, -2, 4, 0, 0, -4, 4, 2], [0, 0, 0, 0, -4, 0, 0, 0], [21, 0, 0, 0, 15, 0, 0, 0], [0, 0, 9, 0, 0, -9, -9, 0], [0, 17, -13, 0, 0, 13, -13, -17], [-29, 0, 0, 0, 1, 0, 0, 0], [0, 0, -13, -26, 0, 13, -13, 26], [0, 7, 7, 14, 0, -7, 7, -7], [0, 0, 32, 0, 0, 9, 9, 0], [-30, 0, 0, 0, -8, 0, 0, 0], [0, 8, 15, 0, 0, -15, 15, -8], [0, -11, -17, -34, 0, 17, -17, 23], [0, -11, -15, 0, 0, 15, -15, 11], [0, 0, -4, 0, 0, 20, 20, 0], [7, 0, 0, 0, -23, 0, 0, 0], [0, 0, 5, 10, 0, -5, 5, -10], [0, -13, -11, 0, 0, 11, -11, 13], [0, 1, 7, 14, 0, -7, 7, -13], [0, 15, 10, 20, 0, -10, 10, -5], [0, 0, 35, 0, 0, -8, -8, 0], [0, -5, 4, 0, 0, -4, 4, 5], [0, 0, 0, 0, 0, -6, -6, 0], [-49, 0, 0, 0, 5, 0, 0, 0], [0, -24, -14, -28, 0, 14, -14, 4], [0, 0, 35, 0, 0, 5, 5, 0], [0, 29, 16, 0, 0, -16, 16, -29], [-60, 0, 0, 0, 2, 0, 0, 0], [0, -7, -14, 0, 0, 14, -14, 7], [0, -26, -4, 0, 0, 4, -4, 26], [-21, 0, 0, 0, -3, 0, 0, 0], [26, 0, 0, 0, 18, 0, 0, 0], [10, 0, 0, 0, -10, 0, 0, 0], [0, -22, -19, 0, 0, 19, -19, 22], [0, 14, 7, 14, 0, -7, 7, 0], [-44, 0, 0, 0, 12, 0, 0, 0], [0, -22, -3, -6, 0, 3, -3, -16], [0, -24, -12, -24, 0, 12, -12, 0], [0, 4, 8, 0, 0, -8, 8, -4], [0, 0, -28, 0, 0, 3, 3, 0], [0, 0, 31, 0, 0, 1, 1, 0], [-7, 0, 0, 0, -1, 0, 0, 0], [0, 15, 4, 8, 0, -4, 4, 7], [0, 27, 9, 0, 0, -9, 9, -27], [0, 0, 21, 0, 0, -10, -10, 0], [0, -16, -5, -10, 0, 5, -5, -6], [7, 0, 0, 0, -11, 0, 0, 0], [0, 4, -14, 0, 0, 14, -14, -4], [0, 0, -30, 0, 0, -6, -6, 0], [0, 1, -11, 0, 0, 11, -11, -1], [0, 3, -7, -14, 0, 7, -7, 17], [0, -17, -1, -2, 0, 1, -1, -15], [0, 4, 13, 0, 0, -13, 13, -4], [0, -6, 10, 0, 0, -10, 10, 6], [0, -3, -13, 0, 0, 13, -13, 3], [-9, 0, 0, 0, -15, 0, 0, 0], [0, 0, 0, 0, -4, 0, 0, 0], [-7, 0, 0, 0, -1, 0, 0, 0], [52, 0, 0, 0, 12, 0, 0, 0], [0, 0, 33, 0, 0, 3, 3, 0], [-1, 0, 0, 0, -7, 0, 0, 0], [0, 6, -5, -10, 0, 5, -5, 16], [0, 0, 42, 0, 0, -5, -5, 0], [0, 0, 29, 0, 0, 5, 5, 0], [0, -28, -5, -10, 0, 5, -5, -18], [0, 0, -21, 0, 0, 2, 2, 0], [0, 0, 7, 0, 0, 18, 18, 0], [0, 0, -21, 0, 0, 3, 3, 0], [49, 0, 0, 0, -5, 0, 0, 0], [0, 18, 19, 38, 0, -19, 19, -20], [0, -14, 7, 0, 0, -7, 7, 14], [0, 0, 26, 0, 0, -3, -3, 0], [0, -7, 16, 0, 0, -16, 16, 7], [0, 0, 21, 0, 0, 14, 14, 0], [0, 27, 13, 26, 0, -13, 13, 1], [0, 0, -1, 0, 0, -1, -1, 0], [0, 25, 19, 0, 0, -19, 19, -25], [0, 0, 9, 0, 0, -3, -3, 0], [41, 0, 0, 0, -1, 0, 0, 0], [0, 12, 5, 10, 0, -5, 5, 2], [-24, 0, 0, 0, 20, 0, 0, 0], [0, 8, -3, 0, 0, 3, -3, -8], [63, 0, 0, 0, -3, 0, 0, 0], [-3, 0, 0, 0, 3, 0, 0, 0], [-10, 0, 0, 0, 18, 0, 0, 0], [7, 0, 0, 0, -23, 0, 0, 0], [0, 15, 14, 28, 0, -14, 14, -13], [0, -10, -4, -8, 0, 4, -4, -2], [12, 0, 0, 0, 8, 0, 0, 0], [0, 20, 12, 24, 0, -12, 12, -4], [-42, 0, 0, 0, -8, 0, 0, 0], [0, -23, -10, -20, 0, 10, -10, -3], [0, 0, 1, 2, 0, -1, 1, -2], [0, -14, -22, 0, 0, 22, -22, 14], [0, 0, -24, 0, 0, 0, 0, 0], [39, 0, 0, 0, -15, 0, 0, 0], [0, -17, -8, 0, 0, 8, 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