
# q-expansion of newform 1120.2.bw.f, downloaded from the LMFDB on 16 October 2025.

# 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 = 1120
weight = 2
poly_data = [1, 0, -3, 0, 8, 0, -3, 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], [[-3, 4, 8, 0, -4, 0, 1, 0], 4], [[0, -13, 0, 0, 0, 0, 0, -1], 4], [[3, 4, -8, 0, 4, 0, -1, 0], 4], [[9, 0, -24, 0, 8, 0, -3, 0], 8], [[-13, -1, 0, 24, 0, -8, -1, 3], 8], [[1, 9, -16, -24, 8, 8, -3, -3], 8], [[0, 15, 0, -40, 0, 16, 0, -5], 4]]

hecke_ring_character_values = [[351, [1, 0, 0, 0, 0, 0, 0, 0]], [421, [1, 0, 0, 0, 0, 0, 0, 0]], [897, [-1, 0, 0, 0, 0, 0, 0, 0]], [801, [0, 0, 0, 0, -1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, -1, 0, 0], [0, -1, 0, 1, 1, 0, 0, 0], [-1, 0, 0, 0, 1, 0, -1, -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], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 3, -1, 3, -1, -3, -3, 0], [-1, -4, 1, -2, 0, 2, -2, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, -1, 0, 1, 1, 3], [5, 1, 2, 0, -10, 0, -1, 2], [0, -3, -2, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -3, -2, 3, 3, 0, 0, -4], [-1, 5, 0, -5, -1, 0, 5, 2], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [4, 3, -2, -3, -14, -3, 0, -2], [8, -4, -2, 4, -3, 8, -8, -4], [0, 0, 0, 0, 0, 0, 0, 0], [-7, 2, -2, 4, 9, 2, -2, -1], [9, 0, -1, 0, -7, -5, -5, 0], [13, 0, -4, 0, -5, 3, 3, 0], [0, 3, 8, 0, -3, -3, -3, 4], [0, 0, 0, 0, 0, 0, 0, 0], [7, 0, -2, 7, 0, 7, -7, -2], [0, 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, 6, -1, -1, 0, 0, 12], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -7, 8, -7, 0, 0, 0, 0], [-10, 3, 5, -3, 14, -3, 0, 5], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-5, 12, -3, 6, 0, -6, 6, 3], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [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, -10, -1, 0, -1, 1, -10], [9, -9, 0, 0, 0, 9, -9, -8], [0, 0, 0, 0, -14, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, -6, -6, -6, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 9, 0, -9, 1, 0, 9, 5], [0, -5, 12, 0, 5, 5, 5, 6], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -14, 0, 14, 14, 0], [5, -5, 0, 0, 0, 5, -5, 8], [0, 0, 0, 0, 0, 0, 0, 0], [-4, 11, 2, -11, -14, -11, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-13, 9, 0, -9, -13, 0, 9, -2], [19, -12, 3, -6, 0, 6, -6, -3], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-14, 0, 0, 1, -13, -1, 0, 7], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [19, -9, 7, -9, -5, 9, 9, 0], [0, -14, 0, 0, 14, 14, 14, 0], [0, 0, 0, 0, 0, 0, 0, 0], [16, -8, -4, 8, 9, 16, -16, -8], [0, 9, -2, 0, -9, -9, -9, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 9, 0, -9, -9, 2], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-29, 3, -8, 3, 13, -3, -3, 0], [0, 0, -9, -5, 0, 5, 5, 9], [0, 0, 0, 0, 0, 0, 0, 0], [-42, 0, 0, 0, 0, 0, 0, 0], [2, -13, -1, 13, 22, 13, 0, -1], [-23, 0, 8, 0, 7, -9, -9, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-7, 7, 0, 0, 0, -7, 7, -10], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-17, -3, -7, 0, 34, 0, 3, -7], [20, -10, -5, 10, 3, 20, -20, -10], [-42, 0, 0, 0, 42, 0, 0, 0], [0, -11, 16, -11, 0, 0, 0, 0], [12, -6, -3, 6, 19, 12, -12, -6], [-16, -1, 8, 1, 34, 1, 0, 8], [0, 0, 0, 0, 0, 0, 0, 0], [7, -15, 0, 15, 7, 0, -15, -4], [0, -14, 0, 14, 14, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [15, 0, -4, 15, 0, 15, -15, -4], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 18, 0, -18, -18, 0], [-33, -5, -7, -5, 19, 5, 5, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -11, -6, 0, 11, 11, 11, -3], [-13, 0, -4, -13, 0, -13, 13, -4], [14, 0, 0, -15, -1, 15, 0, -7], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -9, 16, 0, 9, 9, 9, 8], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-20, 0, 0, 9, -11, -9, 0, 10], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -12, 5, 0, -5, -5, 12], [-12, 6, 3, -6, 23, -12, 12, 6], [0, 0, 0, 0, 0, 0, 0, 0], [10, -17, -5, 17, 14, 17, 0, -5], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [2, 15, -1, -15, -34, -15, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -42, 0, 0, 0], [-14, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [7, 13, 0, -13, 7, 0, 13, 10], [0, 0, 0, 0, 0, 0, 0, 0], [-31, -4, 4, -8, 27, -4, 4, 2], [0, 0, 0, 0, 0, 0, 0, 0], [0, 14, 0, -14, -14, 0, 0, 0], [-22, 0, 0, 5, -17, -5, 0, 11], [-19, -15, -2, 0, 38, 0, 15, -2], [0, -6, 0, 0, 6, 6, 6, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [41, 0, -11, 0, -19, 3, 3, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, -13, 0, 13, 13, -3], [7, 0, 16, 7, 0, 7, -7, 16], [-37, -9, -7, -9, 23, 9, 9, 0], [8, 0, 0, -19, -11, 19, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 11, -9, -11, -11, 0, 0, -18], [0, 0, 0, 0, 0, 0, 0, 0], [42, 0, 0, 0, -42, 0, 0, 0], [-43, 0, 10, 0, 23, 3, 3, 0], [0, 0, 0, 0, 0, 0, 0, 0], [23, 13, 5, 0, -46, 0, -13, 5], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-7, -15, 0, 15, -7, 0, -15, -11], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-29, 24, -6, 12, 0, -12, 12, 6], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [23, 12, -3, 6, 0, -6, 6, 3], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [19, 16, -4, 8, 0, -8, 8, 4], [0, 0, 0, 0, 0, 0, 0, 0], [18, 0, 0, 0, -18, 0, 0, 0], [-25, -11, -7, 0, 50, 0, 11, -7], [0, -3, -16, 3, 3, 0, 0, -32], [-27, 0, 11, 0, 5, -17, -17, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [21, -21, 0, 0, 0, 21, -21, -10], [-35, -2, 2, -4, 33, -2, 2, 1], [0, 0, 0, 0, 0, 0, 0, 0], [-19, 19, 0, 0, 0, -19, 19, 20], [0, 15, -1, -15, -15, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-21, 0, 16, -21, 0, -21, 21, 16], [0, 0, 0, 0, 0, 0, 0, 0], [0, -11, -12, 11, 11, 0, 0, -24], [21, -21, 0, 0, 0, 21, -21, -4], [0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 34, 0, 3, 3, 3, 17], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 26, -3, 0, 0, 0, 0], [17, -10, 10, -20, -27, -10, 10, 5], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-35, -12, 12, -24, 23, -12, 12, 6], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 21, -20, 21, 0, 0, 0, 0], [-13, 32, -8, 16, 0, -16, 16, 8], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -18, 1, 0, -1, -1, 18], [26, -1, -13, 1, -50, 1, 0, -13], [47, -9, 14, -9, -19, 9, 9, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -30, 0, 30, 30, 0, 0, 0], [13, -23, 0, 23, 13, 0, -23, -5], [-7, -16, 16, -32, -9, -16, 16, 8], [0, 21, -22, 21, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -15, 8, 15, 15, 0, 0, 16], [0, 0, 0, 0, 0, 0, 0, 0], [-16, 8, 4, -8, 33, -16, 16, 8], [0, -7, -22, -7, 0, 0, 0, 0], [25, -3, 14, 0, -50, 0, 3, 14], [0, 0, -16, 9, 0, -9, -9, 16], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [33, 0, -13, 0, -7, 19, 19, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 17, -6, 0, -17, -17, -17, -3], [-11, 17, -14, 0, 22, 0, -17, -14], [-14, 0, 0, -15, -29, 15, 0, 7], [0, 3, -19, -3, -3, 0, 0, -38], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [21, -21, 0, 0, 0, 21, -21, 4], [0, 0, 0, 0, 0, 0, 0, 0], [-23, 23, 0, 0, 0, -23, 23, 22], [12, -6, -3, 6, -37, 12, -12, -6], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [11, 25, -7, 0, -22, 0, -25, -7], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-21, 0, 26, -21, 0, -21, 21, 26], [0, 0, 0, 0, 0, 0, 0, 0], [-8, 4, 2, -4, -39, -8, 8, 4], [22, 0, 0, 9, 31, -9, 0, -11], [0, 0, -11, 15, 0, -15, -15, 11], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [17, 18, -18, 36, 1, 18, -18, -9], [28, 0, 0, 1, 29, -1, 0, -14], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [29, -19, 0, 19, 29, 0, -19, 5], [16, 15, -8, -15, -62, -15, 0, -8], [-14, 0, 0, 0, 14, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [54, 0, 0, 0, 0, 0, 0, 0], [42, 0, 0, 0, 0, 0, 0, 0], [-19, 13, -16, 0, 38, 0, -13, -16], [0, 0, 0, 0, 0, 0, 0, 0], [36, -18, -9, 18, 11, 36, -36, -18], [-7, 7, 0, 0, 0, -7, 7, 32], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 25, -2, 25, 0, 0, 0, 0], [0, 21, 8, 21, 0, 0, 0, 0], [-31, -18, 18, -36, 13, -18, 18, 9], [0, 15, 13, -15, -15, 0, 0, 26], [0, 0, 0, 0, 0, 0, 0, 0], [14, 0, 0, -27, -13, 27, 0, -7], [0, 0, 0, 0, 0, 0, 0, 0], [27, 0, -10, 27, 0, 27, -27, -10], [47, -12, 3, -6, 0, 6, -6, -3], [-33, 23, -14, 23, 5, -23, -23, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 10, 0, -10, -10, 0], [0, 11, -36, 0, -11, -11, -11, -18], [7, -27, 0, 27, 7, 0, -27, -10], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 19, -12, 0, -19, -19, -19, -6], [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, 32, -1, 0, -1, 1, 32], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0]]
