# q-expansion of newform 720.2.ct.a, downloaded from the LMFDB on 25 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
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_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 = 720
weight = 2
poly_data = [1, 0, -1, 0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = [[271, [-1, 0, 0, 0]], [181, [0, 0, 0, 1]], [641, [-1, 0, 1, 0]], [577, [0, 0, 0, 1]]]
aps_data = [[-1, 0, 0, 1], [1, 0, -2, 0], [0, 1, 2, -1], [-1, -1, 2, 2], [-3, -2, 1, -1], [-2, 1, -2, 0], [-1, -2, -2, -1], [-3, 0, 0, 3], [-1, 1, 0, -1], [-1, 7, -6, -6], [-4, -1, -4, 0], [2, 0, -4, 6], [-1, -4, -1, 0], [2, 2, -1, -2], [-5, -3, 2, -2], [4, 4, 0, -2], [-8, -8, 7, 1], [-3, 1, 4, -4], [3, -4, 3, 0], [0, -12, 0, 6], [7, -2, -2, 7], [0, 3, 10, 3], [0, 2, 5, 2], [0, -8, 0, 4], [1, -4, 3, 3], [-9, 7, 2, 2], [-9, 9, 2, -7], [0, 20, 0, -10], [1, 2, 2, 1], [4, -4, -9, -5], [-3, -4, 4, 3], [-2, -2, -1, 3], [-5, 8, 13, -13], [-6, -6, -3, 9], [-13, 13, 4, -9], [0, -1, -8, -1], [-2, -5, 2, 10], [-16, -4, 0, 2], [9, 9, -6, -3], [12, -3, -6, 3], [-1, 0, 0, 1], [3, 6, 6, 3], [20, -3, -10, 3], [2, -2, 11, 13], [-4, 0, 8, 6], [4, 0, -8, -20], [-6, -6, 13, -7], [9, -15, 6, 6], [22, 0, -11, 0], [-9, 9, 2, -7], [-5, -8, -8, -5], [6, 13, -6, -26], [0, 6, -13, 6], [-5, -8, 8, 5], [2, 2, -11, 9], [-9, -13, -4, 4], [-15, 2, 2, -15], [-4, 0, 8, -18], [-8, -3, 4, 3], [18, 14, -9, -14], [3, 12, 3, 0], [-12, -5, 12, 10], [6, 0, -12, 14], [2, -13, -2, 26], [-3, 2, 1, 1], [0, -3, -20, -3], [1, 20, 19, -19], [-16, -16, -5, 21], [1, -2, -1, 4], [1, 13, -14, -14], [5, -6, -11, 11], [6, 0, -12, -8], [-15, -15, 0, 0], [28, 1, -28, -2], [-5, -2, -2, -5], [-11, -11, 12, -1], [1, -17, -18, 18], [-2, 0, 0, 0], [-31, 2, 31, -4], [23, 4, -23, -8], [6, -6, 1, 7], [-7, 21, -14, -14], [8, 0, -16, 10], [-3, -10, 10, 3], [-24, 11, 12, -11], [-18, -20, 9, 20], [-12, 0, 24, 18], [-1, -18, -17, 17], [-7, -5, 2, -2], [9, 9, 14, -23], [-12, 0, 24, -4], [0, 5, -4, 5], [11, -4, -4, 11], [-6, 6, 9, 3], [24, -24, -11, 13], [21, 10, -10, -21], [-15, -15, -2, 17], [-4, 0, 8, -32], [10, 0, -20, -8], [-11, -16, 16, 11], [34, -14, -17, 14], [-14, 32, 0, -16], [-15, 8, 15, -16], [0, 4, 3, 4], [-4, 4, -3, -7], [27, -6, -6, 27], [0, 22, 7, 22], [-19, -18, 18, 19], [-10, -3, -10, 0], [2, 2, -1, -2], [-5, 5, 0, -5], [-10, -8, 0, 4], [-18, 18, -11, -29], [9, -14, 5, 5], [-24, 16, 0, -8], [0, 2, 3, 2], [5, -12, -5, 24], [25, -8, -8, 25], [-22, 7, -22, 0], [3, 20, 17, -17], [-15, -15, -2, 17], [5, -6, -11, 11], [-4, 27, 2, -27], [-4, 0, 8, -20], [-3, 12, -9, -9], [9, 14, -14, -9], [-15, 11, 26, -26], [-2, 4, 0, -2], [-3, 11, -8, -8], [14, 9, 14, 0], [-13, -14, 14, 13], [-7, 7, 4, -3], [-22, -1, -22, 0], [0, 0, 0, -10], [-15, 10, -15, 0], [9, -22, 9, 0], [0, 20, 0, -10], [3, 34, 3, 0], [0, 7, 0, -14], [-2, -20, 0, 10], [7, -4, 4, -7], [9, -3, -6, -6], [1, -1, -14, -13], [-12, -36, 0, 18], [17, -10, -10, 17], [16, -27, -8, 27], [0, 11, -16, 11], [-5, 2, 7, -7], [12, 12, -7, -5], [-11, 24, 24, -11], [8, 19, -8, -38], [-32, -16, 0, 8], [-24, -4, 0, 2], [-13, -13, -14, 27], [0, 4, 25, 4], [-28, 11, 14, -11], [-18, 0, 36, -4], [-18, -10, 9, 10], [-17, -22, 22, 17], [-7, 7, 18, 11], [-6, 34, 3, -34], [-21, 38, 38, -21], [31, 31, -26, -5], [-13, -10, 10, 13], [-10, -10, -27, 37], [-21, 11, 32, -32], [24, 0, -48, 2], [-12, -17, 6, 17]]