
# q-expansion of newform 570.2.u.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
    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 = 570
weight = 2
poly_data = [1, 0, 0, -1, 0, 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 = [[191, [1, 0, 0, 0, 0, 0]], [457, [1, 0, 0, 0, 0, 0]], [211, [0, 0, 0, 0, 0, -1]]]
aps_data = [[0, -1, 0, 0, 1, 0], [0, -1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0], [1, 0, -1, -1, 1, 1], [0, -3, 1, -1, 1, -3], [-3, -3, 2, 1, 2, -2], [1, -2, -1, 1, 2, -1], [2, 2, 2, -1, -2, 2], [2, -3, 0, 1, 1, 5], [4, -7, -3, -7, 4, 0], [-6, -1, -1, 6, 2, 2], [-7, 0, 0, 0, -3, 3], [1, -3, -5, -6, 0, 6], [-2, 0, 2, -1, 7, -3], [-5, 2, -7, 2, -5, 0], [-3, 3, 0, 0, 0, 0], [0, 7, -1, 1, -7, 0], [5, -2, 0, -3, -3, -8], [3, 4, -7, 4, 3, 0], [7, 0, -7, 0, 1, 7], [3, 4, -4, -7, 0, 7], [-7, -1, -6, 1, 0, -1], [5, -9, -4, -5, 13, 13], [-1, -1, -3, 3, -2, 3], [-2, 5, 2, -2, -5, 2], [2, 2, 6, 9, -11, -6], [0, 4, 5, -5, 5, 4], [-1, -3, 0, 1, 3, 3], [-8, 0, 8, 7, 1, -1], [-2, 2, 2, 0, 2, -4], [7, 7, -10, -2, -5, 10], [2, 4, -4, 4, -4, -2], [12, -1, 0, -11, -11, 1], [3, 3, 3, 6, -9, -3], [7, -2, 0, -7, 0, 7], [7, 2, 2, 0, 1, -3], [-3, 0, 3, -6, -4, -9], [0, 0, -7, -8, -7, 0], [-4, 3, 0, 1, 1, -9], [-3, -8, -6, 6, 8, 3], [0, 0, 8, -4, 8, 0], [-1, 0, 4, 0, -1, 0], [1, 2, 2, 0, 4, -6], [8, 2, -1, -9, 0, 9], [-8, 9, 2, 8, -11, -11], [-17, 8, -6, 8, -17, 0], [1, 9, -6, 6, -9, -1], [-8, 0, 8, 1, -6, -7], [2, 6, 6, 0, 3, -9], [-3, -6, -6, 0, -1, 7], [-6, 0, 6, 6, -6, 0], [0, -3, 6, 13, 6, -3], [0, 0, -10, -5, 5, 10], [1, -11, 0, 10, 10, 3], [-12, -9, 5, -9, -12, 0], [4, 5, 6, 2, 0, -2], [-7, -5, -12, -5, 0, 5], [5, 0, -5, -5, -5, 0], [0, 2, -4, -17, -4, 2], [5, -10, 0, 5, 5, -15], [10, 6, -5, 5, -6, -10], [0, -1, 1, 8, 1, -1], [0, -1, -12, -12, 0, 12], [-3, -11, 2, 3, 9, 9], [-2, -11, 1, -11, -2, 0], [-3, -3, 13, 1, 2, -13], [0, -4, 7, -14, 7, -4], [-6, 0, 6, 6, 14, 0], [-1, 0, 1, -2, 3, -3], [-1, -15, 1, 1, 14, 14], [0, -8, 14, -4, 14, -8], [13, -5, 11, -11, 5, -13], [-16, -16, -9, 15, 1, 9], [-9, 10, -6, 9, -4, -4], [9, -4, -4, 0, 1, 3], [-5, -7, -14, -9, 0, 9], [0, 7, -16, 7, 0, 0], [0, 7, 7, -7, -7, 0], [1, 6, 11, -11, -6, -1], [-11, 14, 7, 14, -11, 0], [-5, 1, 1, 0, 9, -10], [0, -11, 0, 0, 0, 0], [-22, -22, 0, 16, 6, 0], [7, -9, 0, 2, 2, -24], [4, 15, -7, 7, -15, -4], [6, 6, -5, 14, -20, 5], [-14, -14, 3, 14, 11, 11], [10, 12, 12, 0, -5, -7], [-5, 0, 5, 11, -16, 6], [6, 18, -21, -6, 3, 3], [0, -11, 0, 5, 0, -11], [3, 3, 0, -6, -6, 18], [-7, 13, -12, 7, -1, -1], [12, 12, 3, -9, 0, 9], [3, 0, -3, -5, 14, -2], [-4, 13, -9, 13, -4, 0], [4, 10, 0, -14, -14, -2], [0, 0, 10, -17, 10, 0], [-4, -17, 10, -17, -4, 0], [3, 10, 15, 10, 3, 0], [4, -4, 0, 0, 0, -3], [15, 15, -3, -21, 6, 3], [-5, -1, -12, 5, 13, 13], [-8, 3, 3, 0, -6, 3], [20, -10, -10, 0, 11, -1], [-21, 9, -11, 21, 2, 2], [-23, 8, -6, 6, -8, 23], [2, -19, 0, 17, 17, 3], [-5, 10, -7, 10, -5, 0], [0, -10, 21, 0, -11, -11], [1, 24, 24, 0, -5, -19], [-17, 0, 17, -5, 6, -22], [4, 1, -28, 1, 4, 0], [0, -12, 16, 5, 16, -12], [15, 3, 0, -18, -18, 0], [-1, 0, 1, -7, 11, -8], [19, -17, 10, -9, 0, 9], [-25, -5, -5, 0, -6, 11], [-12, 22, -8, 12, -14, -14], [1, 1, 3, 5, -6, -3], [-23, 10, 0, 13, 13, 11], [0, 20, -9, -11, -9, 20], [5, -8, -15, -5, 23, 23], [18, 3, 3, 0, 5, -8], [-21, 7, -19, 21, 12, 12], [-9, -8, -23, 23, 8, 9], [-15, -15, 31, 7, 8, -31], [3, 36, 2, -1, 0, 1], [1, 0, -1, -6, -30, -5], [0, 7, -19, -14, -19, 7], [11, 7, 13, -13, -7, -11], [-18, -16, -19, 19, 16, 18], [10, -8, -17, -8, 10, 0], [1, 8, 11, 10, 0, -10], [-1, -23, -23, 0, 15, 8], [15, 4, 18, 4, 15, 0], [-19, -19, 13, 22, -3, -13], [0, -6, 9, 13, 9, -6], [7, 9, 9, 0, -20, 11], [0, -3, 5, 14, 5, -3], [1, 1, 10, -8, 7, -10], [-20, 21, 0, -1, -1, -7], [-7, -7, -19, -2, 9, 19], [-7, -8, 27, -8, -7, 0], [16, -2, 22, -16, -20, -20], [-9, -22, -18, -9, 0, 9], [0, -6, 15, -15, 6, 0], [13, -19, 2, -2, 19, -13], [5, 7, 0, -12, -12, -13], [0, -10, 14, -6, 14, -10], [-6, -23, 6, 12, 0, -12], [-15, 14, -10, 15, -4, -4], [14, -17, -3, -17, 14, 0], [-15, -15, -6, 18, -3, 6], [-10, 0, 10, 11, -35, 1], [-1, 24, 24, 0, -17, -7], [-29, -16, 19, 29, -3, -3], [-5, -35, -14, 14, 35, 5], [-1, -1, -6, 3, -2, 6], [-6, -4, -23, -4, -6, 0], [14, -19, -24, -38, 0, 38], [-1, 26, 10, 11, 0, -11], [-19, -1, -20, 20, 1, 19], [-7, -7, 23, -23, 7, 7], [0, 13, -11, 5, -11, 13], [-2, 0, 2, 7, -2, 5], [-15, 32, -25, -10, 0, 10], [12, -2, -12, -2, 12, 0]]
