
# q-expansion of newform 570.2.c.f, downloaded from the LMFDB on 25 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 = 570
weight = 2
poly_data = [5, -26, 54, -66, 59, -34, 16, -4, 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], [[-10, 46, -62, 82, -50, 34, -8, 3], 5], [[25, -51, 67, -82, 50, -34, 8, -3], 5], [[-30, 109, -118, 118, -60, 31, -7, 2], 5], [[50, -138, 191, -186, 110, -57, 14, -4], 5], [[45, -157, 229, -249, 155, -83, 21, -6], 5], [[85, -249, 298, -313, 170, -96, 22, -7], 5]]

hecke_ring_character_values = [[191, [-1, 0, 0, 0, 0, 0, 0, 0]], [457, [-1, 0, 0, 0, 0, 0, 0, 0]], [211, [-1, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, -1, 1, 0, 1, 1], [1, 0, -1, 1, -1, 0, -2, -1], [1, -3, 1, 0, -1, 0, 2, 1], [-1, 0, 0, -2, 2, 0, 1, 2], [0, 0, 0, 2, 0, 0, -1, 0], [0, 1, -3, 0, 1, -2, -1, 1], [-1, 1, -1, 2, 1, 0, -2, -1], [1, 1, -5, 0, 1, -2, -2, 1], [-3, 1, 3, 0, 1, -2, 2, 1], [1, -4, 2, 0, -2, 0, 3, 2], [2, 0, 0, 0, 0, 0, 0, 0], [3, 0, -6, 0, 0, 0, -3, 0], [1, -2, 0, 0, 0, 0, 1, 0], [4, -2, -2, 0, -2, 4, -2, -2], [0, 0, 0, 0, 0, 0, 1, 0], [4, -3, 1, 0, -3, 6, -1, -3], [1, 0, 0, 2, 0, 0, -1, 0], [7, 0, 0, 0, 0, 0, 0, 0], [-1, -2, 8, 0, -2, 4, 3, -2], [0, -3, 3, 0, -3, 0, 3, 3], [-4, 8, 0, 0, 0, 0, -4, 0], [0, -2, 6, 0, -2, 4, 2, -2], [1, -2, 4, 0, -2, 4, 1, -2], [10, 0, 0, 4, 0, 0, -2, 0], [1, 0, 0, 2, -2, 0, 9, -2], [1, 0, 0, -10, 0, 0, 5, 0], [-3, 0, 0, -6, 6, 0, 16, 6], [1, 3, -5, 0, 5, 0, -4, -5], [-1, 0, 0, -2, 2, 0, 7, 2], [-10, 0, 0, 4, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 10, 0], [0, 1, -3, 0, 1, -2, -1, 1], [-10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0], [-4, 6, 2, 0, -2, 0, -2, 2], [-2, 4, 0, 0, 0, 0, -2, 0], [-1, 6, -4, 0, 4, 0, -5, -4], [-3, 0, 0, -6, 6, 0, -9, 6], [4, 0, 0, 8, -8, 0, -8, -8], [6, -2, -6, 0, -2, 4, -4, -2], [4, -6, -2, 0, 2, 0, 2, -2], [0, 0, 0, 0, 0, 0, 5, 0], [6, 0, 0, 0, 0, 0, 0, 0], [-3, 2, 0, 0, 2, -4, 1, 2], [-2, 0, 0, 14, 0, 0, -7, 0], [0, 5, -5, 0, 5, 0, -5, -5], [-10, 0, 0, -8, 0, 0, 4, 0], [-3, 0, 0, -6, 6, 0, -4, 6], [0, 0, 0, 0, 0, 0, 0, 0], [12, -6, -6, 0, -6, 12, -6, -6], [2, 0, 0, 4, -4, 0, 9, -4], [2, -8, 4, 0, -4, 0, 6, 4], [-5, 0, 0, -10, 10, 0, -5, 10], [-3, 0, 0, -6, 6, 0, 11, 6], [2, 4, -16, 0, 4, -8, -6, 4], [4, 2, -14, 0, 2, -4, -6, 2], [6, 0, 0, 2, 0, 0, -1, 0], [4, -10, 2, 0, -2, 0, 6, 2], [5, -4, 2, 0, -4, 8, -1, -4], [-6, 6, 6, 0, -6, 0, 0, 6], [4, 0, 0, 8, -8, 0, -3, -8], [2, 0, 0, 0, 0, 0, 0, 0], [6, 0, 0, 12, -12, 0, -11, -12], [4, 1, -9, 0, 9, 0, -5, -9], [0, 0, 0, 0, 0, 0, -17, 0], [-4, 11, -3, 0, 3, 0, -7, -3], [-5, 0, 0, 14, 0, 0, -7, 0], [12, -8, 0, 0, -8, 16, -4, -8], [2, 0, 0, -4, 0, 0, 2, 0], [-12, 7, 3, 0, 7, -14, 5, 7], [-6, 0, 0, -12, 12, 0, 27, 12], [-2, 4, 0, 0, 0, 0, -2, 0], [-20, 0, 0, 2, 0, 0, -1, 0], [10, -23, 3, 0, -3, 0, 13, 3], [5, 0, 0, 10, -10, 0, -19, -10], [7, 0, 0, 14, -14, 0, -21, -14], [-8, 18, -2, 0, 2, 0, -10, -2], [-9, 4, 6, 0, 4, -8, 5, 4], [1, -2, 0, 0, 0, 0, 1, 0], [-3, 0, 0, -6, 6, 0, 19, 6], [5, -15, 5, 0, -5, 0, 10, 5], [13, -10, 4, 0, -10, 20, -3, -10], [5, 0, 0, 2, 0, 0, -1, 0], [1, -2, 0, 0, 0, 0, 1, 0], [5, -2, -4, 0, -2, 4, -3, -2], [-2, -6, 22, 0, -6, 12, 8, -6], [-6, 5, 7, 0, -7, 0, 1, 7], [-5, 0, 0, -10, 10, 0, 5, 10], [-4, 8, 0, 0, 0, 0, -4, 0], [-7, 0, 14, 0, 0, 0, 7, 0], [0, 0, 0, 0, 0, 0, -14, 0], [10, 0, 0, -16, 0, 0, 8, 0], [5, 0, 0, 10, -10, 0, -35, -10], [25, 0, 0, 10, 0, 0, -5, 0], [9, -5, -3, 0, -5, 10, -4, -5], [-2, 4, -8, 0, 4, -8, -2, 4], [14, -12, 8, 0, -12, 24, -2, -12], [-29, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -18, 0, 0, 9, 0], [12, 0, 0, 0, 0, 0, 0, 0], [-12, 10, -6, 0, 10, -20, 2, 10], [-6, 0, 0, -12, 12, 0, 2, 12], [-2, 4, -8, 0, 4, -8, -2, 4], [-29, 0, 0, 2, 0, 0, -1, 0], [4, -5, -3, 0, 3, 0, 1, -3], [-11, 8, -2, 0, 8, -16, 3, 8], [2, 4, -16, 0, 4, -8, -6, 4], [17, -14, 8, 0, -14, 28, -3, -14], [-7, 18, -4, 0, 4, 0, -11, -4], [-30, 0, 0, 4, 0, 0, -2, 0], [-5, 12, -2, 0, 2, 0, -7, -2], [-10, 6, 2, 0, 6, -12, 4, 6], [-15, 0, 0, -10, 0, 0, 5, 0], [-4, 0, 0, 12, 0, 0, -6, 0], [-5, 6, -8, 0, 6, -12, -1, 6], [2, -16, 12, 0, -12, 0, 14, 12], [1, -1, 1, 0, -1, 2, 0, -1], [2, -6, 14, 0, -6, 12, 4, -6], [-10, 5, 5, 0, 5, -10, 5, 5], [-9, 21, -3, 0, 3, 0, -12, -3], [0, 0, 0, -8, 0, 0, 4, 0], [10, 0, 0, 20, -20, 0, -7, -20], [0, 0, 0, 0, 0, 0, -24, 0], [6, 0, 0, 12, 0, 0, -6, 0], [4, 0, 0, 8, -8, 0, 16, -8], [22, 0, 0, -4, 0, 0, 2, 0], [-6, 0, 0, -12, 12, 0, -3, 12], [-3, 3, 3, 0, -3, 0, 0, 3], [2, -6, 2, 0, -2, 0, 4, 2], [13, 0, 0, 14, 0, 0, -7, 0], [9, 0, 0, 18, -18, 0, -3, -18], [-7, 18, -4, 0, 4, 0, -11, -4], [-4, 12, -4, 0, 4, 0, -8, -4], [-1, 0, 0, -2, 2, 0, 16, 2], [-15, 0, 0, 20, 0, 0, -10, 0], [10, 0, 0, 20, -20, 0, -29, -20], [15, 0, 0, 4, 0, 0, -2, 0], [2, 0, 0, 4, -4, 0, 21, -4], [-3, 0, 0, -6, 6, 0, 14, 6], [10, -15, -5, 0, 5, 0, 5, -5], [0, 0, 0, 0, 0, 0, 30, 0], [5, -13, 3, 0, -3, 0, 8, 3], [-5, 0, 0, -10, 10, 0, 18, 10], [-9, 23, -5, 0, 5, 0, -14, -5], [-10, 10, -10, 0, 10, -20, 0, 10], [14, -34, 6, 0, -6, 0, 20, 6], [0, 0, 0, 0, 0, 0, -12, 0], [-7, 0, 0, -6, 0, 0, 3, 0], [-1, 0, 0, -2, 2, 0, 17, 2], [10, 0, 0, -6, 0, 0, 3, 0], [-6, 0, 0, -12, 12, 0, 16, 12], [2, -16, 12, 0, -12, 0, 14, 12], [-5, 0, 0, -10, 10, 0, -7, 10], [27, 0, 0, 20, 0, 0, -10, 0], [-8, 2, 10, 0, 2, -4, 6, 2], [-8, 0, 0, 6, 0, 0, -3, 0], [-3, 0, 0, -6, 6, 0, 34, 6], [-6, 25, -13, 0, 13, 0, -19, -13], [-1, 3, -7, 0, 3, -6, -2, 3], [-10, 6, 2, 0, 6, -12, 4, 6], [5, 0, 0, 10, -10, 0, -29, -10], [10, -24, 4, 0, -4, 0, 14, 4], [-8, 12, -20, 0, 12, -24, -4, 12], [5, 0, 0, 10, -10, 0, 3, -10], [-10, 0, 0, -20, 20, 0, 26, 20], [-7, 8, 6, 0, -6, 0, -1, 6], [9, -20, 2, 0, -2, 0, 11, 2]]
