
# q-expansion of newform 1170.2.e.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 = 1170
weight = 2
poly_data = [40, -12, 1, 16, -3, -2, 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], 1], [[0, 1, 0, 0, 0, 0], 1], [[240, -966, 120, 183, -70, 4], 445], [[-1000, 643, 390, -184, -5, 13], 445], [[-180, 57, -1, -26, 8, -3], 89], [[-6, -7, -92, 11, 24, -9], 178]]

hecke_ring_character_values = [[911, [1, 0, 0, 0, 0, 0]], [937, [-1, 0, 0, 0, 0, 0]], [1081, [1, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0], [0, -1, -1, 1, -1, 1], [0, 1, 1, 0, -1, 0], [-2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 2, 2, -1, -1, 0], [-2, 2, -2, -2, 0, 0], [0, 0, 0, -2, 2, -2], [-2, 2, -2, -2, 0, 0], [4, -2, -2, 0, -2, 0], [0, 1, 1, -2, 1, 2], [-2, 0, -4, -2, -2, 0], [0, 0, 0, 3, -3, -4], [0, 1, 1, 0, -1, -4], [0, 0, 0, -2, 2, -2], [2, -2, 2, 2, 0, 0], [2, 0, -4, -2, -2, 0], [0, 0, 0, -2, 2, 4], [4, -1, 3, 2, 1, 0], [0, 0, 0, 0, 0, -6], [-8, -2, 2, 2, 0, 0], [0, 2, 2, -2, 0, 4], [-2, 2, -2, -2, 0, 0], [0, -2, -2, 6, -4, 10], [6, -8, -4, 2, -6, 0], [0, 4, 4, -2, -2, -2], [0, 0, 0, -2, 2, 0], [-8, 3, 3, 0, 3, 0], [0, 0, 0, 0, 0, -4], [0, 0, 0, -2, 2, -6], [4, 4, 6, 1, 5, 0], [0, 0, 0, -2, 2, 10], [-4, 4, 2, -1, 3, 0], [0, 0, 0, 0, 0, 0], [4, 3, 7, 2, 5, 0], [0, -2, -2, 2, 0, -6], [0, 2, 2, -6, 4, -4], [0, 0, 0, 8, -8, 0], [0, 0, 0, 2, -2, 14], [-4, 4, 2, -1, 3, 0], [-2, -6, -6, 0, -6, 0], [-8, 6, -2, -4, 2, 0], [0, 0, 0, 6, -6, -2], [0, -1, -1, 6, -5, 14], [0, -10, -2, 4, -6, 0], [8, 4, -2, -3, 1, 0], [0, -5, -5, 12, -7, 12], [0, -4, -4, 0, 4, -4], [-12, 7, -1, -4, 3, 0], [0, 2, 2, 7, -9, 0], [8, -3, 1, 2, -1, 0], [2, 10, -2, -6, 4, 0], [12, -4, -4, 0, -4, 0], [0, 2, 2, 1, -3, 8], [0, 2, 2, 0, -2, 2], [-6, -4, 0, 2, -2, 0], [-4, -9, -5, 2, -7, 0], [0, -6, -6, 8, -2, 6], [-14, 2, 6, 2, 4, 0], [0, 0, 0, -10, 10, 4], [0, -3, -3, -2, 5, -6], [0, -2, -2, 4, -2, 12], [-16, 4, -8, -6, -2, 0], [0, -2, -2, 3, -1, 20], [0, -4, -4, 4, 0, 2], [10, 2, 10, 4, 6, 0], [0, 2, 2, 5, -7, -12], [0, 4, 4, -5, 1, -20], [-16, 11, 3, -4, 7, 0], [0, 0, 0, 10, -10, -14], [12, -2, 6, 4, 2, 0], [0, 4, 4, -4, 0, -22], [0, 6, 6, -8, 2, -10], [-2, 2, -6, -4, -2, 0], [0, -5, -5, 8, -3, 8], [-10, 0, -16, -8, -8, 0], [0, 8, 8, -4, -4, 10], [-6, -6, 2, 4, -2, 0], [10, -10, 6, 8, -2, 0], [-4, 4, 10, 3, 7, 0], [0, -13, 3, 8, -5, 0], [4, -1, -5, -2, -3, 0], [0, 2, 2, -3, 1, -12], [-8, 8, 4, -2, 6, 0], [0, 8, 8, -1, -7, -4], [6, -16, 0, 8, -8, 0], [0, 4, 4, 0, -4, -6], [-8, -9, -1, 4, -5, 0], [0, -8, -8, 4, 4, -4], [0, 0, 0, 2, -2, -24], [-16, -9, 3, 6, -3, 0], [0, 0, 0, 0, 0, -8], [4, 4, 10, 3, 7, 0], [-6, -4, 8, 6, 2, 0], [0, 2, 2, -6, 4, -2], [0, 0, -8, -4, -4, 0], [30, -2, -4, -1, -3, 0], [0, 12, 12, -10, -2, 0], [20, -3, -3, 0, -3, 0], [0, -4, -4, 3, 1, -8], [0, -1, -1, -6, 7, -14], [0, -4, -4, 1, 3, 28], [-14, -6, -4, 1, -5, 0], [0, 12, 6, -3, 9, 0], [0, 0, 0, -8, 8, -10], [0, 2, 2, 6, -8, -12], [0, 2, 2, -6, 4, 18], [8, 12, 16, 2, 14, 0], [2, 10, 4, -3, 7, 0], [0, 6, 6, 2, -8, -2], [0, 8, 8, -16, 8, -6], [0, -2, -2, -4, 6, 6], [30, 0, 12, 6, 6, 0], [-20, 7, -5, -6, 1, 0], [18, 0, -8, -4, -4, 0], [0, -4, -4, -6, 10, -12], [0, 2, 2, -10, 8, -6], [0, 2, 2, 0, -2, 2], [8, 12, -4, -8, 4, 0], [8, 4, 12, 4, 8, 0], [0, 2, 2, 5, -7, 0], [0, -8, -8, 2, 6, -18], [0, 2, 2, -2, 0, 4], [-6, -12, 0, 6, -6, 0], [6, 2, -14, -8, -6, 0], [16, 4, 12, 4, 8, 0], [-16, 6, -2, -4, 2, 0], [0, -4, -4, 0, 4, -30], [0, 5, 5, 10, -15, -6], [10, 0, 12, 6, 6, 0], [0, 13, 13, -12, -1, -8], [0, -8, -16, -4, -12, 0], [0, -4, -4, 2, 2, -2], [-14, 2, 6, 2, 4, 0], [-18, 8, 4, -2, 6, 0], [0, 3, 3, -2, -1, 22], [0, -8, -8, -4, 12, 12], [0, -6, -6, 18, -12, 2], [10, -10, -16, -3, -13, 0], [22, 0, 20, 10, 10, 0], [-24, -3, -11, -4, -7, 0], [0, -2, -2, 18, -16, 10], [0, -10, -10, 16, -6, 12], [22, -2, -6, -2, -4, 0], [8, 2, -14, -8, -6, 0], [0, 11, 11, -2, -9, -6], [0, 12, 12, -8, -4, -8], [36, 4, -4, -4, 0, 0], [0, -5, -5, 4, 1, -28], [0, 1, 1, 10, -11, 10], [18, -10, 4, 7, -3, 0], [0, -8, -8, 15, -7, 20], [0, -10, -10, 16, -6, 2], [0, 16, 16, -9, -7, 4], [-24, 2, 10, 4, 6, 0], [-24, 4, -12, -8, -4, 0], [-10, 20, 16, -2, 18, 0], [0, 4, 4, -16, 12, 0], [12, -9, 15, 12, 3, 0], [0, 4, 4, 2, -6, -12], [0, -2, -2, 17, -15, -12], [0, 5, 5, 4, -9, -32], [-16, -16, 2, 9, -7, 0], [0, -10, -10, -2, 12, 10], [0, 3, 3, 4, -7, -24], [8, -16, -12, 2, -14, 0], [0, 0, 0, 4, -4, -22], [-2, 2, 6, 2, 4, 0], [0, 0, 0, 8, -8, -2], [-26, 16, 8, -4, 12, 0], [-20, 7, 7, 0, 7, 0], [0, 8, 0, -4, 4, 0], [0, -2, -2, 4, -2, 14], [-8, 18, 2, -8, 10, 0], [30, -10, 8, 9, -1, 0], [-18, 0, 4, 2, 2, 0], [0, 8, -8, -8, 0, 0], [0, -2, -2, 0, 2, -2], [6, 14, -10, -12, 2, 0], [0, -8, -8, 24, -16, 24], [-40, 8, 0, -4, 4, 0], [0, 0, 0, 4, -4, -30], [0, -4, -4, 4, 0, 22], [0, 0, 0, 0, 0, -4], [2, 8, 8, 0, 8, 0], [0, 6, 6, 6, -12, -6], [0, -6, -6, -12, 18, 28], [2, 8, 12, 2, 10, 0], [12, -14, 2, 8, -6, 0], [0, -10, -10, 17, -7, -8], [0, 4, 4, 0, -4, -12], [-8, 0, 6, 3, 3, 0], [16, 12, 4, -4, 8, 0], [0, 8, 8, 3, -11, -16], [0, -8, -8, 0, 8, 12], [6, 16, -12, -14, 2, 0], [0, -4, -4, 2, 2, 30], [0, 0, 0, -2, 2, 30], [0, 2, 2, 2, -4, 30], [16, 9, 9, 0, 9, 0], [-8, -10, -6, 2, -8, 0], [0, 1, 1, -14, 13, 34], [2, -22, -8, 7, -15, 0], [-14, 24, 4, -10, 14, 0], [0, 2, 2, 2, -4, 22], [24, -9, -5, 2, -7, 0], [0, 4, 4, -14, 10, -8], [22, -12, 4, 8, -4, 0], [-20, 12, 2, -5, 7, 0], [0, 0, 0, -4, 4, 56], [6, 2, 6, 2, 4, 0], [0, 8, 8, -2, -6, 10], [0, -18, -18, 20, -2, 4], [16, 9, 5, -2, 7, 0], [-22, 14, 14, 0, 14, 0], [0, 2, 2, -8, 6, -34], [-26, 14, 4, -5, 9, 0], [0, 0, 0, -8, 8, -4], [0, 11, 11, 10, -21, -26], [2, 20, 8, -6, 14, 0], [8, -3, -15, -6, -9, 0], [10, -20, -8, 6, -14, 0], [0, -1, -1, 4, -3, 12], [0, 0, 0, 7, -7, 16], [14, -4, 20, 12, 8, 0], [0, -10, -10, -5, 15, -4], [0, 30, 2, -14, 16, 0], [0, -10, -10, 18, -8, -6], [6, -8, 8, 8, 0, 0], [0, 6, 6, -14, 8, -34], [36, 4, -4, -4, 0, 0], [-40, 2, -14, -8, -6, 0], [-14, 22, 20, -1, 21, 0], [0, 12, 12, -1, -11, 16], [0, -13, -13, 8, 5, -4], [34, 6, 10, 2, 8, 0], [0, 5, 5, -2, -3, -42], [8, 12, 20, 4, 16, 0], [40, -22, -6, 8, -14, 0], [0, -2, -2, 0, 2, 20], [12, 0, 18, 9, 9, 0], [0, 6, 6, 0, -6, -18], [-36, -9, -17, -4, -13, 0], [0, 6, 6, 14, -20, -10], [16, 14, -6, -10, 4, 0], [0, 5, 5, -4, -1, 36], [-14, -8, 12, 10, 2, 0], [-2, -8, 0, 4, -4, 0], [0, 10, 10, -2, -8, -22], [0, -9, -9, 2, 7, -2], [38, 20, -4, -12, 8, 0], [0, 5, 5, -16, 11, 24], [14, -14, -4, 5, -9, 0], [0, 0, 0, -18, 18, -26], [34, -4, 4, 4, 0, 0], [-30, 22, 6, -8, 14, 0], [0, 10, 10, -8, -2, -20], [0, 4, 4, -10, 6, 10], [0, -12, -12, 16, -4, 30], [0, 10, 10, -24, 14, -22], [0, 16, 16, -27, 11, -32], [-12, -13, -5, 4, -9, 0], [0, -20, -20, 12, 8, 2], [0, 10, 10, -16, 6, 14], [-16, -4, -10, -3, -7, 0], [-8, -7, -15, -4, -11, 0], [-2, 30, 14, -8, 22, 0], [0, 12, 12, -8, -4, -12], [0, 14, 14, -20, 6, -22], [10, -18, -2, 8, -10, 0], [0, 10, 10, 0, -10, -28], [0, 10, 10, 2, -12, 22], [-8, 18, -10, -14, 4, 0], [0, -4, -4, -8, 12, -12], [0, 4, 4, 0, -4, -44], [0, 6, 6, -16, 10, -28], [-16, -24, 0, 12, -12, 0], [22, -20, 0, 10, -10, 0], [-8, 16, -10, -13, 3, 0], [0, 8, 8, -6, -2, 26], [0, 17, 5, -6, 11, 0], [0, -4, -4, 12, -8, -6], [32, 0, 0, 0, 0, 0], [0, -2, -2, 10, -8, 36], [16, -8, 12, 10, 2, 0], [0, 10, 10, -13, 3, -48], [0, 17, 17, -2, -15, -14], [-8, 13, 1, -6, 7, 0], [-14, -16, 8, 12, -4, 0], [6, -8, 4, 6, -2, 0], [0, 0, 0, 3, -3, -60], [0, 10, 10, -12, 2, -42], [-30, 8, 8, 0, 8, 0], [0, 0, 0, -8, 8, 2], [-50, 20, 16, -2, 18, 0], [32, -10, -18, -4, -14, 0], [0, 6, 6, -28, 22, 2], [-16, -4, 14, 9, 5, 0], [0, 0, 0, -10, 10, -28], [0, -14, -14, 7, 7, 20], [0, -9, -9, -18, 27, 34], [-32, 22, 22, 0, 22, 0]]
