
# q-expansion of newform 360.2.k.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 = 360
weight = 2
poly_data = [8, -8, 6, -6, 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], [[-8, 2, 0, 3, 0, 1], 4], [[-8, 2, -4, 3, 0, 1], 4], [[4, 0, 2, -1, 0, -1], 2], [[5, -2, 3, -2, 1, -1], 1]]

hecke_ring_character_values = [[271, [1, 0, 0, 0, 0, 0]], [181, [-1, 0, 0, 0, 0, 0]], [281, [1, 0, 0, 0, 0, 0]], [217, [1, 0, 0, 0, 0, 0]]]
aps_data = [[0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [1, -1, -1, 1, 1, -1], [0, -2, 0, -2, 0, 2], [1, -3, 1, -1, -1, 1], [-1, -1, 1, 1, 3, -1], [-1, 1, -1, -1, 1, 1], [1, 1, 1, -1, -1, 1], [0, 0, 0, -2, 0, 0], [-3, 1, -1, -1, -3, 1], [-1, -1, -1, 1, 1, 3], [4, -2, -2, 2, 2, -2], [-2, 2, -2, 2, 2, 2], [-3, 1, -3, -1, -5, 1], [0, 0, 0, 2, 0, 0], [0, 2, 0, -2, 0, -2], [0, 4, 0, 4, 0, -4], [0, 0, 0, 4, 0, 0], [0, 0, -4, 0, -4, 0], [-6, 0, 0, 0, 0, 0], [5, 1, -1, -1, -3, 1], [-2, 2, -2, 6, 2, 2], [4, 2, 2, -2, -2, 2], [4, 2, -2, -2, -6, 2], [0, 0, 0, 2, 0, 0], [-3, 3, -1, -3, -7, 3], [-4, 8, -4, -4, 4, 0], [2, -2, 2, -2, -2, -2], [-9, 3, 5, -3, -1, 3], [-5, -3, 1, 3, 7, -3], [2, 0, 2, 0, -2, -4], [11, -1, 1, 1, 3, -1], [-3, 7, -3, -7, 3, -1], [0, 0, 0, -2, 0, 0], [3, 3, 1, -3, -5, 3], [3, -5, 3, -11, -3, -1], [2, -2, 2, 6, -2, -2], [-7, 1, -3, -1, -5, 1], [-2, -2, -2, 4, 2, 6], [2, -4, 2, 4, -2, 0], [-2, 2, -2, -14, 2, 2], [8, 0, 0, 0, 0, 0], [8, 2, 6, -2, 2, 2], [2, -6, 2, -16, -2, 2], [-11, -3, -1, 3, 5, -3], [-1, -3, -1, 11, 1, 5], [-11, -1, 3, 1, 5, -1], [2, -6, 2, -2, -2, 2], [4, -8, 4, 4, -4, 0], [15, -1, 1, 1, 3, -1], [-10, -2, -2, 2, 2, -2], [4, -2, -6, 2, -2, -2], [6, -12, 6, -8, -6, 0], [-9, 3, -3, -3, -9, 3], [15, -1, 3, 1, 5, -1], [-4, 4, -4, 10, 4, 4], [23, -1, 5, 1, 7, -1], [5, -11, 5, -1, -5, 1], [4, -2, -6, 2, -2, -2], [0, 0, 0, -20, 0, 0], [-4, 12, -4, 6, 4, -4], [-2, 10, -2, 18, 2, -6], [-6, 2, 10, -2, 6, 2], [8, -2, 2, 2, 6, -2], [-6, 2, -6, 12, 6, 10], [1, -9, 1, -15, -1, 7], [6, 0, 8, 0, 8, 0], [2, 2, 2, 6, -2, -6], [-4, 0, -4, -8, 4, 8], [-3, 5, 3, -5, -7, 5], [0, 0, -4, 0, -4, 0], [-3, -1, -9, 1, -7, -1], [1, 1, 1, -9, -1, -3], [5, -9, 5, -7, -5, -1], [-3, 1, 1, -1, -1, 1], [4, -12, 4, -14, -4, 4], [-3, 1, -3, -1, 3, 5], [6, 0, 4, 0, 4, 0], [-10, 4, -4, -4, -12, 4], [-2, 8, -2, 4, 2, -4], [-4, 8, -4, -20, 4, 0], [0, 0, -4, 0, -4, 0], [-6, 4, -4, -4, -12, 4], [1, 1, 3, -1, 1, 1], [-6, 10, -6, -2, 6, 2], [22, -4, 4, 4, 12, -4], [16, -6, -2, 6, 10, -6], [-4, 4, -4, -10, 4, 4], [19, 1, 1, -1, -1, 1], [-2, 10, -2, 22, 2, -6], [12, 4, 0, -4, -8, 4], [-9, 1, 5, -1, 3, 1], [0, -2, 0, -14, 0, 2], [3, -11, 3, 3, -3, 5], [11, -1, -5, 1, -3, -1], [12, -20, 12, 2, -12, -4], [-10, 0, 0, 0, 0, 0], [4, 4, 4, 0, -4, -12], [-6, 14, -6, 14, 6, -2], [-2, -6, -2, -6, 2, 10], [2, 2, 2, 16, -2, -6], [6, -10, 6, -14, -6, -2], [-14, 0, 4, 0, 4, 0], [1, -1, 1, -15, -1, -1], [4, 2, 6, -2, 2, 2], [4, -8, 4, -12, -4, 0], [3, -1, 1, 1, 3, -1], [-20, 4, -8, -4, -16, 4], [-2, 4, 12, -4, 4, 4], [-19, 3, -5, -3, -11, 3], [-1, 3, -1, -3, 1, -1], [-21, 3, -3, -3, -9, 3], [-3, 7, -3, -15, 3, -1], [-17, 3, 5, -3, -1, 3], [12, -2, 10, 2, 14, -2], [-6, 14, -6, 6, 6, -2], [1, 5, 17, -5, 7, 5], [6, -2, 6, 4, -6, -10], [-4, -2, -4, 10, 4, 10], [4, 0, 4, -16, -4, -8], [-18, 0, 0, 0, 0, 0], [2, 10, 2, -4, -2, -14], [-2, 2, -2, -26, 2, 2], [5, -5, 5, 5, -5, -5], [0, -8, 0, -18, 0, 8], [0, -4, 0, 16, 0, 4], [-14, 2, 6, -2, 2, 2], [3, -3, -15, 3, -9, -3], [7, -1, 7, -11, -7, -13], [-1, 17, -1, 23, 1, -15], [17, 5, -3, -5, -13, 5], [9, 5, -5, -5, -15, 5], [-3, 5, -3, 23, 3, 1], [2, -4, 12, 4, 20, -4], [8, -6, 6, 6, 18, -6], [-2, 6, -2, 4, 2, -2], [-4, 20, -4, 16, 4, -12], [0, 0, 0, 10, 0, 0], [-8, -10, 2, 10, 22, -10], [-9, 21, -9, 19, 9, -3], [8, -16, 8, 18, -8, 0], [-7, 3, -9, -3, -15, 3], [0, 0, 0, -20, 0, 0], [2, -2, 2, -34, -2, -2], [8, 8, 12, -8, -4, 8], [-1, -13, -1, 5, 1, 15], [-7, 1, 7, -1, 5, 1], [-7, 11, -7, 13, 7, 3], [-9, -1, -13, 1, -11, -1], [-5, 3, -5, 5, 5, 7], [-4, 6, -10, -6, -22, 6], [10, -26, 10, -2, -10, 6], [9, 1, -11, -1, -13, 1], [8, -16, 8, -20, -8, 0], [2, -6, 10, 6, 22, -6], [27, -1, -7, 1, -5, -1], [10, 0, -8, 0, -8, 0], [2, 12, 12, -12, -12, 12], [4, -4, 4, -18, -4, -4], [-4, 12, -4, 0, 4, -4], [-29, 7, 1, -7, -13, 7], [-21, 9, 5, -9, -13, 9], [-10, 20, -10, -8, 10, 0], [11, -5, -3, 5, 7, -5], [9, -11, -11, 11, 11, -11], [-7, -7, -5, 7, 9, -7], [-7, 21, -7, 11, 7, -7]]
