
# q-expansion of newform 243.2.a.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 = 243
weight = 2
poly_data = [-1, -3, 0, 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], 1], [[0, 1, 0], 1], [[-2, 0, 1], 1]]

hecke_ring_character_values = None
aps_data = [[1, -1, 0], [0, 0, 0], [2, 1, 0], [-1, 1, -2], [1, 2, -3], [-1, 1, 1], [3, 0, 0], [-1, 0, 3], [2, -2, 3], [4, -1, 3], [-4, 1, -2], [-1, -3, 0], [-1, -5, 3], [-4, -2, 1], [-2, 5, 0], [6, 0, -3], [-7, 1, 0], [2, 1, -5], [2, -5, 1], [-3, -9, 3], [2, 6, -3], [2, -5, 4], [-2, -1, -3], [0, 9, -6], [5, -8, 4], [4, -4, -3], [5, 4, 4], [3, 0, 3], [-10, -3, 0], [5, 1, -6], [-1, 3, -3], [-4, 7, 6], [7, -1, 6], [5, 7, -5], [5, 4, -9], [8, 4, -5], [-7, -11, 10], [-1, 3, -12], [-1, 1, -3], [7, -7, 3], [-3, 0, -6], [-1, -6, -3], [-11, -1, -3], [-10, 1, 4], [12, 0, 6], [-1, -6, 6], [-7, 1, 4], [-1, 7, 4], [-11, 2, 3], [-10, 10, -8], [3, 9, -9], [-16, -2, -6], [2, 10, 1], [-15, -9, 6], [-1, -5, -6], [13, 5, -3], [9, 0, 9], [-7, -3, 6], [-1, 13, -17], [-2, -16, 12], [-1, 7, 7], [8, 16, -9], [11, -12, -3], [2, 4, 6], [5, -2, 1], [25, -4, 0], [-1, -20, 4], [8, -8, 10], [11, 7, -6], [20, 4, 4], [-17, 8, -9], [15, 9, -9], [-19, 4, -5], [11, -5, -11], [-1, 18, -15], [5, 10, 3], [4, 11, -9], [8, -9, -6], [-4, -2, 6], [-4, 1, 7], [14, -14, 0], [-4, -5, 16], [3, -9, -15], [-19, -15, 6], [2, -14, 16], [16, -7, 12], [-18, 9, -21], [-7, -2, 7], [13, -10, -6], [5, -5, -17], [21, -9, -3], [-14, -13, 15], [8, 9, 3], [8, -8, 6], [2, 7, -5], [24, 0, 3], [-4, -11, 15], [-27, 0, 3], [2, -12, 15], [17, -3, 18], [-16, -5, 7], [15, -9, -6], [14, -11, 15], [-2, 8, 15], [23, -2, 13], [2, 15, -12], [-17, -22, 12], [6, -9, -12], [-1, 1, 6], [-19, -2, -5], [-10, -20, 16], [11, -3, -3], [-7, -11, 12], [-13, 7, -23], [8, 27, -15], [-5, 14, -12], [-13, 10, -17], [3, 9, -6], [23, 19, -15], [1, 11, 6], [2, 10, -20], [-16, 7, -11], [1, -1, -18], [-9, 9, 3], [-13, 19, 13], [15, 0, 24], [-1, -2, 13], [-21, 0, 12], [14, -8, -2], [-13, -2, -14], [-19, -15, 15], [23, -14, 0], [-25, 10, 13], [-25, 15, 6], [-25, 10, 9], [-10, 16, -11], [0, -27, 33], [8, -20, 19], [-16, 4, 15], [-18, 9, -9], [17, 0, 3], [13, -16, 9], [17, -5, 4], [-9, 18, -3], [8, 24, -18], [4, -10, 21], [17, 7, -11], [-17, -4, 3], [29, 10, -14], [3, 9, 12], [-10, -5, -5], [3, 0, 18], [-1, 24, 0], [-1, -11, 9], [-4, -8, 4], [-2, 5, 9], [11, 6, 6], [-23, 11, 0], [8, -15, 6], [-22, 16, -21], [10, -19, 9], [-3, -18, 0], [23, 13, 4], [24, 18, -3], [-16, 1, 18], [-14, 2, 0], [2, 0, 0], [5, -5, 28]]
