
# q-expansion of newform 2736.2.bm.m, downloaded from the LMFDB on 18 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 = 2736
weight = 2
poly_data = [9, -27, 84, 3, 10, -1, 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], [[270, -27, 84, -100, 10, -1], 813], [[-783, 2436, 190, 290, -29, 30], 813], [[-5310, -14916, -1923, -1466, -16, -161], 2439], [[-10107, 15726, -597, 2027, -284, 191], 2439]]

hecke_ring_character_values = [[1711, [-1, 0, 0, 0, 0, 0]], [2053, [1, 0, 0, 0, 0, 0]], [1217, [1, 0, 0, 0, 0, 0]], [1009, [0, 0, 0, -1, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [-1, 1, 0, -1, 0, 0], [0, 0, 0, 0, 1, -1], [-1, 0, 0, -2, -1, 1], [4, 0, 0, 2, 0, 0], [1, -1, 0, 1, 2, -1], [3, -1, 0, 0, -1, 1], [0, 0, 0, 0, -1, 0], [2, 1, 1, 1, 0, 0], [0, 0, 2, 0, 1, 1], [-1, -2, 1, -2, 0, 0], [0, 1, -2, 0, 0, -3], [2, 0, 0, -2, 0, 0], [4, 0, 0, 2, 1, 0], [8, 0, 0, 4, 0, 0], [-9, 0, 0, -9, 2, -1], [0, 1, -1, 3, 0, 0], [0, 2, -2, 3, -1, 2], [-6, 0, 0, -6, 0, 0], [-2, 1, 0, -2, 2, -1], [-6, 2, 0, -6, 0, 0], [0, 2, -1, 0, 2, -2], [6, -1, -1, 3, 3, 0], [3, 2, -4, -3, 0, 0], [0, 5, -5, -5, 2, -4], [0, 0, -2, 0, 2, 2], [0, 0, 0, 0, 2, 2], [2, 2, -4, -2, 0, 0], [1, 0, 0, 2, 0, 0], [0, 2, -2, 6, -2, 4], [2, -1, 2, -2, 0, -2], [0, 2, -2, -11, 0, 0], [-6, 0, 0, -3, 1, 0], [-1, 1, 0, -1, 4, -2], [0, 0, 4, 0, -3, -3], [0, 2, 0, 0, 4, -2], [-9, 4, -2, -18, 1, -1], [0, -6, 6, 12, -2, 4], [-10, 0, 0, 10, 0, 0], [12, 0, -3, 0, -2, -2], [-6, -1, -1, -3, -6, 0], [-4, 0, 0, -8, 4, -4], [5, -1, 2, -5, 0, -3], [5, 0, 1, 0, -2, -2], [8, -4, -4, 4, 0, 0], [9, -5, 0, 9, -2, 1], [-6, 2, 0, -6, -6, 3], [6, 0, 3, 0, 0, 0], [0, 0, 2, 0, 0, 0], [8, -5, 0, 8, -2, 1], [-4, 4, -2, -8, -2, 2], [0, 3, 3, 0, 3, 0], [4, 1, 1, 2, -4, 0], [8, -1, -1, 4, 3, 0], [-2, -4, 8, 2, 0, 7], [4, -2, 4, -4, 0, -6], [6, 2, -4, -6, 0, -3], [3, 0, 5, 0, 4, 4], [16, 1, 1, 8, -3, 0], [-8, 1, -2, 8, 0, 0], [-1, -2, 1, -2, -6, 6], [9, 2, 0, 9, 6, -3], [-8, -4, 2, -16, -1, 1], [0, 1, -1, -6, 1, -2], [-4, 2, 2, -2, 6, 0], [-6, 0, -5, 0, -2, -2], [1, -2, 4, -1, 0, 6], [-5, 4, -8, 5, 0, 5], [-4, 0, -6, 0, 0, 0], [-7, 0, -2, 0, -2, -2], [-10, 2, -4, 10, 0, 1], [-12, -4, -4, -6, 3, 0], [2, 0, 0, 4, 6, -6], [15, 0, 1, 0, 1, 1], [12, 0, 0, 12, 2, -1], [0, 2, -2, 4, -2, 4], [11, -7, 0, 11, -8, 4], [-1, 2, -4, 1, 0, 6], [4, -3, -3, 2, 3, 0], [6, 8, -4, 12, 2, -2], [-9, -5, 10, 9, 0, 6], [0, 0, 0, -6, -4, 8], [10, -7, -7, 5, -3, 0], [0, -4, 0, 0, -2, 1], [20, -5, -5, 10, -4, 0], [2, -6, 3, 4, -3, 3], [16, 0, 3, 0, -3, -3], [-10, -8, 0, -10, 4, -2], [-8, 12, -6, -16, 0, 0], [-1, 4, -2, -2, -1, 1], [0, 0, 0, 0, -2, 0], [0, 0, 8, 0, -1, -1], [-12, 2, -4, 12, 0, -8], [0, -5, 10, 0, 0, -4], [-12, -2, -2, -6, -13, 0], [-4, 0, 0, -2, -6, 0], [-1, -4, 2, -2, 0, 0], [0, 2, -2, 6, 6, -12], [0, 2, -2, 20, 2, -4], [0, -2, 2, 18, 4, -8], [0, 8, -8, -8, 0, 0], [24, 0, 3, 0, 2, 2], [-1, -10, 5, -2, -3, 3], [-6, -2, 1, -12, -12, 12], [-21, 0, -2, 0, 0, 0], [-4, 2, -4, 4, 0, 2], [0, -5, 5, 26, 1, -2], [0, -6, 6, 0, 3, -6], [-4, -6, 3, -8, -9, 9], [-6, 0, 2, 0, 3, 3], [0, -4, 0, 0, -4, 2], [0, -4, 4, -11, 0, 0], [6, 8, -4, 12, -10, 10], [-12, -4, -4, -6, -7, 0], [4, -3, 6, -4, 0, -9], [-10, 1, -2, 10, 0, -6], [8, 4, -2, 16, -4, 4], [2, 0, 10, 0, 0, 0], [0, 12, -12, -12, 4, -8], [6, -1, -1, 3, 12, 0], [9, -6, 3, 18, 3, -3], [-8, -8, 4, -16, -6, 6], [24, 0, 0, 0, -4, -4], [-10, 8, -4, -20, 0, 0], [19, -7, 0, 19, -8, 4], [0, -1, 1, -1, -4, 8], [-8, 6, -12, 8, 0, 4], [-16, -2, 4, 16, 0, 6], [5, 0, 11, 0, -4, -4], [9, 2, -4, -9, 0, -1], [6, -12, 0, 6, -2, 1], [0, -2, 2, -24, 0, 0], [30, -2, 0, 30, -8, 4], [-13, 0, 4, 0, 2, 2], [0, 1, -1, -9, -1, 2], [2, -9, -9, 1, 0, 0], [0, 0, -7, 0, -4, -4], [-15, -2, 1, -30, -12, 12], [-28, 0, 7, 0, -1, -1], [0, 10, -10, 6, 6, -12], [0, -2, 2, 14, -6, 12], [-12, -2, -2, -6, -10, 0], [0, -9, 9, -6, -2, 4], [2, 8, -4, 4, -12, 12], [6, 0, 0, 6, -10, 5], [20, 2, 0, 20, 4, -2], [9, -4, 8, -9, 0, 0], [-12, -5, -5, -6, 2, 0], [-24, 0, 6, 0, 3, 3], [-3, -5, 10, 3, 0, -6], [-26, 0, -1, 0, 3, 3], [12, -5, -5, 6, -2, 0], [0, -6, 6, 18, -2, 4], [-15, 4, 0, -15, 2, -1], [30, 0, 0, 0, 3, 3], [0, -4, 2, 0, 8, -8], [23, -2, 0, 23, 8, -4], [0, -9, 9, -16, 3, -6], [-12, 0, 0, -6, 6, 0], [-12, -2, 4, 12, 0, -2], [0, 1, -2, 0, 0, 3], [-6, -6, 12, 6, 0, 4], [15, -6, 0, 15, -6, 3], [-7, 12, -6, -14, 6, -6], [18, 0, 0, 18, 12, -6], [36, 4, 0, 36, -4, 2], [0, 13, -13, 3, 6, -12], [-13, 1, -2, 13, 0, 9], [0, 10, -10, -16, -4, 8], [15, 0, -18, 0, -3, -3], [13, 9, -18, -13, 0, -6], [12, -4, 8, -12, 0, -1], [-12, 0, -1, 0, -1, -1], [0, -4, 4, -12, 5, -10], [0, 8, -8, 25, 4, -8], [12, -1, -1, 6, 4, 0], [4, 14, 0, 4, -4, 2], [0, 0, 8, 0, -2, -2], [37, -9, 0, 37, 0, 0], [-8, 0, 0, -4, -12, 0], [-6, 0, -6, 0, 4, 4], [-28, 6, 6, -14, 12, 0], [20, 5, -10, -20, 0, -3], [4, 4, -2, 8, 4, -4], [19, 0, -13, 0, 0, 0], [-4, -2, -2, -2, -6, 0], [-42, 10, 0, -42, 4, -2], [-8, 6, -3, -16, -9, 9], [-18, 16, -8, -36, 8, -8], [-10, 4, 4, -5, -12, 0], [-26, -5, -5, -13, 11, 0], [-12, 0, -5, 0, -2, -2], [30, 0, 0, -30, 0, 0], [6, 8, 8, 3, -7, 0], [18, 5, 5, 9, -3, 0], [0, 0, 0, -13, 6, -12], [5, -3, 0, 5, 0, 0], [10, 0, -7, 0, 5, 5], [-16, -8, 4, -32, 5, -5], [8, -2, -2, 4, -12, 0], [0, -4, 4, 36, 4, -8], [8, -2, 4, -8, 0, 12], [-16, 1, -2, 16, 0, -9], [3, 8, -16, -3, 0, -11], [0, 1, -1, -13, 2, -4], [52, 0, 0, 26, 3, 0], [0, 0, 0, -15, -3, 6], [31, -10, 0, 31, 0, 0], [-21, 0, 10, 0, -1, -1], [18, 5, 0, 18, -2, 1], [0, 7, -7, 35, 4, -8], [20, -16, 8, 40, -6, 6], [0, -18, 18, 9, -5, 10], [-18, 0, 6, 0, 4, 4], [0, 0, 0, 0, 0, 0], [-12, -6, 12, 12, 0, 0], [3, 6, -3, 6, -9, 9], [36, 0, 0, 0, -3, -3], [-19, -5, 0, -19, 4, -2], [34, 1, 1, 17, -6, 0], [0, 0, -8, 0, 6, 6], [-14, -3, 6, 14, 0, 3], [12, 10, -20, -12, 0, -10], [-51, 0, 0, -51, 2, -1], [0, -8, 8, -18, 8, -16], [-11, 8, -4, -22, 12, -12], [6, -6, 0, 6, 16, -8], [-12, -8, 0, -12, -6, 3], [-8, -12, 6, -16, -8, 8], [0, -1, 1, -21, -8, 16], [0, 10, -10, 3, 3, -6], [30, 0, 2, 0, 9, 9], [-15, 4, -2, -30, 0, 0], [9, 8, -4, 18, 5, -5], [2, -4, 8, -2, 0, -10], [4, 0, -3, 0, 3, 3], [-20, 0, -4, 0, -2, -2], [-18, 1, -2, 18, 0, -2], [0, 12, -12, 0, -5, 10], [33, 0, 0, 33, 6, -3], [-14, -18, 9, -28, 6, -6], [0, -2, -2, 0, 0, 0], [6, -3, -3, 3, -12, 0], [21, -1, 2, -21, 0, -3], [8, 8, -4, 16, -1, 1], [-12, 10, 10, -6, 5, 0], [0, -6, 6, 21, 3, -6], [-36, -7, 0, -36, 8, -4], [-28, 4, 4, -14, -10, 0], [3, 0, 17, 0, 2, 2], [-17, 5, 0, -17, -18, 9], [-6, -8, 4, -12, 5, -5], [-26, 0, 0, -13, 12, 0], [25, 5, 0, 25, 0, 0], [-48, 6, 6, -24, 4, 0], [0, 2, -2, 26, 2, -4], [45, 0, -2, 0, -1, -1], [8, -2, 4, -8, 0, 12], [0, 1, -1, 27, 1, -2], [0, -2, 2, -6, 4, -8], [-6, 9, 0, -6, -8, 4], [13, 13, 0, 13, -16, 8], [-19, 5, -10, 19, 0, 0], [0, -5, 5, -22, -1, 2], [-15, 0, 20, 0, 1, 1], [12, 4, -2, 24, 12, -12], [-16, 0, -17, 0, -3, -3], [0, -2, 2, 18, -4, 8], [0, -4, 4, 34, 6, -12], [-3, 18, -9, -6, 6, -6], [0, 0, -4, 0, -6, -6], [9, 10, 0, 9, 0, 0], [12, -8, 4, 24, 5, -5], [-24, 4, 4, -12, -6, 0], [-12, 6, -12, 12, 0, -7], [17, -24, 12, 34, -11, 11], [3, -7, 14, -3, 0, 6], [-10, 4, 4, -5, -6, 0], [-20, 4, 4, -10, -2, 0], [42, 0, 6, 0, 5, 5], [-6, 12, -6, -12, 7, -7], [-32, -4, -4, -16, -22, 0], [-18, 0, 0, -36, -12, 12], [-17, -7, 14, 17, 0, 15], [52, 4, 4, 26, 8, 0], [-27, 0, 2, 0, 0, 0], [-8, 6, 6, -4, 6, 0], [-12, 4, -8, 12, 0, -10], [1, -30, 15, 2, 3, -3], [-34, 0, 16, 0, -2, -2], [-20, -4, 2, -40, 2, -2], [4, -7, -7, 2, -21, 0], [-30, 0, 6, 0, -4, -4], [3, -5, 10, -3, 0, 6], [-20, 0, 14, 0, -4, -4], [0, -8, 8, 12, -10, 20], [5, 1, 0, 5, -8, 4], [51, 9, 0, 51, -6, 3], [-12, -6, 3, -24, 2, -2], [-12, 23, 0, -12, -6, 3], [-29, 1, -2, 29, 0, 6], [8, -6, -6, 4, 3, 0]]
