
# q-expansion of newform 1512.2.s.m, 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 = 1512
weight = 2
poly_data = [2401, -1372, 245, 84, -76, 12, 5, -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], [[343, -1911, 3290, -1282, -194, -26, -3, -1], 12348], [[-42532, 2205, 574, -1520, 506, 128, 18, 13], 37044], [[3773, -8967, -518, 334, -346, 74, 15, 19], 12348], [[-1960, 1169, -468, -146, 48, -10, -4, 3], 1764], [[27783, -5929, -2506, 762, -866, 114, 53, -43], 24696], [[154693, -34398, 224, 5882, -2384, -746, 405, -124], 37044]]

hecke_ring_character_values = [[1135, [1, 0, 0, 0, 0, 0, 0, 0]], [757, [1, 0, 0, 0, 0, 0, 0, 0]], [785, [1, 0, 0, 0, 0, 0, 0, 0]], [1081, [0, 0, 0, 1, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 1, -1, 0], [0, -1, 0, -1, -1, 0, 0, 0], [-1, 1, 0, -1, 0, 0, 0, -1], [-1, 1, -1, 1, 0, -2, 0, 1], [1, 0, 0, 1, 0, 0, 3, 0], [-1, 1, 1, 1, 0, 2, -2, 0], [1, -1, -1, 0, 1, 1, -1, -1], [3, -2, 0, -2, -2, -1, 0, -2], [-2, 1, 0, -2, 0, 0, 2, -1], [2, -2, -2, 1, -1, -2, 2, 1], [-1, 0, -2, 0, -2, -3, 0, 0], [-1, 0, 2, 0, 2, -2, 0, 0], [0, 0, 0, 5, 2, -1, 1, -2], [-2, 1, -2, 0, 2, 0, -3, 1], [5, 1, 0, 5, 0, 0, -2, -1], [1, -1, -1, 0, 1, 4, -4, -1], [0, 0, 1, -1, -1, 0, -4, -1], [6, 0, -1, 0, -1, 2, 0, 0], [-3, 2, -1, -2, 1, 0, 0, -1], [-2, 2, 2, -3, 1, 0, 0, -1], [-4, -2, 0, -2, -2, 2, 0, -2], [2, -2, -2, 5, 0, 1, -1, 0], [1, 0, -2, 0, -2, 2, 0, 0], [-3, 3, 0, -3, 0, 0, 0, -3], [0, 0, 0, 5, 1, -2, 2, -1], [2, -2, -2, 4, -4, -4, 4, 4], [4, 1, -2, 6, 2, 0, -4, 1], [-2, 2, -1, 2, 1, 4, 0, 2], [-7, 0, -1, 0, -1, 0, 0, 0], [1, -1, -1, 2, -1, -3, 3, 1], [-8, 1, 2, -10, -2, 0, 3, -3], [1, -1, 2, -1, 1, 4, 0, -1], [0, 0, 0, 5, 0, 1, -1, 0], [-2, 1, 0, -2, 0, 0, -2, -1], [-4, -2, 0, -4, 0, 0, 6, 2], [-2, 2, 2, -11, 0, 0, 0, 0], [-8, 2, 2, 2, 4, 2, 0, 2], [-1, 1, 1, 18, 1, -1, 1, -1], [-13, 0, -2, -11, 2, 0, 1, 2], [2, 3, -2, 3, 1, -2, 0, 3], [-2, 2, 2, -3, 2, -5, 5, -2], [-4, -2, -1, -3, 1, 0, 0, 3], [8, 4, -2, 4, 2, -4, 0, 4], [12, 1, 2, 10, -2, 0, -8, -3], [-5, -2, 0, -2, -2, -6, 0, -2], [-2, 2, 0, 2, 2, -2, 0, 2], [-2, 1, 2, -4, -2, 0, 1, -3], [1, -1, -1, -10, -3, -2, 2, 3], [1, -1, -1, 0, -5, -1, 1, 5], [1, 0, -2, 0, -2, -9, 0, 0], [-3, 2, -6, 3, 6, 0, -2, 4], [-3, 2, -1, 2, 1, 1, 0, 2], [-2, 2, 2, -6, 0, -2, 2, 0], [-9, -3, 2, -11, -2, 0, -4, 1], [-2, -4, 0, -2, 0, 0, 6, 4], [3, -3, -3, 2, 3, 4, -4, -3], [2, 3, 0, 2, 0, 0, 0, -3], [14, -2, 3, -2, 1, 8, 0, -2], [1, 0, 0, 1, 0, 0, 2, 0], [-8, 2, -1, 2, 1, 2, 0, 2], [-3, 4, 0, 4, 4, -8, 0, 4], [15, -3, 0, 15, 0, 0, 4, 3], [1, -1, -1, -3, 2, -6, 6, -2], [-1, 1, 1, 1, -1, 8, -8, 1], [-1, 1, 1, -9, -4, -4, 4, 4], [12, 5, 0, 5, 5, -2, 0, 5], [-24, 0, 0, -24, 0, 0, -4, 0], [-1, -6, 1, -6, -5, -2, 0, -6], [0, -6, 4, -4, -4, 0, -2, 2], [-2, 2, 2, -4, 4, 4, -4, -4], [13, 0, 1, 12, -1, 0, 4, -1], [-3, 3, 3, -1, 0, 14, -14, 0], [-15, 3, 2, 3, 5, 8, 0, 3], [2, -2, -2, -8, -4, 4, -4, 4], [-16, 5, 0, -16, 0, 0, -5, -5], [-3, 3, 3, -12, 1, -4, 4, -1], [3, -3, -3, 15, -3, 0, 0, 3], [-6, 0, 3, -9, -3, 0, -8, -3], [-9, -2, 0, -2, -2, -5, 0, -2], [4, -3, 4, -3, 1, 4, 0, -3], [-13, -5, 2, -15, -2, 0, 2, 3], [-1, -4, 2, -4, -2, 2, 0, -4], [4, -4, -4, 2, -2, -10, 10, 2], [3, -3, -3, -12, 1, -7, 7, -1], [7, -2, 0, -2, -2, 7, 0, -2], [4, -4, -4, 17, -4, 2, -2, 4], [4, 2, 3, 2, 5, -6, 0, 2], [-12, -1, -4, -1, -5, -4, 0, -1], [3, -3, -3, 9, 3, -4, 4, -3], [-3, -2, 4, -7, -4, 0, 1, -2], [-1, 6, -3, 2, 3, 0, -6, -3], [-8, 2, 2, 2, 4, 10, 0, 2], [0, 0, 0, 11, -4, 0, 0, 4], [-3, 2, -3, 2, -1, -15, 0, 2], [0, 0, 0, -6, 6, 8, -8, -6], [6, 6, -2, 8, 2, 0, -8, -4], [-3, 3, 3, 0, -4, -6, 6, 4], [-3, 3, 3, 11, 2, 4, -4, -2], [-17, 6, -2, 6, 4, 4, 0, 6], [-7, 2, 0, -7, 0, 0, -11, -2], [16, 3, -2, 18, 2, 0, 1, -1], [-3, 3, 3, 5, 1, 2, -2, -1], [3, 4, -3, 6, 3, 0, 0, -1], [-18, 0, 1, -19, -1, 0, -12, -1], [-3, 2, -2, 2, 0, 9, 0, 2], [-3, 3, 3, 11, 1, -4, 4, -1], [-24, 1, 0, -24, 0, 0, 7, -1], [-21, 4, 4, 4, 8, 6, 0, 4], [3, -3, -3, 15, 4, -2, 2, -4], [16, -5, 0, 16, 0, 0, 0, 5], [-18, 0, -3, 0, -3, -6, 0, 0], [0, 0, -1, 1, 1, 0, -12, 1], [-3, -4, 1, -4, -3, 4, 0, -4], [-9, -2, -4, -5, 4, 0, 11, 6], [19, 0, 2, 0, 2, 0, 0, 0], [-2, 2, 0, -2, 0, 0, 4, -2], [1, -1, -1, -2, -3, -13, 13, 3], [25, -6, 2, -6, -4, 1, 0, -6], [-3, -8, 1, -4, -1, 0, 2, 7], [30, 1, -2, 1, -1, 8, 0, 1], [6, -6, -6, 26, -2, 12, -12, 2], [-3, 3, 2, -5, -2, 0, 12, -5], [4, -4, -4, 7, -2, -2, 2, 2], [15, -4, 3, -4, -1, 1, 0, -4], [-1, 1, 1, 4, 3, -8, 8, -3], [-3, 3, 3, 23, 7, -6, 6, -7], [13, -2, -1, -2, -3, 0, 0, -2], [-1, 1, 1, -20, 3, 4, -4, -3], [-9, 2, -2, -7, 2, 0, 0, 0], [-9, -2, -2, -2, -4, 3, 0, -2], [3, -3, -3, 1, 4, 12, -12, -4], [-1, 8, -3, 8, 5, -12, 0, 8], [0, 0, 0, -20, -4, -8, 8, 4], [-20, 3, -2, 3, 1, -8, 0, 3], [-2, 3, -4, 2, 4, 0, -5, 1], [11, -6, 6, 5, -6, 0, 8, 0], [-10, -2, -1, -2, -3, -4, 0, -2], [-8, 2, 6, -14, -6, 0, -2, -8], [-16, 4, -4, 4, 0, -2, 0, 4], [10, -10, -10, 8, -8, -12, 12, 8], [22, -3, 0, 22, 0, 0, -8, 3], [10, -4, -2, -4, -6, -18, 0, -4], [-39, -2, -1, -38, 1, 0, 6, 3], [8, -2, 4, -2, 2, -6, 0, -2], [-4, -1, 0, -1, -1, -8, 0, -1], [9, -3, -6, 15, 6, 0, -10, 9], [8, -8, -8, 9, -8, -8, 8, 8], [1, -1, -1, 6, -5, -7, 7, 5], [-1, 1, 1, -4, -3, -14, 14, 3], [-1, -6, 3, -6, -3, 11, 0, -6], [2, 5, -6, 5, -1, -14, 0, 5], [-4, 4, 4, -22, 10, 16, -16, -10], [18, -2, -1, 19, 1, 0, 18, 3], [33, 2, -2, 2, 0, 5, 0, 2], [-1, 1, 1, -16, -1, -22, 22, 1], [-2, 2, 2, 19, 4, -13, 13, -4], [-11, -7, 2, -7, -5, 0, 0, -7], [9, 9, -6, 15, 6, 0, -6, -3], [2, -2, -2, -1, -6, 5, -5, 6], [11, -6, 1, -6, -5, 7, 0, -6], [-1, -5, -3, -5, -8, 2, 0, -5], [-3, 3, 3, 4, -5, -1, 1, 5], [-18, -6, 6, -24, -6, 0, -4, 0], [-17, -1, -4, -13, 4, 0, 22, 5], [-1, 7, -9, 8, 9, 0, -14, 2], [-12, -1, -4, -8, 4, 0, -8, 5], [31, -1, -2, -1, -3, -4, 0, -1], [7, -7, -7, -10, -9, -3, 3, 9], [1, 0, 2, -1, -2, 0, 7, -2], [0, -6, 6, -6, 0, -10, 0, -6], [4, -4, -4, 6, 0, 14, -14, 0], [-15, 2, -7, -8, 7, 0, -8, 5], [-17, -1, 3, -20, -3, 0, -10, -2], [1, -6, 1, -6, -5, 5, 0, -6], [-1, -6, 6, -6, 0, -6, 0, -6], [-5, 5, -4, -1, 4, 0, 16, -1], [-9, -2, 13, -2, 11, 2, 0, -2], [-13, 13, 13, -8, 9, 4, -4, -9], [-1, 1, 1, -17, 6, -6, 6, -6], [11, 8, -3, 8, 5, -7, 0, 8], [15, 4, -9, 4, -5, 0, 0, 4], [8, -8, -8, -7, 0, -11, 11, 0], [12, -6, 0, 12, 0, 0, 14, 6], [-26, -4, 8, -34, -8, 0, 2, -4], [26, -1, 0, 26, 0, 0, 10, 1], [3, -2, 3, 0, -3, 0, 8, -1], [-3, 3, 3, -4, 10, 12, -12, -10], [7, -7, 8, -1, -8, 0, -4, -1], [-5, 5, 5, 5, 12, 2, -2, -12], [-6, 18, -9, 18, 9, -4, 0, 18], [0, 0, 0, -9, 2, 14, -14, -2], [-1, 1, 1, 9, -9, -8, 8, 9], [5, -4, 4, 1, -4, 0, -13, 0], [22, -10, 0, 22, 0, 0, 0, 10], [-23, 4, -11, -12, 11, 0, 0, 7], [1, -1, -1, 2, 9, -2, 2, -9], [5, 12, -9, 12, 3, 3, 0, 12], [1, -1, -1, -6, 5, 15, -15, -5], [-12, 5, -8, -4, 8, 0, 7, 3], [-1, -5, -3, -5, -8, 2, 0, -5], [5, -5, -5, -18, -7, -10, 10, 7], [13, -14, -2, 15, 2, 0, 10, 16], [12, -6, 3, -6, -3, -2, 0, -6], [15, -3, 6, 9, -6, 0, 4, -3], [-1, 1, 1, -12, 5, 0, 0, -5], [4, -4, -4, -10, -8, 2, -2, 8], [1, 0, 13, 0, 13, 15, 0, 0], [3, 2, -4, 7, 4, 0, -8, 2], [7, -7, -7, 32, -2, 12, -12, 2], [2, 10, -2, 10, 8, -8, 0, 10], [-21, -2, 7, -2, 5, 10, 0, -2], [11, -11, -11, 23, -5, 6, -6, 5], [-10, 3, 2, -12, -2, 0, 15, -5], [11, -11, -11, 8, -14, -16, 16, 14], [12, 0, 10, 2, -10, 0, -18, -10], [-1, 0, -6, 5, 6, 0, -7, 6], [3, -3, -3, -3, -3, -10, 10, 3], [-31, 6, -8, 6, -2, -7, 0, 6], [10, -10, -10, -15, -9, -10, 10, 9], [27, 10, -3, 10, 7, 2, 0, 10], [-3, 3, 3, 2, -1, -3, 3, 1], [4, -4, -4, -24, -12, 2, -2, 12], [5, 0, -2, 0, -2, 11, 0, 0], [1, 4, 1, 4, 5, 14, 0, 4], [-2, 2, 2, 8, -8, -4, 4, 8], [15, 2, 6, 9, -6, 0, -21, -8], [-1, 1, 1, -21, 4, -10, 10, -4], [-11, 11, 11, -23, -1, -12, 12, 1], [15, 8, -3, 18, 3, 0, 8, -5], [16, 12, -2, 18, 2, 0, 0, -10], [-19, 2, 9, 2, 11, -2, 0, 2], [3, -2, 8, -5, -8, 0, 19, -6], [-9, 1, 0, 1, 1, -6, 0, 1], [-36, 6, -2, -34, 2, 0, -10, -4], [13, -13, -13, 25, -8, 6, -6, 8], [5, -5, -5, -21, -9, 10, -10, 9], [-25, -6, -3, -6, -9, 3, 0, -6], [22, 2, 9, 2, 11, 4, 0, 2], [39, 2, -6, 45, 6, 0, -5, 4], [-1, 1, 1, 15, 2, -20, 20, -2], [-31, -10, 3, -34, -3, 0, 0, 7], [11, -11, -11, 32, -7, 0, 0, 7], [-8, 8, 5, 8, 13, 0, 0, 8], [-12, 12, 12, -1, 6, 17, -17, -6], [-27, 1, -5, 1, -4, -12, 0, 1], [32, -9, -8, 40, 8, 0, -7, 17], [24, -12, 8, 16, -8, 0, 6, 4], [-5, 0, 4, 0, 4, 29, 0, 0], [3, -4, -3, -4, -7, 18, 0, -4], [-7, 7, 7, -12, 1, 5, -5, -1], [-10, -1, 2, -12, -2, 0, 3, -1], [-11, 5, -6, 5, -1, 0, 0, 5], [-32, -3, 2, -34, -2, 0, -1, 1], [-12, 12, 12, -12, 6, -10, 10, -6], [33, -5, -5, 38, 5, 0, 0, 10], [-9, 4, -4, -5, 4, 0, 2, 0], [34, 2, -4, 2, -2, -12, 0, 2], [-4, 4, 4, 24, 6, -14, 14, -6], [5, -4, -3, 8, 3, 0, 8, 7], [8, 4, -5, 4, -1, 0, 0, 4], [21, 3, 5, 16, -5, 0, -10, -8], [7, -3, -1, -3, -4, 12, 0, -3], [-20, 2, -2, -18, 2, 0, -6, 0], [-2, 2, 2, 13, 10, -6, 6, -10], [-9, 2, -7, 2, -5, -21, 0, 2], [17, -6, 0, -6, -6, -7, 0, -6], [-32, -4, -2, -4, -6, 10, 0, -4], [8, 4, -4, 12, 4, 0, -20, 0], [-1, 1, 1, -37, 8, 8, -8, -8], [-5, 4, -12, 7, 12, 0, 20, 8], [35, -4, 6, 29, -6, 0, 4, -2], [-10, 10, 10, -27, -3, -2, 2, 3], [-18, 0, 4, 0, 4, 2, 0, 0], [-5, 5, 5, -32, -5, -4, 4, 5], [-14, 14, 14, -17, 0, -1, 1, 0], [6, 5, 4, 2, -4, 0, 4, -9], [-2, 2, 2, 23, 12, -12, 12, -12], [4, -4, -4, -8, 2, -18, 18, -2], [1, 11, -14, 15, 14, 0, 4, 3], [6, -5, 2, 4, -2, 0, 24, 3], [34, -10, -9, -10, -19, -16, 0, -10], [-5, 5, -5, 5, 0, 2, 0, 5], [0, 0, 0, -19, 0, 14, -14, 0], [8, -8, -8, -18, 0, -14, 14, 0], [-14, 8, -9, -5, 9, 0, -12, 1], [-25, -6, 4, -6, -2, 3, 0, -6], [35, -5, -7, 42, 7, 0, -14, 12], [42, 0, 2, 0, 2, -6, 0, 0], [31, 11, -2, 33, 2, 0, -6, -9], [-21, 4, -6, -15, 6, 0, -3, 2], [10, -10, -10, 15, -8, 3, -3, 8], [41, -6, 1, -6, -5, 9, 0, -6], [-31, -12, 7, -12, -5, 2, 0, -12], [24, 1, 2, 22, -2, 0, -1, -3], [4, -4, -4, -3, -3, -6, 6, 3], [15, -8, 9, -8, 1, 33, 0, -8], [1, -1, -1, -19, 7, -4, 4, -7], [23, -3, -12, -3, -15, -18, 0, -3], [4, -4, -4, 13, 10, -12, 12, -10], [0, 0, 0, 16, -4, 24, -24, 4], [12, 0, -2, 14, 2, 0, -22, 2]]
