
# q-expansion of newform 3024.2.r.m, downloaded from the LMFDB on 04 September 2025.

# 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 = 3024
weight = 2
poly_data = [81, -27, 9, 9, -15, 3, 1, -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, 0, 0], 1], [[0, -9, 9, 6, -6, 2, -2, -1], 27], [[9, 9, -6, 3, 1, -1, 1, 0], 9], [[54, 9, -18, 3, -3, -2, 2, 1], 27], [[-54, 72, -9, -12, 15, -7, -2, 2], 27], [[-81, 72, 36, -21, 12, -4, -5, 2], 27], [[-36, 36, 6, -12, 5, -1, -2, 1], 9], [[216, 9, -72, 12, -15, -8, 8, 1], 27]]

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, -1, 0, 0, 0, 0]], [2593, [1, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, -1, 0, -2, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, -1], [0, -1, -1, 0, 1, 1, 1, 0], [1, -1, -1, 0, -1, 0, -1, 0], [0, 0, 0, 0, -1, 0, 0, 1], [0, 0, 0, -3, 0, 2, 1, 2], [-1, 0, 1, -1, -2, 0, 0, -2], [1, -1, -1, 0, 0, 0, 0, 0], [-2, 0, -1, -2, 2, 0, 0, 2], [0, -1, 0, 1, 0, 1, 2, 1], [0, -1, 0, -1, 0, 1, -1, 1], [0, -4, -4, 0, 3, -1, 3, 0], [3, 0, 0, 3, 3, 0, 0, 1], [0, 0, 0, -6, 0, 3, 1, 3], [-4, 0, 1, -4, -3, 0, 0, 1], [-1, 0, 0, 0, -1, 0, -1, 0], [-5, -1, -1, 0, 3, 0, 3, 0], [0, -1, 0, 5, 0, -3, 3, -3], [0, 1, 0, -2, 0, 0, 1, 0], [0, -3, -3, 0, 2, 1, 2, 0], [0, 2, 0, -3, 0, -1, -1, -1], [0, 1, 0, -1, 0, -5, 3, -5], [-4, 0, 4, -4, -3, 0, 0, 1], [-9, -2, -2, 0, 2, 0, 2, 0], [-7, -1, -1, 0, 1, 1, 1, 0], [4, 0, -4, 4, 2, 0, 0, 0], [2, -3, -3, 0, 2, -3, 2, 0], [-7, 0, 0, -7, 3, 0, 0, 2], [0, 1, 0, 3, 0, 1, 4, 1], [-11, 0, -3, -11, 1, 0, 0, 0], [6, 0, -1, 6, 0, 0, 0, -5], [0, -2, 0, 9, 0, 0, -1, 0], [9, 0, -7, 9, 3, 0, 0, 3], [0, -5, -5, 0, 7, -2, 7, 0], [1, 0, 2, 1, -2, 0, 0, 1], [0, -3, 0, -9, 0, 0, 2, 0], [-5, 3, 3, 0, 3, -1, 3, 0], [-11, 1, 1, 0, -2, 2, -2, 0], [0, -3, 0, -5, 0, 2, 6, 2], [5, 0, 2, 5, -7, 0, 0, -1], [-3, 2, 2, 0, -1, 2, -1, 0], [0, 8, 8, 0, -5, 1, -5, 0], [3, 0, -6, 3, 4, 0, 0, 1], [0, 4, 0, 9, 0, 5, 0, 5], [0, 6, 0, -1, 0, -1, -3, -1], [-1, 0, 0, -1, -2, 0, 0, 3], [-7, 7, 7, 0, -4, 3, -4, 0], [-7, 0, 2, -7, -2, 0, 0, -5], [0, -2, 0, -8, 0, 4, 5, 4], [0, -5, -5, 0, 5, -5, 5, 0], [-14, 0, 0, -14, -4, 0, 0, 1], [0, -3, 0, -3, 0, -5, 7, -5], [-5, 2, 2, 0, -3, 0, -3, 0], [6, 1, 1, 0, -4, 1, -4, 0], [0, -7, 0, -8, 0, 0, 1, 0], [0, 1, 0, 0, 0, -2, -9, -2], [1, 0, -7, 1, 4, 0, 0, 0], [15, 0, 2, 15, 3, 0, 0, 2], [8, 1, 1, 0, 9, 3, 9, 0], [2, 0, 5, 2, 4, 0, 0, 7], [0, -4, 0, 1, 0, -3, 3, -3], [0, 7, 0, -12, 0, 1, -8, 1], [0, 5, 0, -8, 0, 2, 5, 2], [-11, 0, 6, -11, -1, 0, 0, 3], [3, 0, -5, 3, 3, 0, 0, 6], [0, 4, 0, -12, 0, 4, -2, 4], [0, -4, 0, -2, 0, 4, 1, 4], [0, -9, -9, 0, 1, 2, 1, 0], [0, 1, 0, -1, 0, -4, -6, -4], [5, 0, 5, 5, -10, 0, 0, 0], [-5, 6, 6, 0, 5, 1, 5, 0], [3, 0, -5, 3, -4, 0, 0, 0], [0, -5, 0, -5, 0, -5, 9, -5], [13, -3, -3, 0, 2, 4, 2, 0], [6, 0, 9, 6, 4, 0, 0, 0], [2, 0, -8, 2, 4, 0, 0, -3], [2, 0, 0, 2, 7, 0, 0, 3], [0, -3, 0, 7, 0, 1, -3, 1], [-19, -2, -2, 0, -1, 0, -1, 0], [7, 0, 0, 0, -8, 8, -8, 0], [0, -2, 0, -3, 0, -4, -7, -4], [0, -4, 0, -12, 0, 1, 8, 1], [-8, 5, 5, 0, -3, -3, -3, 0], [0, 3, 0, 19, 0, -2, -7, -2], [0, 5, 0, 6, 0, -6, -7, -6], [-4, 0, 0, -4, 1, 0, 0, 7], [-1, 5, 5, 0, -2, 7, -2, 0], [0, 3, 0, -3, 0, 5, -11, 5], [-5, 10, 10, 0, -11, -2, -11, 0], [9, 0, -3, 9, 8, 0, 0, 5], [8, 0, -4, 8, 2, 0, 0, 7], [-10, 1, 1, 0, -2, -5, -2, 0], [8, 0, 2, 8, -5, 0, 0, 1], [-9, 4, 4, 0, 4, -6, 4, 0], [5, -6, -6, 0, -3, -2, -3, 0], [5, 8, 8, 0, -6, 1, -6, 0], [0, 0, 0, -18, 0, -8, 3, -8], [-31, 0, 0, 0, -1, 2, -1, 0], [2, 0, 3, 2, -2, 0, 0, -12], [0, -2, 0, 5, 0, -4, -8, -4], [-11, 0, 6, -11, -5, 0, 0, -11], [10, -2, -2, 0, 1, 2, 1, 0], [0, 3, 0, 6, 0, 2, -2, 2], [13, -11, -11, 0, 3, -3, 3, 0], [6, 0, 8, 6, -6, 0, 0, 4], [0, 11, 0, -16, 0, 2, -12, 2], [11, 0, 14, 11, -2, 0, 0, 3], [-15, 6, 6, 0, -8, -5, -8, 0], [2, 0, 5, 2, 2, 0, 0, -4], [0, -2, 0, 2, 0, 10, -7, 10], [-12, 9, 9, 0, -2, 7, -2, 0], [0, -6, 0, -18, 0, 2, 9, 2], [-5, 0, -6, -5, 3, 0, 0, -3], [-25, 7, 7, 0, -9, -9, -9, 0], [14, 0, -9, 14, 7, 0, 0, 6], [0, 3, 0, 15, 0, -3, -9, -3], [7, 0, -2, 7, 5, 0, 0, -8], [0, -3, 0, 10, 0, 3, -10, 3], [0, -4, 0, 15, 0, 2, 1, 2], [-6, -8, -8, 0, 0, -3, 0, 0], [0, 3, 0, 2, 0, 3, 2, 3], [6, 6, 6, 0, -2, 6, -2, 0], [0, 0, 0, 6, 0, 0, 6, 0], [13, 6, 6, 0, -9, -2, -9, 0], [0, 3, 0, 5, 0, 0, -2, 0], [-23, 0, 8, -23, 0, 0, 0, -5], [5, 4, 4, 0, -11, -3, -11, 0], [20, 0, 3, 20, -11, 0, 0, -2], [13, 0, 1, 13, -3, 0, 0, -3], [28, -1, -1, 0, 1, -4, 1, 0], [-6, 0, -4, -6, 2, 0, 0, -6], [28, 0, -8, 28, 7, 0, 0, -1], [-31, 0, 0, 0, 2, 1, 2, 0], [5, 0, -5, 5, 10, 0, 0, 12], [0, 0, -1, 0, 11, 0, 0, -6], [20, -11, -11, 0, 3, 3, 3, 0], [20, -6, -6, 0, 1, 4, 1, 0], [0, -5, 0, -31, 0, 9, 3, 9], [12, 0, -4, 12, -12, 0, 0, 2], [7, 4, 4, 0, 3, 4, 3, 0], [5, 5, 5, 0, 11, 3, 11, 0], [0, 0, 0, -21, 0, -3, -6, -3], [0, -1, 0, -1, 0, -7, 3, -7], [0, 6, 0, 0, 0, -5, -5, -5], [5, 0, 12, 5, -2, 0, 0, 2], [-16, -7, -7, 0, -3, 4, -3, 0], [-12, 0, -3, -12, -1, 0, 0, 0], [-9, 15, 15, 0, -6, 0, -6, 0], [-25, -3, -3, 0, 2, -5, 2, 0], [-22, 0, 6, -22, -1, 0, 0, -5], [0, 0, 0, 1, 0, 0, 6, 0], [0, 0, 0, 9, 0, 2, 1, 2], [-2, 1, 1, 0, -6, 1, -6, 0], [0, -8, 0, 7, 0, -4, 15, -4], [-14, 1, 1, 0, 7, 0, 7, 0], [21, 0, 7, 21, 8, 0, 0, 2], [0, 4, 0, -22, 0, -2, -1, -2], [24, 10, 10, 0, -2, -1, -2, 0], [44, 0, -5, 44, 7, 0, 0, -4], [6, -5, -5, 0, 1, -14, 1, 0], [-11, 0, -8, -11, -8, 0, 0, -4], [0, -8, 0, -18, 0, 7, 1, 7], [7, -7, -7, 0, 1, 3, 1, 0], [0, 0, 0, 10, 0, 4, -4, 4], [8, -1, -1, 0, 3, 12, 3, 0], [-15, 0, -3, -15, -7, 0, 0, -1], [0, 6, 0, 15, 0, 7, 3, 7], [8, 0, 5, 8, 10, 0, 0, 7], [9, 0, -4, 9, -4, 0, 0, 7], [0, -12, 0, -7, 0, 10, 4, 10], [7, 0, 4, 7, -4, 0, 0, -5], [-18, 0, 11, -18, -2, 0, 0, 3], [0, 17, 0, -1, 0, -7, -2, -7], [-23, 1, 1, 0, 2, -8, 2, 0], [18, -4, -4, 0, 11, -3, 11, 0], [0, 7, 0, 23, 0, 4, -4, 4], [0, 4, 0, -22, 0, -3, -1, -3], [0, 7, 0, 39, 0, 2, -7, 2], [-20, 0, 12, -20, -6, 0, 0, -4], [15, 5, 5, 0, 7, -4, 7, 0], [-10, 0, 0, -10, -9, 0, 0, -5], [0, 5, 0, 15, 0, 1, -1, 1], [-9, 1, 1, 0, 0, -10, 0, 0], [0, -9, 0, -10, 0, 6, 5, 6], [-10, 0, -9, -10, 4, 0, 0, -4], [25, -1, -1, 0, 9, 11, 9, 0], [-7, -6, -6, 0, -11, -1, -11, 0], [0, -11, 0, -19, 0, 3, 6, 3], [5, -12, -12, 0, 3, 3, 3, 0], [0, 5, 0, 12, 0, -5, -4, -5], [-3, -17, -17, 0, 12, -1, 12, 0], [-18, 0, 6, -18, 11, 0, 0, -4], [7, 0, 3, 7, -19, 0, 0, -2], [0, -11, 0, 9, 0, -1, 11, -1], [0, -7, 0, 14, 0, 3, -9, 3], [15, -14, -14, 0, 18, 3, 18, 0], [17, 0, -4, 17, 0, 0, 0, -5], [0, 10, 0, 25, 0, -1, -2, -1], [-9, 0, 12, -9, -5, 0, 0, -8], [0, -13, 0, -18, 0, -6, 10, -6], [22, 13, 13, 0, -4, 2, -4, 0], [4, 7, 7, 0, -14, -1, -14, 0], [-6, -4, -4, 0, -6, -4, -6, 0], [-10, 0, 0, -10, 5, 0, 0, 10], [0, -6, 0, 18, 0, 11, 0, 11], [-30, 0, -5, -30, 12, 0, 0, 12], [20, 1, 1, 0, 2, -8, 2, 0], [-21, 0, -1, -21, 3, 0, 0, 5], [0, 2, 0, -8, 0, 0, 8, 0], [0, -5, 0, 36, 0, -1, 4, -1], [6, 0, -13, 6, 13, 0, 0, 12], [0, -8, 0, 11, 0, -5, 11, -5], [-24, 0, 2, -24, 1, 0, 0, 7], [0, 9, 0, 3, 0, -3, 2, -3], [18, -1, -1, 0, 11, -6, 11, 0], [17, 0, 0, 17, 11, 0, 0, 0], [-11, 0, -4, -11, -13, 0, 0, -12], [-2, 0, -8, -2, 4, 0, 0, 8], [-10, 0, -2, -10, -10, 0, 0, 5], [-21, 11, 11, 0, -8, 6, -8, 0], [-44, 0, -6, -44, 4, 0, 0, -1], [0, -13, 0, -6, 0, 2, 11, 2], [0, 5, 0, 48, 0, 1, -6, 1], [-20, 0, 0, 0, 1, -7, 1, 0], [0, 1, 0, -16, 0, 3, 6, 3], [0, 3, 0, 5, 0, -4, 0, -4], [19, 0, 15, 19, -10, 0, 0, 0], [6, 4, 4, 0, 10, -4, 10, 0], [8, 0, -14, 8, 12, 0, 0, 4], [-1, 0, -9, -1, 9, 0, 0, 8], [0, -10, 0, 14, 0, -2, -2, -2], [0, -5, 0, -23, 0, 10, 4, 10], [-2, 0, -6, -2, -20, 0, 0, -2], [-4, -22, -22, 0, 13, 5, 13, 0], [15, 0, -1, 15, 8, 0, 0, 7], [-17, 9, 9, 0, 1, 3, 1, 0], [0, -5, 0, 10, 0, -4, 18, -4], [-11, 4, 4, 0, 3, 14, 3, 0], [-34, 0, -5, -34, -8, 0, 0, -9], [-7, -8, -8, 0, 19, -6, 19, 0], [-12, 0, -8, -12, -11, 0, 0, -6], [0, 9, 0, 18, 0, 8, 7, 8], [31, -8, -8, 0, 11, -6, 11, 0], [-2, 0, -3, -2, -2, 0, 0, 2], [-17, 0, 11, -17, 1, 0, 0, -5], [33, -6, -6, 0, 8, -9, 8, 0], [-16, 0, -17, -16, -4, 0, 0, -7], [24, 8, 8, 0, 8, 3, 8, 0], [22, 0, -6, 22, 1, 0, 0, -13], [0, -9, 0, -58, 0, 0, 5, 0], [0, 15, 0, -9, 0, 1, -15, 1], [-12, 15, 15, 0, 1, 5, 1, 0], [13, -1, -1, 0, -9, 19, -9, 0], [0, 1, 0, -23, 0, 7, -3, 7], [-1, -9, -9, 0, 4, 14, 4, 0], [3, 10, 10, 0, 0, 6, 0, 0], [0, -16, 0, 2, 0, 3, 11, 3], [0, 1, 0, 18, 0, 9, -5, 9], [20, 0, 5, 20, 8, 0, 0, 5], [-19, -16, -16, 0, 4, 1, 4, 0], [-4, 0, -8, -4, 0, 0, 0, -10], [0, 0, 0, 20, 0, -9, 4, -9], [-13, -1, -1, 0, 1, -17, 1, 0], [0, 9, 0, 17, 0, 5, -18, 5], [24, 0, -7, 24, -12, 0, 0, 2], [4, 0, 4, 4, 1, 0, 0, -3], [20, 0, -9, 20, 13, 0, 0, 6], [-33, -3, -3, 0, 3, -11, 3, 0], [0, -12, 0, 11, 0, 4, -6, 4], [10, 0, -1, 10, -9, 0, 0, -16], [3, 0, -2, 3, -1, 0, 0, 3], [37, -12, -12, 0, 8, 11, 8, 0], [50, 0, 0, 50, 5, 0, 0, -3], [0, 5, 0, -41, 0, -1, -1, -1], [-20, 8, 8, 0, 8, 11, 8, 0], [0, -7, 0, -9, 0, 3, 10, 3], [21, 0, -14, 21, -4, 0, 0, 3], [28, 0, -23, 28, 4, 0, 0, 9], [0, 24, 0, 26, 0, 1, -11, 1], [0, -4, 0, -25, 0, 0, 7, 0], [0, 0, -1, 0, -5, 0, 0, 9], [37, -5, -5, 0, -7, -1, -7, 0], [-31, 3, 3, 0, -7, -15, -7, 0], [-31, 0, -4, -31, 3, 0, 0, 12], [0, -12, 0, -17, 0, -10, 7, -10], [46, -13, -13, 0, 9, 1, 9, 0], [0, -4, 0, -18, 0, 3, -9, 3], [36, -6, -6, 0, -10, -3, -10, 0], [11, 0, 17, 11, -15, 0, 0, -6], [-15, 0, 11, -15, -3, 0, 0, -6], [0, 20, 0, 16, 0, 7, -11, 7], [-1, 0, 5, -1, -8, 0, 0, -14], [0, -4, 0, -43, 0, 8, 7, 8], [0, -16, 0, -18, 0, -2, 14, -2], [-5, 5, 5, 0, -2, 0, -2, 0], [0, -18, 0, 7, 0, -3, 8, -3], [-3, 0, 6, -3, -12, 0, 0, 6], [-8, 4, 4, 0, 19, 3, 19, 0], [43, -2, -2, 0, 15, 8, 15, 0]]
