
# q-expansion of newform 3072.2.d.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 = 3072
weight = 2
poly_data = [2, -12, 30, -44, 43, -28, 14, -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], [[-5, 6, -7, 2, -1, 0, 0, 0], 1], [[4, -12, 19, -15, 10, -3, 1, 0], 1], [[-5, 16, -24, 17, -11, 3, -1, 0], 1], [[31, -126, 223, -256, 175, -98, 28, -8], 1], [[-42, 166, -285, 325, -220, 123, -35, 10], 1], [[-43, 168, -285, 325, -220, 123, -35, 10], 1], [[109, -440, 774, -890, 610, -342, 98, -28], 1]]

hecke_ring_character_values = [[2047, [1, 0, 0, 0, 0, 0, 0, 0]], [2053, [-1, 0, 0, 0, 0, 0, 0, 0]], [1025, [1, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, 0], [-1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 1, 1], [0, 0, 0, 0, -2, 0, 1, 1], [0, 1, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 1, -1], [0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 2, 1, 0], [3, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 2, -1, 1], [0, -1, 3, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, -3, -1], [0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, 2, 1, 0], [0, 0, 0, 0, 0, -4, 0, 0], [0, 0, 0, 0, -4, -2, -1, 1], [0, 0, 0, 0, 4, 2, 2, 2], [2, -2, 0, 0, 0, 0, 0, 0], [2, 2, 4, 0, 0, 0, 0, 0], [3, 1, 4, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, -5, -3, 1], [2, -2, 6, 2, 0, 0, 0, 0], [0, 2, 6, -2, 0, 0, 0, 0], [0, 0, 0, 0, 5, 0, -1, -2], [-1, 1, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, -4, 0, -4], [0, 0, 0, 0, -6, 0, -1, -1], [-2, 0, 4, -4, 0, 0, 0, 0], [3, 1, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, -2, 2, 2], [0, -1, 3, 3, 0, 0, 0, 0], [0, 0, 0, 0, -4, 6, 2, 2], [0, 0, 0, 0, 7, -4, -1, 0], [-1, 1, -8, -4, 0, 0, 0, 0], [0, 0, 0, 0, -8, 4, -1, -3], [0, 0, 0, 0, 0, 1, -1, -3], [10, 2, -2, 4, 0, 0, 0, 0], [0, 0, 0, 0, -9, -6, 1, 2], [0, 0, 0, 0, 0, -2, 6, -2], [0, 0, 0, 0, 6, 2, -1, 3], [-10, 2, 6, 0, 0, 0, 0, 0], [-4, -6, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 9, 2, -3, 0], [-7, -1, 4, 4, 0, 0, 0, 0], [0, 0, 0, 0, 4, 10, -2, -2], [-3, -1, -8, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, -5, -3, 1], [0, 0, 0, 0, -2, 6, 5, -3], [2, 6, 2, -2, 0, 0, 0, 0], [-14, 2, -10, -4, 0, 0, 0, 0], [-4, 4, -2, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, -5, 3], [10, 2, 2, -2, 0, 0, 0, 0], [16, 4, 4, -4, 0, 0, 0, 0], [0, 0, 0, 0, 5, 8, -1, 6], [13, -1, 8, 4, 0, 0, 0, 0], [0, 0, 0, 0, 2, 2, 1, 5], [-10, 2, -2, 2, 0, 0, 0, 0], [0, 0, 0, 0, 12, 4, 0, 0], [0, 0, 0, 0, 1, -6, 5, 8], [0, 0, 0, 0, 4, 2, -6, 2], [4, 0, 12, -4, 0, 0, 0, 0], [4, -6, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, -8, 1, 6], [0, 0, 0, 0, 12, 6, 2, -6], [10, -2, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, -16, -9, 1, -3], [0, 0, 0, 0, 8, -2, 5, 7], [-2, -2, -6, 6, 0, 0, 0, 0], [6, -6, -4, -8, 0, 0, 0, 0], [7, -3, -12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 4, 5, -5], [0, 0, 0, 0, 0, -3, 7, 1], [-10, -6, 10, 0, 0, 0, 0, 0], [0, 0, 0, 0, -11, 10, -1, 2], [0, 0, 0, 0, 8, -4, 3, 1], [0, -5, -5, -5, 0, 0, 0, 0], [-4, 6, -2, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -7, 7, 3], [0, 0, 0, 0, -2, 6, -7, 1], [-4, -8, -10, 4, 0, 0, 0, 0], [4, 0, -8, 0, 0, 0, 0, 0], [-11, -5, 12, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, -13, 5, 1], [-16, -1, -1, -5, 0, 0, 0, 0], [16, 2, -2, -2, 0, 0, 0, 0], [0, 0, 0, 0, 5, 10, 3, -2], [-9, -3, 12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 16, -9, 9, 5], [-6, 2, -4, -12, 0, 0, 0, 0], [-19, 3, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 4], [0, 0, 0, 0, -12, 0, -4, -4], [18, -2, 12, 8, 0, 0, 0, 0], [0, 0, 0, 0, 19, -2, -5, -4], [0, -1, -17, 3, 0, 0, 0, 0], [0, 0, 0, 0, -8, -15, -1, 1], [0, 0, 0, 0, 2, -10, -9, -1], [0, 0, 0, 0, 0, 7, -3, 7], [0, 0, 0, 0, 9, -8, -3, 4], [0, 0, 0, 0, 16, -13, -3, -3], [16, -1, 3, 7, 0, 0, 0, 0], [0, 0, 0, 0, -4, 12, 0, 0], [-2, -2, -16, -4, 0, 0, 0, 0], [0, 0, 0, 0, 20, -10, 2, -2], [-2, 8, -12, -4, 0, 0, 0, 0], [12, -4, -14, 0, 0, 0, 0, 0], [2, -2, -4, 0, 0, 0, 0, 0], [-1, 5, -8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 1, 3], [6, -8, -4, 4, 0, 0, 0, 0], [0, 0, 0, 0, 12, 0, 0, 8], [-15, -1, 16, 0, 0, 0, 0, 0], [8, 1, -11, -3, 0, 0, 0, 0], [0, 0, 0, 0, -8, 7, -7, -5], [-12, -4, 2, 8, 0, 0, 0, 0], [0, 0, 0, 0, -3, 10, -1, -6], [0, 0, 0, 0, -20, 4, -4, 0], [0, 0, 0, 0, -16, 0, 3, 1], [-12, 4, -12, -4, 0, 0, 0, 0], [0, 0, 0, 0, -9, -20, 3, -4], [0, 0, 0, 0, -16, -7, -1, 3], [0, 0, 0, 0, 8, 5, 3, -7], [0, 0, 0, 0, -3, -4, 7, -2], [0, 0, 0, 0, -18, -6, -5, 7], [-10, -10, 8, 4, 0, 0, 0, 0], [1, 7, -8, 4, 0, 0, 0, 0], [0, 0, 0, 0, 10, -10, 9, -3], [0, 0, 0, 0, 4, -4, 8, 0], [22, -6, -2, 0, 0, 0, 0, 0], [-3, -1, -4, -12, 0, 0, 0, 0], [0, 0, 0, 0, 2, -10, 11, 3], [-8, 5, -11, 1, 0, 0, 0, 0], [12, -2, 8, 12, 0, 0, 0, 0], [0, 0, 0, 0, -7, 24, -1, 2], [0, 0, 0, 0, -8, 9, 3, 5], [0, 0, 0, 0, 3, 8, 7, 12], [8, 5, 1, -11, 0, 0, 0, 0], [0, 0, 0, 0, -8, 3, 1, -9], [0, 0, 0, 0, 7, 4, 5, -6], [25, 7, 12, -8, 0, 0, 0, 0], [0, 0, 0, 0, -16, 6, -2, 2], [0, 0, 0, 0, 18, 16, -1, 7], [14, -2, 4, -12, 0, 0, 0, 0], [0, 0, 0, 0, 16, 12, 5, 7], [8, -3, -7, -7, 0, 0, 0, 0], [0, 0, 0, 0, -32, 9, -1, -3], [-6, -10, -10, 8, 0, 0, 0, 0], [0, 0, 0, 0, -8, 22, -5, -7], [2, -10, -2, 10, 0, 0, 0, 0], [0, 0, 0, 0, -8, -11, 7, 5], [-8, -8, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -32, 13, 3, 1], [2, 2, 6, -12, 0, 0, 0, 0], [-19, 3, 0, -8, 0, 0, 0, 0], [8, 1, 1, -3, 0, 0, 0, 0], [-10, 0, 0, -16, 0, 0, 0, 0], [0, 0, 0, 0, -11, -14, 7, 2], [0, 0, 0, 0, 8, -2, -10, 6], [8, -1, -9, 3, 0, 0, 0, 0], [23, 1, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 8, 11, 5, 1], [-16, -3, -3, 9, 0, 0, 0, 0], [12, -8, 0, -4, 0, 0, 0, 0], [5, 7, 4, -12, 0, 0, 0, 0], [0, 0, 0, 0, -4, 6, 15, 1], [-12, 2, 10, -6, 0, 0, 0, 0], [0, 0, 0, 0, 5, -2, -9, 2], [0, 0, 0, 0, 20, 16, 8, -4], [0, 0, 0, 0, -12, 12, 17, 7], [-10, -2, -6, 4, 0, 0, 0, 0], [-22, -4, -12, -4, 0, 0, 0, 0], [-3, 7, 8, 4, 0, 0, 0, 0], [-26, -2, 18, -2, 0, 0, 0, 0], [0, 0, 0, 0, 28, 8, -4, -4], [0, 0, 0, 0, 11, -20, 7, 4], [-15, -9, 8, -4, 0, 0, 0, 0], [0, 0, 0, 0, 16, -6, 13, -1], [-5, 1, 12, -4, 0, 0, 0, 0], [0, 0, 0, 0, 24, 15, -7, 5], [0, 0, 0, 0, 18, -4, -7, -11], [-10, -2, -6, 14, 0, 0, 0, 0], [-14, -2, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, -11, -10, 3, 14], [0, 0, 0, 0, -6, -2, 11, -5], [0, 0, 0, 0, -4, -20, -8, 0], [4, 0, 34, 6, 0, 0, 0, 0], [24, -8, -8, -8, 0, 0, 0, 0], [-6, 0, -4, -20, 0, 0, 0, 0], [0, 0, 0, 0, -24, 16, 12, 4], [0, 0, 0, 0, -28, -16, 4, 4], [0, 0, 0, 0, 3, 10, -3, -6], [0, 0, 0, 0, -4, 2, -2, -2], [-18, -4, -16, 16, 0, 0, 0, 0], [-2, 10, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -42, 6, -1, -5], [-14, -2, 10, 6, 0, 0, 0, 0], [-8, 0, -4, 16, 0, 0, 0, 0], [0, 0, 0, 0, 13, 16, 1, 12], [1, 3, -4, 8, 0, 0, 0, 0], [0, 0, 0, 0, 2, 4, -3, -7], [10, -6, 4, 8, 0, 0, 0, 0], [0, 0, 0, 0, 8, 3, -19, -3], [0, 0, 0, 0, 5, -18, -11, 4], [21, -1, 4, 4, 0, 0, 0, 0], [0, 0, 0, 0, -32, 7, 9, 1], [18, 8, 20, -4, 0, 0, 0, 0], [0, 0, 0, 0, -8, 1, 11, 9], [-12, -6, 12, -8, 0, 0, 0, 0], [0, 0, 0, 0, 23, -4, 3, -4], [25, -1, 8, 12, 0, 0, 0, 0], [0, 0, 0, 0, 32, 0, 4, 4], [-30, 6, -8, -8, 0, 0, 0, 0], [16, 4, 8, -8, 0, 0, 0, 0], [-25, 5, 8, 0, 0, 0, 0, 0], [-8, -9, 3, -9, 0, 0, 0, 0], [-38, -2, 14, 0, 0, 0, 0, 0], [0, 0, 0, 0, -35, 4, -7, 4], [0, 0, 0, 0, 42, 0, -7, 5], [-3, 3, -12, -8, 0, 0, 0, 0], [-8, -1, 35, 3, 0, 0, 0, 0], [11, -7, -8, -20, 0, 0, 0, 0], [0, 0, 0, 0, -32, -1, 9, -3], [0, 0, 0, 0, 4, 4, -7, -17], [-6, 8, 0, 8, 0, 0, 0, 0], [2, -2, 4, 8, 0, 0, 0, 0], [15, 1, -4, -8, 0, 0, 0, 0], [0, 0, 0, 0, -20, 8, -8, -4], [0, 0, 0, 0, -10, 24, 3, -1], [0, 0, 0, 0, 8, -33, -11, -1], [-17, -3, -20, 8, 0, 0, 0, 0], [-18, 18, 14, -6, 0, 0, 0, 0], [0, 0, 0, 0, 16, 1, 19, -7], [50, 2, 0, 4, 0, 0, 0, 0], [-8, -10, 18, -6, 0, 0, 0, 0], [0, 0, 0, 0, 51, -8, -5, 0], [0, 0, 0, 0, -16, 17, -17, 3], [8, -8, 10, 8, 0, 0, 0, 0], [0, 0, 0, 0, -32, 3, -3, 5], [0, 0, 0, 0, 20, -22, 10, 2], [13, -5, -4, 4, 0, 0, 0, 0], [0, 0, 0, 0, -8, -4, 13, -9], [26, -4, 0, 16, 0, 0, 0, 0], [-10, 10, 12, 8, 0, 0, 0, 0], [21, -9, -8, -12, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 0, -4], [0, 0, 0, 0, 4, 10, -6, -6], [20, 4, 4, -16, 0, 0, 0, 0], [0, 0, 0, 0, -10, -8, -11, -3], [-2, -12, -8, 16, 0, 0, 0, 0], [-24, -8, 10, -8, 0, 0, 0, 0], [-6, -8, 20, 4, 0, 0, 0, 0], [0, 0, 0, 0, -17, 18, 3, 0], [0, 0, 0, 0, 16, 22, -2, 10], [0, 0, 0, 0, 18, -4, 11, 11], [0, 0, 0, 0, -28, 0, 16, 0], [0, 0, 0, 0, 19, -10, 7, 16], [4, 0, 8, -16, 0, 0, 0, 0], [-15, 11, -8, -8, 0, 0, 0, 0], [0, 0, 0, 0, -8, 0, 4, -4], [0, 0, 0, 0, -2, -4, 15, 11], [0, 0, 0, 0, -48, 0, -7, 3], [8, 1, -3, 1, 0, 0, 0, 0], [0, 0, 0, 0, 4, -6, -14, 10], [0, 0, 0, 0, 11, -24, -5, -8], [10, -4, -8, -16, 0, 0, 0, 0], [0, 0, 0, 0, 0, 15, 13, 3], [0, 0, 0, 0, 35, 20, 7, -4], [0, 0, 0, 0, -20, -18, -7, 15], [0, 0, 0, 0, 28, -12, 4, -4], [-26, 0, -12, 4, 0, 0, 0, 0], [-15, -5, 8, 8, 0, 0, 0, 0], [6, 8, 8, 8, 0, 0, 0, 0], [-11, -5, -12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 24, 7, -7, 1], [0, 0, 0, 0, 40, 12, 3, 1], [-14, 2, 8, 20, 0, 0, 0, 0], [0, 0, 0, 0, -8, 22, 6, 10], [-6, 14, 12, -16, 0, 0, 0, 0], [21, 3, 12, -20, 0, 0, 0, 0], [-16, -8, -22, 8, 0, 0, 0, 0], [0, 0, 0, 0, 16, -24, 5, -1], [0, 0, 0, 0, 40, 7, -3, 3], [24, 16, -20, 8, 0, 0, 0, 0], [20, 4, -10, 18, 0, 0, 0, 0], [0, 0, 0, 0, 13, 8, -11, 0], [-1, -7, 16, -8, 0, 0, 0, 0], [-10, 10, 22, -6, 0, 0, 0, 0], [0, 0, 0, 0, -11, -10, 15, 2], [0, 0, 0, 0, 8, -10, -2, -18], [-30, -8, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, -48, 2, -6, 2], [0, 0, 0, 0, -24, 18, 11, 1], [0, 0, 0, 0, 37, 6, 11, -10], [-9, 5, 16, -4, 0, 0, 0, 0], [0, 0, 0, 0, 21, 8, 11, 10], [0, 0, 0, 0, -8, 21, 3, -5], [0, 0, 0, 0, 8, 13, -9, -23], [36, -10, -28, -8, 0, 0, 0, 0], [0, 0, 0, 0, -25, 2, 5, -2], [37, 15, -16, 4, 0, 0, 0, 0]]
