
# q-expansion of newform 4016.2.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 = 4016
weight = 2
poly_data = [-5, -12, 20, 15, -8, -3, 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], [[1, -9, -15, 4, 4, -1], 4], [[-17, 5, 19, -4, -4, 1], 4], [[-3, -27, 17, 6, -6, 1], 4], [[-4, 15, 12, -7, -3, 1], 2]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [-1, 1, 0, -1, 0, 0], [-1, 0, 0, 0, 0, 1], [-1, 1, -1, -1, 0, 0], [-1, 1, 0, 1, 0, 0], [2, -1, -1, 0, 1, 0], [0, 0, -1, 2, -1, -1], [-3, -1, 1, -1, -1, 0], [0, 0, 3, 0, -1, -1], [0, -2, 1, 0, 0, -1], [-1, -1, 0, 2, 0, 0], [1, 0, -3, 1, 0, -1], [-1, 3, 0, 0, -2, 0], [0, -2, 0, 2, -2, -2], [1, -1, -1, 3, 0, -2], [-1, 1, 4, -4, 0, 2], [-11, 1, -1, -1, 0, 0], [3, -1, -2, 1, 0, -4], [-2, 0, -2, 4, 2, 0], [1, -1, 4, 0, -3, 2], [-5, -1, 3, -1, -3, -1], [-4, 4, 0, 2, -2, -2], [5, -1, 0, 4, 1, 2], [2, -2, 0, -4, -2, 0], [-2, -4, 4, 2, -3, 0], [1, -2, 5, -5, -2, 3], [-4, 2, 3, -2, 1, 1], [-5, -1, -4, 4, 0, -4], [4, 0, 2, 1, 0, 2], [0, 0, -4, 2, 2, -2], [3, -5, 2, 1, 2, 2], [8, -6, 0, 0, 0, 0], [-3, 3, 4, 2, -4, -2], [-4, 2, 0, -4, 5, 2], [-2, 0, -4, 2, 4, 4], [-9, -1, -5, 1, 4, 0], [-6, 2, -1, -8, 1, 3], [-6, -2, 0, 0, 0, -2], [6, 4, -2, 0, 2, 2], [2, -5, 0, -2, 0, 3], [-2, 1, 2, 2, -6, -1], [-10, 6, -2, -2, 2, -4], [0, 2, 2, 0, -6, -4], [1, 3, 4, -5, 0, 0], [0, 2, -4, -4, 2, 2], [-9, 3, 0, -1, 2, 2], [-4, 2, 0, -4, 4, 4], [0, 1, 4, -4, 2, 5], [5, -1, 0, 2, 6, 6], [0, 2, 1, -4, 0, 0], [-2, -6, 6, -2, -2, 0], [7, -4, -5, 3, 0, -3], [1, 0, 0, 0, 0, 0], [-2, 2, 2, -6, -4, 0], [0, -1, 1, 0, 3, 4], [4, -4, 4, -6, -6, 2], [-5, 4, 3, -3, 6, 3], [2, 4, -9, 2, 5, 1], [4, 4, -2, 0, 6, 2], [12, -10, 4, 4, 2, -2], [1, 5, 4, -4, 0, 0], [3, 3, -3, 3, 2, -6], [-5, 3, 0, 0, 1, 4], [2, 4, -2, -2, 6, 0], [18, -3, -2, -2, 2, -1], [2, -2, 6, -6, 1, 4], [-10, 1, 3, -4, -3, -6], [9, -1, -12, 8, 2, 2], [-10, -2, -4, 0, 2, 2], [10, -2, 4, -2, -2, 2], [-7, -2, 1, 7, -2, -1], [8, 0, 0, 0, -4, 0], [4, 2, 0, -2, -5, -4], [16, -6, 4, 8, 0, 2], [2, 2, -6, -8, 4, 4], [2, 0, 1, 2, -3, -1], [2, -6, 1, -2, -1, -7], [-10, 0, -6, 2, -2, 2], [0, -6, 4, -2, 2, 2], [7, -1, 5, -5, 6, 10], [-6, -10, 5, 6, -1, 1], [4, -2, 7, 8, -4, 1], [0, -10, 6, 2, -2, 4], [6, 8, -8, -4, 6, 0], [-16, -4, 2, 6, -8, -4], [2, -1, -9, 4, 1, -4], [-20, 6, 8, -10, 0, 4], [3, -7, 4, 2, -2, 2], [-6, -2, -2, 6, -6, 0], [10, -6, 5, 8, -3, -3], [6, 4, 0, 0, -4, -10], [4, -6, -2, 8, 4, 2], [9, 1, -12, 3, 0, 0], [9, -3, -1, -7, 6, 2], [9, -9, 2, 3, 0, 4], [-23, 7, 4, -7, 2, 6], [10, -8, 6, 0, -4, 0], [0, 0, -2, 4, -7, -2], [8, 2, 0, 10, 1, 0], [4, 4, -2, 0, -7, -6], [1, 5, 1, 3, -6, 4], [7, 1, -8, 0, -6, -2], [9, 3, 0, -4, 4, 2], [8, 0, 2, -2, 12, 4], [-3, 5, -1, -5, -8, 0], [-9, 7, 2, -5, 0, 8], [-1, -5, -1, -5, 3, 4], [0, -2, 4, 2, 0, -2], [-2, 6, 0, 2, 4, 2], [-17, 5, -1, 3, -8, -4], [5, 3, -3, -9, 12, 10], [7, -9, -1, 3, 3, -4], [5, 1, 2, 1, -6, 2], [-14, 2, 14, -10, -4, 4], [0, 12, -2, 2, 2, 2], [0, 10, -8, -8, 8, 2], [-4, 0, -4, -2, -6, -10], [15, -5, 4, 10, -2, -4], [4, -12, -3, 14, -3, -7], [-16, 12, -5, -8, 1, -5], [-8, 0, 8, 2, -10, -8], [-2, 0, -10, 4, 13, 0], [-3, 5, -2, -7, 0, -4], [-11, -7, 4, 1, -12, 0], [-14, 12, -4, -8, -4, -4], [14, 1, 2, 4, 8, -3], [-7, 2, -8, 8, 0, -3], [-24, 3, -4, -2, -2, -9], [16, -2, -5, 4, 3, 1], [0, 9, 4, -4, -2, -3], [4, -6, -7, 8, 0, 3], [-9, 4, 8, 0, 0, 5], [10, -8, 4, -2, 3, 8], [8, 0, 0, 4, 8, 2], [-3, -1, 1, -1, -7, -5], [-20, 6, 0, -8, -1, 6], [-7, 9, 1, -7, 4, -2], [-2, 6, -5, 2, -1, -1], [-4, 2, -10, 8, -4, -4], [-6, -1, -8, 4, -10, 1], [1, -5, -2, 3, -8, -12], [-4, -2, -2, -4, 8, 14], [16, 12, 0, -12, 8, 8], [-8, 6, 0, 6, 2, 0], [12, 4, -11, -2, 8, -4], [-23, 3, 0, -9, 4, -4], [0, 4, -4, 4, 10, 2], [-14, 1, -2, -2, 2, -9], [-12, -5, -4, 2, 10, -1], [10, 3, 2, -6, 6, 1], [6, -4, 0, 14, -6, -10], [-7, 5, 9, -7, 4, 2], [24, 0, -2, 8, -4, -6], [8, -16, 4, 6, 6, -2], [28, 6, -6, 2, 6, -2], [-2, -2, 0, -4, -4, 4], [-2, 0, -8, 10, 0, -10], [8, 2, -2, -6, 10, 10], [-20, 0, -1, -4, -11, 3], [-4, 1, 12, 0, -10, -3], [8, -2, 0, 6, -12, -4], [-33, 7, 4, -4, 1, -4], [7, -5, 0, 9, -12, -12], [4, -6, 8, -6, 8, 6], [-16, -4, 6, -4, 2, 8], [4, -14, 0, 12, 2, 2], [-17, 1, 0, 0, 8, 10], [17, -10, 12, 0, 4, 1], [-4, -6, 0, 4, -8, 4], [5, 9, -4, 1, -16, -4], [-11, -9, 10, -1, -8, -4], [-3, 3, 14, -17, 0, 12], [4, 4, 0, -6, 10, 10], [4, -14, 7, 0, 2, 9], [-10, -2, 1, -12, 16, 5], [34, -10, -8, 10, -6, -8], [-1, -3, -4, 0, 2, 2], [-2, 2, 12, 0, -4, 4], [-6, 4, 4, -10, 9, 4], [-19, -3, -5, -3, 13, 3], [11, -7, 7, 7, -6, -8], [-21, -3, 10, -15, 2, 2], [2, 14, 9, -12, -4, -3], [-20, 4, -2, -2, -2, 4], [-33, 9, 8, -7, 6, 2], [0, -10, -8, 20, -8, -12], [8, 0, 10, 0, -1, -6], [-1, 11, -5, 7, 5, -8], [7, 3, -12, -2, 18, -4], [4, 4, 2, 7, -12, -2], [1, 9, 7, 5, -8, 2], [-13, 15, -8, 12, 6, -4], [-5, 15, -9, 5, -4, 0], [-13, 11, -4, -14, -2, 0], [6, 1, -5, 8, 1, -6], [19, -1, -1, 11, -5, -8], [-2, 4, 16, -2, -13, 4], [4, -6, 2, 14, -10, 0], [-6, 20, -10, 3, 6, -4], [0, 2, -4, -6, 10, 6], [-7, 7, -18, 11, 0, -4], [14, -19, -2, 8, -4, 9], [9, -4, 8, 4, 4, -5], [-22, -2, 2, 4, -5, 6], [22, -4, 0, 8, -9, -14], [10, -6, -10, -6, 10, 0], [7, -1, 4, 3, 12, 12], [6, 0, 6, -6, 8, 6], [-14, -11, -4, 12, -2, -1], [-12, 14, -6, -14, 6, 0], [-4, -4, 5, -4, 3, 5], [-9, -4, -8, 0, 16, 5], [-12, 8, -2, -6, 9, 4], [-1, 14, -8, -4, 8, -5], [12, 12, -5, 0, 5, -1], [2, -8, -8, 0, 12, 2], [-14, 2, -2, -1, -16, -2], [-9, -14, 7, -11, -2, -3], [-16, 14, 2, -6, -5, 10], [14, -10, 7, -10, 9, 7], [-11, -5, 1, -5, -3, -4], [10, -16, -1, 2, -8, 0], [10, 12, -8, -4, 12, 8], [-7, -7, -13, -3, 12, 2], [-26, 3, 6, 6, -6, -3], [-28, 10, 2, 8, -6, -4], [32, 6, 4, -2, -2, 0], [-5, 3, 3, -21, 1, 3], [16, -4, -2, 10, -7, -12], [17, -17, 0, -9, -2, 2], [12, -8, 0, 6, 9, 2], [-10, -2, -4, -10, 10, 6], [-6, 0, 0, -12, 8, 2], [15, -5, -7, 7, -10, 2], [10, -10, -3, 12, -8, -12], [-6, 0, 8, 6, -10, -2], [1, 5, -20, 12, 0, 0], [2, -6, -7, 4, -1, 5], [12, -3, -11, 0, 3, -14], [-36, -5, 0, -2, 8, 1], [-12, 7, -12, 12, 2, -9], [-10, -4, -6, -2, 4, 0], [-21, -5, -3, 1, -6, -4], [-18, -4, -8, -8, 14, 2], [0, -6, 0, -14, 6, 0], [16, -6, 8, -2, -10, -4], [16, -4, 2, 10, -8, 0], [-7, 13, -6, -13, 4, 8], [-2, -4, 0, -2, -6, 0], [-19, -1, 16, -1, -14, 2], [-8, -8, 14, 12, 4, 4], [-1, -5, -11, 15, 7, -3], [5, 3, 19, -17, -13, -3], [19, -11, -5, 13, -8, -10], [-26, 6, 6, 0, 15, 10], [0, 2, 2, -4, -10, 6], [-8, 0, 16, 10, -3, -2], [6, 7, 12, -14, -4, 11], [6, 2, -10, 8, 0, -12], [-14, 10, -2, 2, -8, -6], [-18, -5, -4, 10, 4, -1], [-3, 19, -8, -16, 2, 6], [-24, -6, 3, 0, 5, 9], [2, -10, -2, 10, -20, -6], [-15, 1, 8, 6, 4, -4], [-4, 0, -8, 10, 4, -2], [-18, -6, 12, -6, -8, 2], [0, 15, -6, 8, 0, -15], [0, 14, 0, 0, 4, 0], [38, 18, -17, -6, 17, 3], [0, -8, -12, 16, -15, -8], [-18, -2, 18, -20, -6, 0], [-20, 2, -4, 8, -14, -10], [-9, -3, -16, 8, -1, -2], [16, -24, 4, 4, 0, 8], [-20, -14, -11, 6, 17, 3], [6, 0, -1, -14, -13, 1], [10, -2, -14, 16, 0, 2], [13, -17, 7, 5, -2, 8], [0, 0, -9, 10, -12, -8], [0, -2, -9, 0, 6, -11], [-6, 6, -24, 8, -8, -6], [3, 11, -5, 13, 4, 0], [-28, 8, -2, -14, 11, 4], [0, 4, 0, -10, 7, 4], [2, -8, 10, 0, -4, 20], [19, -11, -3, -7, -8, -2], [-28, 16, 7, -2, 8, 16], [0, -12, 12, 2, 2, 10], [-7, -5, -16, 3, 14, 10], [14, 0, 19, 0, -1, -3], [0, 12, -2, -7, 4, 14], [39, 3, 6, 9, -16, 0], [-40, -6, 0, -2, 6, 0], [-20, 18, -8, -26, 6, 6], [20, -14, -18, 7, 6, 0], [9, 15, 0, -4, -6, -6], [-28, 8, -10, -12, 5, 2], [-4, 10, -11, -2, 3, -5], [-6, 14, -4, -2, 12, 4], [-32, 6, 2, -10, -9, -2], [-16, -12, 0, 4, -16, -10], [-18, -22, 13, 4, -9, -7], [-20, 19, -16, -16, 2, -1], [-24, 9, -4, 4, -8, 3], [5, 3, 17, 9, -14, 4], [-4, -9, -24, 16, 10, -9], [-20, 2, 2, -19, 14, 8], [-34, 1, 4, 2, -4, -3], [-14, 8, 4, 2, -2, 0], [36, -24, 1, 0, -3, -3], [-9, 9, -5, -3, -9, -3], [-14, -9, 18, 6, -10, 5], [-19, 19, -11, -9, -7, 4], [-32, 10, 0, -16, 13, 8], [-16, 3, -14, -10, 10, 7], [-3, -7, 4, 0, 15, 0], [-2, -10, -5, 0, 16, -1], [-4, -7, 20, -20, -12, -1], [-19, 17, -14, 21, 8, -8], [-24, 2, 22, -22, -10, 4], [11, -9, 16, -6, 2, -4], [18, 14, -7, 0, 12, -12], [12, 18, -14, -4, 14, 8], [-10, -10, 14, -13, 8, 2], [-5, 3, 4, -8, 2, 8], [-9, 3, -12, -1, -2, -2], [-30, 0, 10, -16, -4, 6], [-4, -1, -18, 0, 4, 1], [-19, 5, -19, 3, -6, -16], [13, 13, -19, 27, 16, -4], [-20, 2, 20, -18, -8, 10], [-25, 15, -21, 11, -2, -2], [-17, -1, 12, -37, -8, 12], [8, -10, 20, 12, -4, 6], [1, -7, -15, 11, -6, -8], [-16, -8, -8, 14, -4, -12], [-16, 12, 10, -1, -4, -6], [-1, 13, -13, 15, 4, 0], [-17, 25, -9, -7, 0, -14], [-3, 13, -10, -23, 12, 8], [-16, 14, 4, 6, -8, 4], [-15, -13, 1, -1, 3, 0], [12, -3, 14, -24, 12, 15], [7, -17, 2, 7, -4, 4], [-22, -4, -5, 2, 11, -7], [30, -14, 26, -6, -5, 12], [32, -25, 0, -2, 2, 7], [2, -2, -4, 12, -16, 2], [-36, 8, 6, -12, -2, -2], [26, -9, -9, 12, 1, -16], [32, -14, -10, 32, -2, -10], [8, -22, 2, -3, -6, -12], [-29, 3, 1, -3, -14, -4], [-4, 4, -16, -14, 8, 10], [16, -9, -13, 0, 1, 6], [21, -19, 0, 13, 2, -14], [-44, 10, 6, -6, 0, 8], [-8, 10, 1, 0, -8, -9], [-31, -3, -7, 11, -2, 0], [-22, 2, -26, 20, 10, -4], [22, 2, 8, -18, 4, 4], [-6, -10, 16, 6, 6, 4], [9, -7, 2, 7, -14, -14], [39, -12, -12, 16, 8, -19], [0, 18, -6, 6, -3, 2], [34, 2, -6, 10, 0, -8], [-10, 2, 6, 0, 12, 10], [34, -2, -18, 8, -11, -6], [17, -17, -7, 13, 0, 8], [-42, 16, -2, 2, -6, -10], [16, 14, -20, -2, 13, 0], [-8, -9, 11, -12, -11, -6], [-4, -18, 17, -4, -5, 7], [-12, 12, -8, 2, 11, 6], [-11, 2, -4, 0, -20, -15], [-51, 2, 5, -3, -8, 5], [2, 10, 10, -20, 6, 6], [-8, 22, -6, 4, -10, -12], [26, -4, -1, 14, -3, 1], [28, 14, -8, 4, -1, 0], [-8, -14, 8, -2, -6, -2], [-31, 10, -11, 17, 0, -3], [8, -12, -12, 16, 8, 8], [-9, -10, 16, -8, -12, 19], [-51, 1, 0, -12, 9, 8], [28, 2, 17, -12, -15, -9], [14, -14, 20, 8, -1, 12], [-30, 14, 4, 4, -10, -12], [-16, 10, 1, -4, 20, 12], [18, -8, 17, 6, -3, -9], [-21, 23, 4, 7, -12, -16], [-30, -1, 0, 0, 14, 5], [16, 7, -8, 2, 18, -1], [14, -16, 16, 10, 0, 8], [-6, 8, 14, -21, -10, 12], [5, -7, -19, 29, -1, -16], [-20, 32, -14, -18, -1, -4], [-43, 11, 20, -20, 6, 6], [-13, 7, 3, -15, 20, 8], [-5, 2, -8, 0, -8, -13], [36, -6, 12, 26, -16, -10], [-1, 9, 0, 4, -15, -10], [0, 23, 10, -12, -8, 5], [-32, 24, 0, -10, 4, -2], [40, 2, 6, 8, -5, -8], [-18, -2, 4, -6, -16, -6], [11, -7, 1, -5, 12, -4], [-42, -2, 4, -8, 20, 4], [-16, 20, -16, -8, -6, 4], [18, -18, 0, -4, 8, 14], [24, 28, -4, -16, 4, -4], [12, -10, 11, 2, 5, 5], [-4, 8, -12, 12, 10, 2], [-18, 5, 2, 20, 0, -7], [9, -15, 1, -11, -1, 17], [48, -4, 10, 16, -14, -8], [22, -24, 10, 10, -2, -2], [-12, -4, -4, 10, -17, -12], [-21, -21, 0, 12, 16, 12], [-6, -12, -4, 8, -4, 4], [-22, 12, -4, 6, -12, -26], [9, -3, 12, 2, 22, 8], [29, -9, -7, 7, -7, 3], [31, 9, -14, 9, 16, 8], [-8, 12, -3, 2, 15, 1], [-23, -3, 2, -7, 12, 0], [-7, -9, 9, -27, 2, 12], [-18, 16, -10, 18, -4, -20]]
