
# q-expansion of newform 616.2.a.f, downloaded from the LMFDB on 22 January 2026.

# 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
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_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 = 616
weight = 2
poly_data = [-4, -1, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = None
aps_data = [[0, 0], [0, -1], [-2, 1], [-1, 0], [1, 0], [-2, 2], [-2, 0], [0, -2], [-4, 1], [-2, 0], [-8, -1], [2, 1], [-10, 0], [4, 0], [4, -4], [6, -4], [-8, 1], [6, -4], [0, -1], [-4, 3], [6, -4], [0, 2], [-8, 0], [-10, -1], [6, 1], [-2, 6], [-4, 8], [12, 2], [6, -6], [-2, 3], [0, -2], [8, 2], [-10, 3], [0, -2], [-10, 0], [0, 2], [14, 1], [-4, 8], [8, -6], [-18, -2], [-8, -5], [-10, 7], [-4, -9], [2, 0], [-10, -4], [-4, -4], [20, 2], [-24, -1], [0, -4], [-10, 3], [18, -8], [-8, 0], [-10, 6], [-8, 9], [-14, 0], [-16, 10], [18, -4], [8, 4], [14, -2], [-6, 14], [8, 8], [-2, -4], [0, 8], [-12, 12], [-2, 7], [10, -9], [0, 5], [26, 0], [12, -2], [-2, -4], [-18, -1], [-8, 4], [-8, 3], [14, -10], [-8, -7], [-8, 7], [-6, -9], [10, -8], [2, 0], [14, 8], [-12, 12], [-10, 16], [16, -12], [6, -3], [-8, 10], [-16, 11], [-10, -3], [2, 14], [14, 0], [4, -17], [8, -15], [0, 8], [-20, 1], [-12, -2], [-12, -8], [-16, 12], [-2, -5], [30, -3], [-8, 4], [-10, -2], [4, 12], [30, 4], [0, -12], [34, 0], [-20, -8], [14, -3], [4, 0], [-2, 4], [16, 0], [38, 2], [8, -22], [-2, 10], [-6, -8], [40, -5], [-20, -7], [22, 5], [0, 17], [-8, -11], [10, 13], [4, -10], [14, -19], [18, 6], [-18, -4], [-12, -8], [0, -11], [-10, 18], [-30, 13], [-8, -13], [16, -5], [-18, -2], [-36, 6], [16, 0], [-4, -13], [14, -16], [6, -18], [-34, 6], [34, -8], [-16, 2], [30, -7], [10, -12], [0, -2], [-26, -4], [-4, -19], [-4, 20], [22, -11], [0, 3], [46, 0], [38, -2], [24, 5], [0, 20], [-34, -4], [-2, -7], [-52, 0], [-16, -6], [12, -4], [0, 0], [-8, -12], [-14, 0], [-18, 2], [-26, 12], [0, 7], [2, -22], [-32, -2], [-24, -9], [-34, 15], [40, 3], [-24, 8], [-2, 6]]
