# q-expansion of newform 85.4.d.a, downloaded from the LMFDB on 26 June 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 = 85
weight = 4
poly_data = [1, 0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = [[52, [1, 0]], [71, [-1, 0]]]
aps_data = [[1, 0], [0, 8], [0, 5], [0, -14], [0, 20], [-58, 0], [-17, 68], [80, 0], [0, 118], [0, 126], [0, 70], [0, -134], [0, -100], [272, 0], [-464, 0], [642, 0], [-180, 0], [0, 110], [-924, 0], [0, 90], [0, 828], [0, -1334], [552, 0], [1490, 0], [0, 1376], [642, 0], [-1248, 0], [0, -564], [0, -1214], [0, 1388], [1136, 0], [0, -2620], [1886, 0], [0, -4], [870, 0], [-1368, 0], [-3374, 0], [0, 1048], [0, 1266], [0, -362], [1720, 0], [0, 1910], [-4208, 0], [0, -2112], [0, -2074], [0, 4466], [0, 2540], [3832, 0], [0, 6096], [2650, 0], [0, 4228], [-80, 0], [0, 5980], [312, 0], [-434, 0], [1752, 0], [0, 1206], [5112, 0], [0, -3154], [4702, 0], [0, 1388], [-7318, 0], [216, 0], [0, -7170], [0, -1472], [0, 1526], [-2708, 0], [0, 10516], [0, -2704], [9670, 0], [-7938, 0], [2240, 0], [0, -6594], [-4258, 0], [0, -8984], [2912, 0], [-7950, 0], [0, -12634], [0, 10740], [2950, 0], [0, -5944], [13682, 0], [0, 410], [-2558, 0], [0, 5946], [10852, 0], [0, -3284], [4106, 0], [8462, 0], [3992, 0], [-3164, 0], [0, 4866], [0, 2926], [-15128, 0], [0, -18684], [0, -8562], [16950, 0], [0, -17820], [-13548, 0], [0, 7970], [0, -22044], [20166, 0], [1712, 0], [-7270, 0], [0, 26200], [-1794, 0], [27876, 0], [9362, 0], [13600, 0], [0, -5200], [0, -21854], [-3558, 0], [0, -804], [0, -4344], [-24488, 0], [0, -15500], [0, 10928], [12416, 0], [0, -8782], [6020, 0], [6342, 0], [0, -18632], [0, 21186], [0, 15408], [0, 3940], [-14798, 0], [0, 14926], [0, 7146], [19136, 0], [37922, 0], [-34480, 0], [0, -7062], [0, -16630], [-3314, 0], [22902, 0], [-22690, 0], [-2418, 0], [0, -8484], [5346, 0], [0, 11236], [0, 28880], [0, -21110], [0, 33958], [0, 8436], [-8210, 0], [0, 4126], [0, 20838], [0, -2864], [-11980, 0], [-13728, 0], [0, 8506], [0, 48600], [-24668, 0], [0, 14086], [0, -52364], [0, -34570], [45480, 0], [0, -1284], [10166, 0], [0, 47310], [0, -1284], [40602, 0], [-39624, 0], [8772, 0], [-17974, 0], [0, 47558], [0, 5490], [0, 45286]]