
# q-expansion of newform 64.6.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
    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 = 64
weight = 6
poly_data = [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 = None
aps_data = [[0], [12], [-54], [-88], [-540], [418], [594], [-836], [-4104], [594], [4256], [298], [17226], [12100], [-1296], [-19494], [7668], [34738], [-21812], [-46872], [67562], [-76912], [-67716], [29754], [-122398], [-11286], [-27256], [-122364], [-99902], [-29646], [336512], [-100980], [-317142], [148324], [-196614], [74360], [-120878], [111340], [-491832], [-707454], [-493668], [559450], [-724032], [7106], [530442], [56168], [339196], [779360], [744876], [272746], [-153846], [1154736], [657074], [-1341900], [132354], [943272], [-967518], [-518320], [-2222726], [-196614], [1552276], [1072170], [-1585892], [-730728], [584858], [2482866], [-377948], [639122], [2904660], [3991570], [1429218], [1161864], [-1089232], [-3505766], [-4043852], [5187456], [950346], [520738], [764370], [2640506], [4980204], [237994], [-3882384], [-66958], [-6501352], [4607604], [3776706], [-3180694], [-6685470], [-4351216], [-7079940], [3221856], [2297096], [-2821500], [4136284], [8332632], [-4341006], [-6741846], [7721956], [682066], [-2157716], [2675970], [3553308], [-12922470], [6083572], [-15824062], [-4602204], [8611218], [-7982280], [10174010], [-9958432], [-4195862], [9125514], [-6457340], [-14051368], [8471682], [-488564], [2481192], [5291298], [-4724676], [6174202], [-9409246], [-15008598], [12970692], [-22655588], [-19016910], [-15131126], [-15032304], [-7412296], [2779282], [12104620], [4469256], [28846256], [-9608678], [4545882], [-21592318], [14840010], [24878524], [-31607982], [-3100086], [-1871804], [20018394], [15311768], [-9593100], [-25220942], [-17762328], [486970], [-1926342], [-22353836], [18583776], [29104834], [-3146958], [-15999524], [-34587432], [-17439644], [-2595888], [-17641096], [39678498], [39341258], [-46250622], [37902492], [-26646246], [40979048], [27203364], [25355538], [11913912], [29193056], [17300074]]
