
# q-expansion of newform 256.7.d.f, downloaded from the LMFDB on 23 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 = 256
weight = 7
poly_data = [16, 0, -7, 0, 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], 1], [[0, -3, 0, 1], 2], [[0, 22, 0, -2], 1], [[-112, 0, 32, 0], 1]]

hecke_ring_character_values = [[255, [-1, 0, 0, 0]], [5, [-1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [0, 0, -1, 0], [0, -5, 0, 0], [0, 0, 0, 5], [0, 0, 31, 0], [0, 733, 0, 0], [-4766, 0, 0, 0], [0, 0, 243, 0], [0, 0, 0, -169], [0, 12749, 0, 0], [0, 0, 0, -676], [0, -997, 0, 0], [-29362, 0, 0, 0], [0, 0, 695, 0], [0, 0, 0, 122], [0, 96427, 0, 0], [0, 0, 2531, 0], [0, -5459, 0, 0], [0, 0, -12721, 0], [0, 0, 0, 8589], [-288626, 0, 0, 0], [0, 0, 0, 5014], [0, 0, -6589, 0], [-310738, 0, 0, 0], [-1457086, 0, 0, 0], [0, 319579, 0, 0], [0, 0, 0, 22417], [0, 0, 37095, 0], [0, 767869, 0, 0], [-601694, 0, 0, 0], [0, 0, 0, -27024], [0, 0, 91807, 0], [-3810034, 0, 0, 0], [0, 0, -4481, 0], [0, 1637131, 0, 0], [0, 0, 0, -90265], [0, 408397, 0, 0], [0, 0, -59577, 0], [0, 0, 0, 128537], [0, -2563267, 0, 0], [0, 0, -75333, 0], [0, -4845781, 0, 0], [0, 0, 0, 184232], [-2430334, 0, 0, 0], [0, 1115323, 0, 0], [0, 0, 0, 79005], [0, 0, 126179, 0], [0, 0, 0, -53756], [0, 0, -435817, 0], [0, -797989, 0, 0], [-8043826, 0, 0, 0], [0, 0, 0, -181598], [5051042, 0, 0, 0], [0, 0, -152205, 0], [23455234, 0, 0, 0], [0, 0, 0, -350171], [0, -14713891, 0, 0], [0, 0, 0, 137406], [0, -13811317, 0, 0], [-8640082, 0, 0, 0], [0, 0, 411019, 0], [0, 22270747, 0, 0], [0, 0, -1578405, 0], [0, 0, 0, 807227], [-1127186, 0, 0, 0], [0, 17244077, 0, 0], [0, 0, -1299697, 0], [-34253086, 0, 0, 0], [0, 0, -1760581, 0], [0, 23500429, 0, 0], [-12723134, 0, 0, 0], [0, 0, 0, -326223], [0, 0, 0, 179218], [0, -343573, 0, 0], [0, 0, -479277, 0], [0, 0, 0, 542032], [0, 50560891, 0, 0], [0, -17413283, 0, 0], [-68839838, 0, 0, 0], [-45995858, 0, 0, 0], [0, 0, 875143, 0], [0, 47103899, 0, 0], [0, 0, 0, 837834], [84021026, 0, 0, 0], [0, 0, 0, -2390197], [0, 0, -2594045, 0], [-88092542, 0, 0, 0], [37542286, 0, 0, 0], [0, 57629981, 0, 0], [0, 0, 0, 166886], [0, 0, 5300259, 0], [0, 0, 0, -1252636], [0, 0, 0, -174399], [0, 0, 5972999, 0], [0, 0, -2206645, 0], [0, 0, 0, 212307], [0, 67093229, 0, 0], [198564878, 0, 0, 0], [0, 0, -6955617, 0], [0, 7209421, 0, 0], [0, 0, 1368871, 0], [0, 44542397, 0, 0], [0, 0, -7796837, 0], [25614062, 0, 0, 0], [0, 0, -3575453, 0], [107271554, 0, 0, 0], [0, 0, -698417, 0], [200340706, 0, 0, 0], [0, 0, 0, -2220385], [190305838, 0, 0, 0], [0, 0, 0, -2024620], [0, 49555547, 0, 0], [-37082674, 0, 0, 0], [0, 0, 13080167, 0], [0, 0, 0, -407709], [-421292798, 0, 0, 0], [0, 0, 2639135, 0], [0, 0, 0, -2973867], [0, 32192093, 0, 0], [0, 0, 17367059, 0], [0, -141948661, 0, 0], [277692866, 0, 0, 0], [0, 46151323, 0, 0], [0, 0, 8742039, 0], [0, 0, -621509, 0], [0, 176116781, 0, 0], [0, -231366469, 0, 0], [0, 0, 0, -7429434], [0, 0, 0, 7781039], [0, -254135027, 0, 0], [0, 0, 4123671, 0], [0, 0, 0, -4571935], [0, 0, 0, 3474530], [0, 151992683, 0, 0], [-863611282, 0, 0, 0], [-19789822, 0, 0, 0], [0, 282024187, 0, 0], [0, 0, 13045427, 0], [0, -10799923, 0, 0], [-651310258, 0, 0, 0], [0, 0, -17844489, 0], [0, 447432619, 0, 0], [0, 0, 0, 13627771], [0, 0, 1004067, 0], [0, -20272211, 0, 0], [0, 0, 0, -6742547], [0, 214497227, 0, 0], [-220597714, 0, 0, 0], [0, 0, 37041211, 0], [0, 0, 0, -1040844], [0, 257626237, 0, 0], [-118518878, 0, 0, 0], [0, 0, -4152789, 0], [0, 0, 0, 8229251], [0, 0, 17532095, 0], [0, 0, 0, 9675758], [0, 0, 0, -15596601], [722626498, 0, 0, 0], [412016206, 0, 0, 0], [0, 393833021, 0, 0], [0, 0, -17089669, 0], [-1183226386, 0, 0, 0], [0, 0, 0, 7724797], [0, 0, 22686447, 0], [-921322526, 0, 0, 0], [0, 0, 0, 3944143], [0, 0, 0, -28754268], [0, -683950117, 0, 0]]
