# q-expansion of newform 8256.2.a.bk, downloaded from the LMFDB on 15 July 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 = 8256
weight = 2
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], [1], [1], [-1], [-3], [3], [0], [-5], [-4], [5], [-2], [10], [0], [1], [-3], [-4], [-4], [8], [-2], [0], [-16], [-14], [9], [14], [-7], [-6], [2], [-13], [-9], [-19], [8], [-6], [-5], [-10], [6], [-8], [2], [7], [-3], [2], [0], [-3], [18], [14], [-18], [-4], [15], [17], [-6], [-18], [11], [9], [-14], [3], [13], [-26], [12], [-12], [-2], [18], [-24], [-20], [-4], [-13], [-4], [8], [-11], [-5], [16], [-12], [-12], [0], [-6], [30], [0], [-4], [11], [-27], [-2], [28], [16], [32], [-39], [2], [14], [-15], [-15], [6], [-6], [-39], [-26], [33], [4], [26], [43], [-36], [0], [21], [35], [-10], [-32], [24], [36], [34], [-27], [6], [16], [42], [-39], [-2], [1], [-22], [8], [-10], [8], [-46], [-18], [26], [39], [-36], [-23], [-48], [22], [-4], [-20], [40], [-5], [36], [7], [20], [1], [-30], [17], [-38], [-2], [-13], [-35], [26], [42], [34], [-4], [12], [24], [-33], [18], [46], [-31], [-50], [-51], [22], [-14], [20], [6], [18], [-8], [-10], [38], [39], [-4], [38], [-45], [-19], [22], [-60], [4], [-46], [-9], [10], [-16], [-14], [-10], [32], [60], [-29], [16], [62], [-24], [-10], [-44], [-34], [56], [-4], [-16], [-17], [44], [27], [-16], [28], [45], [8], [-26], [22], [20], [47], [30], [-6], [-67], [21], [-34], [18], [38], [-17], [-4], [6], [-22], [61], [-49], [24], [-9], [38], [-18], [28], [-52], [20], [-6], [-65], [23], [-35], [50], [-2], [-32], [-2], [-17], [-26], [0], [-9], [10], [32], [53], [-48], [-4], [46], [-38], [-11], [-36], [-4], [-60], [-4], [6], [-9], [72], [53], [-48], [35], [-55], [35], [63], [77], [38], [65], [29], [20], [-3], [-6], [-78], [-22], [-40], [26], [-38], [-31], [-20], [10], [39], [-21], [-22], [-47], [59], [6], [61], [-24], [-34], [17], [-74], [12], [-17], [-7], [-11], [64], [-19], [0], [-39], [46], [40], [4], [59], [28], [63], [54], [-67], [82], [-20], [71], [70], [28], [13], [6], [68], [16], [-20], [80], [85], [-7], [-48], [-23], [44], [-80], [65], [-26], [-70], [50], [16], [51], [50], [45], [-81], [-10], [-29], [77], [26], [-51], [-28], [60], [-7], [40], [87], [-73], [-29], [16], [-39], [87], [80], [2], [-47], [-94], [26], [-3], [-72], [21], [4], [41], [-91], [6], [10], [85], [66], [-61], [-24], [-16], [-78], [-82], [22], [62], [-12], [29], [97], [81], [52], [46], [-58], [-49], [80], [-90], [-8], [-18], [-82], [32], [-89], [-94], [-48], [64], [-4], [0], [-51], [4], [-82], [42], [-52], [-14], [-23], [38], [-38], [18], [52], [31], [-6], [15], [55], [-31], [24], [48], [34], [-10], [30], [-68], [-22], [-41], [-78], [-34], [-1], [-5], [35], [-60], [-60], [-4], [-47], [-67], [40], [-50], [-32], [12], [-74], [33], [76], [0], [96], [-66], [21], [10], [12], [-18], [57], [-34], [-24], [-60], [-30], [-5], [-26], [-35], [78], [-94]]