# q-expansion of newform 8256.2.a.o, 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]]