# q-expansion of newform 8256.2.a.r, 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], [2], [-2], [0], [-2], [6], [4], [-6], [2], [-4], [-4], [-2], [1], [-6], [4], [-8], [12], [4], [0], [-14], [-8], [4], [10], [-2], [12], [-12], [20], [-10], [18], [0], [-12], [-10], [-20], [-18], [-2], [24], [0], [6], [12], [4], [-6], [8], [14], [-12], [-14], [-16], [14], [-28], [10], [-6], [-10], [-10], [-24], [-18], [12], [-4], [-4], [-12], [-22], [-4], [-4], [28], [26], [14], [-8], [16], [-26], [4], [-24], [2], [-34], [-32], [-4], [-20], [16], [14], [26], [-10], [-26], [28], [-20], [22], [38], [-28], [-36], [-6], [22], [-12], [38], [-12], [6], [12], [-20], [28], [-12], [16], [-26], [-28], [-38], [-12], [-12], [-12], [-42], [24], [-22], [-28], [18], [-22], [-30], [-46], [46], [18], [20], [38], [30], [-36], [12], [-26], [8], [-46], [-10], [-22], [-40], [8], [36], [38], [-22], [2], [-36], [-12], [-36], [-14], [48], [26], [30], [-46], [4], [-20], [-26], [44], [24], [-16], [48], [-4], [-12], [42], [-42], [-48], [0], [-10], [10], [4], [-8], [-4], [-12], [52], [-34], [50], [36], [40], [-30], [-12], [-12], [6], [-20], [-22], [-16], [-2], [-24], [44], [-44], [-16], [26], [10], [-62], [40], [30], [60], [-28], [-22], [-12], [28], [-50], [60], [54], [-40], [32], [22], [12], [34], [36], [60], [-34], [36], [-50], [34], [26], [-22], [24], [-16], [-34], [52], [50], [-12], [-22], [-30], [44], [46], [-4], [-2], [16], [40], [0], [-4], [26], [-22], [-18], [24], [-48], [8], [28], [-14], [4], [4], [-2], [34], [-12], [62], [28], [12], [-52], [56], [6], [12], [66], [14], [-36], [4], [42], [44], [28], [8], [54], [-2], [-42], [38], [48], [28], [-50], [-46], [-14], [-46], [-14], [-2], [-36], [-4], [28], [66], [18], [-46], [36], [14], [6], [-2], [-36], [6], [26], [16], [-44], [62], [-28], [-30], [14], [38], [58], [32], [8], [70], [12], [46], [4], [42], [-44], [8], [0], [-14], [34], [62], [2], [24], [60], [-6], [12], [26], [-84], [68], [36], [-84], [52], [-78], [-46], [-40], [-60], [16], [-58], [12], [-16], [-64], [-44], [-20], [-50], [-74], [80], [42], [-2], [-4], [82], [14], [-18], [-20], [-58], [-46], [16], [26], [-34], [-8], [28], [18], [-34], [68], [-88], [2], [-12], [-68], [48], [-4], [2], [-62], [38], [-30], [58], [54], [2], [72], [24], [24], [-12], [0], [8], [-76], [62], [44], [-74], [18], [42], [4], [20], [6], [30], [-32], [46], [-36], [60], [-36], [-94], [-82], [-44], [46], [-36], [56], [-50], [90], [-60], [-20], [60], [42], [46], [10], [-26], [-32], [22], [36], [98], [52], [-102], [-102], [-22], [-52], [-90], [-58], [-26], [-20], [76], [4], [-22], [56], [-26], [32], [62], [70], [-66], [-28], [102], [-6], [30], [-8], [-10], [-4], [16], [70], [2], [-44], [-4], [-6], [-16], [-30], [88], [-58], [76], [-78], [26], [-36], [-4], [-2], [-58], [-76], [-70], [-52], [84]]