
# q-expansion of newform 8281.2.a.f, downloaded from the LMFDB on 19 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 = 8281
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 = [[-1], [0], [0], [0], [3], [0], [7], [7], [-6], [-5], [0], [-8], [0], [2], [-7], [-3], [7], [-7], [3], [5], [-14], [-6], [0], [0], [14], [-14], [14], [8], [-4], [9], [2], [14], [-4], [0], [6], [3], [7], [13], [7], [7], [-10], [-7], [-20], [-4], [-2], [-14], [-26], [-21], [28], [-14], [-27], [19], [-28], [-14], [-14], [-24], [-21], [7], [-17], [12], [-14], [-14], [-21], [0], [-14], [6], [20], [23], [4], [-14], [-14], [-8], [14], [15], [12], [0], [-3], [-14], [-22], [28], [14], [-30], [24], [21], [14], [-20], [12], [6], [28], [-16], [14], [-7], [-25], [30], [-8], [-28], [28], [-14], [-14], [-8], [2], [-18], [28], [1], [4], [-14], [-21], [0], [32], [-7], [14], [-32], [-30], [28], [-16], [18], [7], [-42], [-6], [-40], [0], [-26], [-35], [24], [-35], [30], [-50], [-42], [28], [-42], [-4], [-9], [-20], [9], [0], [14], [-42], [-7], [-42], [-31], [28], [-36], [4], [5], [35], [-7], [42], [21], [-56], [20], [48], [42], [-40], [42], [-38], [-54], [50], [-14], [7], [-28], [27], [9], [5], [-28], [38], [7], [4], [-7], [-16], [-49], [3], [0], [22], [52], [56], [-21], [-9], [-60], [28], [7], [13], [-42], [9], [-28], [-53], [-42], [25], [-28], [48], [-21], [21], [33], [50], [42], [-66], [21], [24], [-34], [-14], [-14], [-32], [28], [28], [-70], [-28], [-42], [-35], [-6], [-30], [42], [29], [-7], [16], [-63], [-56], [-70], [4], [-21], [-15], [-2], [-55], [28], [6], [-8], [-42], [-5], [-42], [60], [42], [69], [-31], [-14], [-23], [-45], [-14], [28], [42], [-22], [30], [-42], [45], [-56], [56], [62], [-56], [42], [7], [63], [-25], [16], [68], [0], [45], [-21], [63], [-36], [-17], [49], [21], [42], [-10], [2], [14], [-14], [14], [-14], [40], [-42], [12], [11], [42], [-11], [28], [-34], [14], [21], [27], [52], [34], [-84], [-42], [-32], [70], [-14], [21], [36], [11], [54], [35], [49], [-3], [0], [-22], [-77], [2], [21], [-28], [21], [-70], [28], [49], [48], [-24], [-68], [8], [-16], [-66], [-35], [-15], [85], [14], [-18], [50], [-10], [-23], [42], [-84], [31], [42], [-61], [28], [48], [-63], [-23], [-46], [-21], [20], [-49], [78], [-89], [-8], [74], [-28], [-63], [-25], [-7], [-2], [70], [56], [70], [-74], [-47], [-42], [-68], [-36], [-12], [-21], [27], [21], [42], [56], [-60], [-16], [-28], [43], [42], [49], [7], [1], [-37], [12], [63], [4], [55], [-42], [1], [-28], [-81], [-33], [60], [56], [41], [-54], [-56], [22], [-21], [72], [56], [-42], [63], [42], [26], [-51], [-88], [35], [-28], [-67], [7], [-43], [70], [-16], [0], [-4], [28], [41], [67], [63], [-21], [16], [-74], [70], [-90], [-71], [28], [84], [0], [38], [-44], [-49], [-49], [-7], [78], [2], [64], [9], [-35], [48], [-42], [-28], [-28], [-66], [-56], [96], [14], [-56], [0], [-20], [-30], [-28], [42]]
