
# q-expansion of newform 4864.2.a.f, downloaded from the LMFDB on 24 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 = 4864
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], [1], [-4], [1], [-5], [-1], [9], [5], [-2], [10], [12], [-10], [-12], [9], [-9], [-2], [-13], [-10], [-7], [-10], [8], [-6], [-6], [-4], [8], [-11], [15], [-6], [-6], [14], [-13], [2], [8], [8], [2], [-10], [-24], [-14], [12], [2], [-23], [-4], [-12], [5], [-13], [16], [-1], [14], [-10], [-19], [-16], [6], [-6], [-16], [-6], [-3], [6], [-20], [24], [-21], [-12], [33], [21], [31], [1], [-2], [22], [26], [-35], [3], [16], [-13], [-17], [8], [12], [-26], [36], [30], [-34], [-15], [18], [8], [-20], [20], [-20], [17], [-24], [40], [6], [-8], [12], [-20], [-10], [-11], [-6], [-32], [17], [4], [-4], [22], [-20], [0], [20], [3], [-48], [-42], [-28], [-34], [10], [-12], [2], [10], [8], [-24], [-34], [37], [46], [-19], [15], [-20], [15], [4], [-30], [20], [-10], [-3], [29], [-6], [20], [50], [20], [-2], [27], [1], [-9], [11], [5], [5], [9], [50], [3], [19], [7], [22], [-40], [12], [16], [-30], [-41], [10], [26], [18], [-51], [-30], [43], [13], [37], [-3], [-38], [-8], [32], [12], [-18], [-2], [10], [-50], [6], [54], [-15], [5], [-40], [-37], [32], [-18], [14], [-36], [-14], [16], [-9], [-43], [38], [30], [17], [0], [63], [4], [6], [17], [40], [-24], [-4], [-21], [-66], [18], [-39], [16], [-61], [15], [62], [-18], [13], [28], [-12], [36], [-39], [27], [-3], [-25], [13], [-62], [44], [-47], [66], [8], [21], [36], [-34], [24], [-46], [52], [18], [-19], [33], [20], [-54], [-48], [30], [6], [22], [-25], [36], [14], [42], [-25], [-2], [36], [0], [-56], [-44], [54], [-25], [65], [56], [56], [48], [-36], [-40], [-23], [-64], [-65], [-43], [-72], [-44], [44], [-58], [52], [-18], [22], [70], [36], [-56], [-19], [-18], [7], [-29], [27], [-19], [0], [49], [21], [21], [-33], [38], [11], [-24], [-83], [-50], [6], [-6], [-3], [-43], [49], [42], [56], [-7], [-34], [11], [64], [-66], [56], [-4], [0], [14], [42], [-42], [-18], [-19], [50], [14], [39], [1], [63], [80], [-34], [15], [23], [-32], [34], [-8], [-54], [-34], [-4], [-24], [0], [18], [-48], [-73], [-78], [-23], [-73], [-3], [64], [-29], [-22], [71], [17], [-12], [12], [-74], [-81], [9], [66], [0], [42], [35], [66], [19], [7], [75], [11], [21], [-42], [57], [8], [30], [-3], [-44], [36], [4], [4], [-64], [-26], [86], [72], [5], [54], [-21], [8], [48], [18], [24], [27], [-34], [-44], [-18], [2], [-8], [32], [53], [-23], [-75], [-80], [-20], [57], [20], [-66], [45], [-8], [90], [90], [-25], [-62], [-1], [-68], [47], [-90], [84], [-44], [16], [63], [-102], [34], [-72], [4], [-80], [-46], [-20], [64], [38], [-54], [42], [-60], [-38], [-71], [-20], [-36], [33], [-42], [26], [46], [-45], [34], [-63], [26], [-60], [82], [-9], [-12], [-22], [7], [27], [-87], [-2], [-37], [-61], [-45], [40], [-63]]
