
# q-expansion of newform 2160.2.a.m, downloaded from the LMFDB on 23 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 = 2160
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], [0], [1], [-2], [-4], [-2], [5], [5], [-1], [-2], [-7], [-6], [0], [-4], [-4], [9], [-14], [-11], [-14], [0], [-12], [3], [1], [0], [16], [12], [-4], [12], [-19], [6], [-6], [18], [-17], [-12], [-16], [-8], [16], [14], [3], [-13], [4], [-19], [-14], [10], [-5], [16], [19], [10], [-3], [-29], [6], [6], [11], [24], [27], [24], [-10], [9], [-4], [-14], [-8], [7], [-22], [22], [8], [-27], [-12], [22], [-20], [-3], [26], [-18], [22], [-32], [23], [27], [-36], [-14], [-30], [25], [-36], [3], [-14], [26], [17], [-39], [6], [28], [42], [18], [13], [-28], [-40], [6], [25], [15], [28], [40], [-10], [22], [28], [-10], [-4], [-10], [5], [-16], [15], [-9], [-18], [-11], [36], [0], [33], [-32], [-13], [0], [-6], [-33], [41], [-18], [-42], [30], [-2], [9], [19], [-10], [26], [-46], [-32], [0], [15], [0], [3], [2], [-30], [-35], [-15], [40], [3], [-24], [-44], [46], [20], [39], [-2], [16], [-28], [21], [-41], [-33], [-32], [-54], [22], [23], [-12], [2], [-32], [-8], [-56], [34], [53], [6], [30], [-14], [2], [3], [55], [-48], [35], [-25], [4], [-17], [58], [-4], [-17], [30], [-55], [26], [-16], [-14], [28], [0], [50], [-38], [1], [20], [-58], [-32], [-31], [-34], [-34], [39], [-5], [56], [-31], [11], [-3], [24], [14], [-56], [-48], [-28], [14], [-21], [-6], [-46], [44], [-31], [36], [4], [-50], [-28], [-56], [-36], [-28], [45], [8], [56], [-57], [-23], [50], [-55], [34], [48], [-3], [31], [35], [40], [34], [-20], [4], [-3], [-32], [10], [-18], [-27], [9], [3], [50], [-12], [-4], [41], [-70], [61], [2], [10], [8], [-74], [-52], [-71], [66], [22], [33], [41], [29], [0], [-7], [-6], [-3], [-30], [-16], [-5], [3], [44], [59], [-28], [-8], [-78], [-54], [-26], [23], [2], [24], [1], [-4], [34], [11], [-19], [17], [24], [-72], [-3], [43], [19], [-76], [-10], [2], [-54], [15], [24], [-8], [-21], [18], [-66], [16], [60], [1], [54], [56], [-42], [24], [-15], [16], [1], [87], [-32], [12], [-5], [-36], [-16], [-55], [-60], [0], [-58], [-11], [-53], [-40], [72], [-2], [-38], [-17], [32], [60], [32], [-83], [54], [71], [88], [16], [-81], [50], [-62], [32], [-52], [-64], [36], [47], [-75], [-26], [12], [56], [57], [-86], [79], [-26], [32], [7], [-52], [-12], [39], [20], [30], [70], [16], [-70], [-35], [90], [-10], [-74], [19], [-10], [-98], [40], [90], [-60], [-2], [-63], [-74], [-46], [54], [-11], [37], [-36], [-63], [8], [18], [10], [-28], [30], [-10], [-28], [19], [-18], [33], [-60], [44], [35], [78], [12], [-40], [34], [-3], [-6], [-22], [-74], [-16], [79], [-18], [19], [36], [18], [30], [-46], [6], [102], [-27], [-26], [-42], [-28], [-78], [-74], [27], [-89], [-67], [-38], [18], [-22], [32], [75], [-99], [60], [58], [-43], [14], [6], [-33], [-21], [-18], [-17], [-80]]
