
# q-expansion of newform 5184.2.a.m, 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 = 5184
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], [-2], [-1], [3], [-2], [6], [-1], [-8], [-1], [-2], [-10], [4], [10], [-4], [-9], [14], [10], [-9], [10], [-12], [11], [-2], [6], [-12], [-16], [-5], [-17], [-2], [-14], [-9], [20], [-21], [12], [-17], [4], [-2], [-21], [0], [-14], [18], [-9], [11], [16], [-2], [2], [-6], [7], [15], [12], [-5], [18], [-29], [-12], [15], [-6], [2], [-9], [-16], [-21], [0], [0], [7], [3], [14], [14], [-10], [-34], [2], [-34], [-24], [10], [-8], [-14], [18], [-13], [-25], [-29], [40], [7], [12], [19], [-28], [-12], [14], [-37], [-30], [28], [28], [-42], [-16], [34], [-6], [0], [18], [-18], [30], [-13], [12], [11], [-36], [-5], [-20], [27], [30], [15], [-32], [-17], [22], [-6], [15], [48], [-20], [35], [-36], [6], [34], [30], [31], [23], [34], [36], [10], [-45], [-1], [-14], [6], [-14], [-8], [-24], [22], [42], [39], [31], [-9], [-18], [39], [-37], [16], [23], [8], [22], [-38], [22], [42], [-13], [44], [46], [35], [-42], [20], [54], [-8], [-30], [42], [27], [-25], [43], [-54], [-21], [-18], [-38], [-2], [24], [-38], [15], [43], [-14], [24], [-21], [-4], [-5], [-58], [38], [-58], [-45], [-52], [59], [28], [-38], [59], [-26], [36], [-5], [-45], [14], [26], [30], [-5], [-56], [-22], [55], [-22], [42], [55], [-22], [7], [34], [63], [22], [22], [15], [-14], [-54], [50], [-28], [-10], [26], [14], [15], [42], [6], [64], [-42], [-38], [23], [44], [15], [-5], [-50], [-57], [-4], [-24], [-5], [-18], [-66], [44], [4], [47], [-46], [-10], [15], [-64], [68], [-13], [54], [24], [38], [-30], [6], [36], [42], [14], [-34], [2], [-54], [28], [16], [-57], [7], [-12], [15], [-73], [18], [-9], [58], [-57], [-2], [32], [0], [-41], [-34], [18], [16], [19], [-17], [-16], [27], [31], [58], [-58], [-24], [-65], [-36], [16], [-17], [-17], [-4], [60], [40], [-56], [-33], [4], [24], [31], [-22], [-46], [-13], [-54], [-42], [15], [8], [87], [-21], [-64], [-57], [18], [-28], [43], [62], [34], [46], [-56], [-26], [18], [-30], [-24], [19], [6], [58], [31], [-74], [-12], [-50], [60], [-16], [-29], [-66], [84], [-58], [-49], [-88], [-66], [-29], [-78], [42], [-22], [34], [26], [-37], [16], [-34], [34], [28], [15], [-37], [-58], [-12], [-5], [38], [19], [32], [71], [78], [-93], [56], [-80], [23], [-94], [-25], [-53], [-42], [15], [75], [-36], [18], [22], [2], [-85], [75], [-34], [28], [-92], [79], [-78], [-88], [-38], [30], [-38], [48], [-21], [68], [-65], [-30], [-76], [23], [18], [-81], [87], [-41], [-28], [-9], [14], [62], [72], [19], [74], [-64], [-85], [-53], [78], [2], [92], [39], [94], [27], [-38], [23], [-1], [102], [70], [-93], [31], [38], [91], [79], [-6], [32], [7], [-25], [-78], [36], [-5], [43], [32], [-64], [-1], [52], [62], [-26], [-68], [48], [23], [-37], [-48], [43], [-16], [-60]]
