
# q-expansion of newform 1584.2.a.f, downloaded from the LMFDB on 25 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 = 1584
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], [-2], [4], [-1], [-6], [-2], [-4], [4], [-6], [0], [6], [6], [-4], [-12], [-2], [12], [-14], [-4], [-12], [-6], [4], [4], [-10], [-14], [-14], [0], [4], [-6], [-2], [-12], [4], [-2], [4], [10], [-4], [-10], [20], [0], [10], [20], [-2], [-12], [10], [2], [16], [4], [0], [12], [14], [-10], [8], [10], [-4], [-2], [-24], [-26], [-20], [26], [22], [-4], [-22], [-4], [4], [26], [-18], [-20], [18], [-4], [-6], [-18], [0], [16], [-14], [28], [-4], [30], [22], [38], [-14], [-12], [-10], [0], [2], [-4], [28], [22], [34], [-14], [24], [12], [0], [16], [-28], [4], [32], [22], [-18], [-20], [-38], [28], [-30], [-20], [-10], [20], [-46], [-36], [30], [36], [10], [-28], [-6], [22], [4], [8], [-42], [28], [-28], [-18], [-36], [-18], [26], [-46], [12], [12], [-30], [-18], [12], [0], [-22], [-44], [-32], [-32], [38], [-42], [10], [-18], [20], [54], [-10], [28], [2], [40], [-52], [46], [12], [-6], [-26], [-4], [20], [42], [14], [-4], [0], [-28], [12], [4], [-42], [-38], [42], [12], [6], [44], [12], [-26], [36], [-32], [-14], [10], [22], [-52], [-10], [-48], [-22], [-48], [38], [52], [-42], [-12], [-22], [-64], [36], [6], [54], [-36], [-26], [2], [-36], [-30], [-32], [-46], [60], [-60], [54], [-4], [22], [-30], [38], [-66], [24], [-46], [-52], [-2], [-30], [-36], [-2], [-56], [4], [6], [4], [42], [6], [56], [-60], [48], [-70], [68], [14], [4], [-34], [-38], [-4], [-10], [32], [44], [-30], [-10], [36], [-52], [-60], [62], [68], [-20], [-58], [-20], [48], [50], [42], [60], [28], [12], [-20], [8], [-10], [-2], [-16], [72], [-20], [-44], [8], [-14], [30], [-68], [-6], [-14], [-12], [-58], [-52], [6], [-78], [44], [-12], [66], [-6], [6], [-20], [-34], [54], [-12], [34], [-10], [12], [10], [-28], [-14], [16], [52], [2], [-38], [44], [-48], [-64], [0], [2], [36], [60], [-62], [18], [-48], [14], [-26], [-12], [-42], [-12], [10], [-14], [8], [-50], [12], [52], [-46], [2], [-4], [60], [20], [18], [84], [14], [12], [50], [24], [-42], [78], [-12], [40], [-38], [44], [-64], [66], [-18], [-84], [-38], [2], [-40], [86], [50], [-36], [-68], [-24], [2], [-30], [6], [-68], [-4], [60], [36], [-42], [-50], [42], [-28], [46], [78], [-70], [8], [54], [52], [-50], [28], [-40], [62], [-92], [26], [6], [-12], [-6], [38], [44], [-76], [-18], [28], [-70], [-58], [-20], [-28], [28], [-22], [-14], [12], [50], [36], [-84], [48], [-30], [60], [14], [-100], [-8], [98], [14], [-94], [22], [-18], [28], [-50], [-44], [-84], [80], [-74], [-36], [84], [82], [6], [-20], [36], [-4], [-94], [12], [62], [60], [-30], [-14], [54], [92], [-82], [-70], [52], [62], [30], [-4], [76], [-46], [-70], [-60], [12], [10], [-50], [-88], [32], [-74], [-96], [30], [-6], [-100], [-12], [26], [6], [-36], [70], [-4], [72]]
