
# q-expansion of newform 2592.2.a.m, downloaded from the LMFDB on 16 May 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, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_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 = 2592
weight = 2
poly_data = [-6, 0, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0], 1], [[0, 2], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0], [0, 0], [-1, 0], [0, 1], [0, -1], [3, 0], [-5, 0], [0, -1], [0, -1], [-5, 0], [0, 0], [-5, 0], [2, 0], [0, -1], [0, 2], [-2, 0], [0, 2], [-13, 0], [0, -1], [0, 1], [3, 0], [0, -3], [0, 2], [-13, 0], [-6, 0], [-10, 0], [0, 2], [0, 4], [-9, 0], [-1, 0], [0, -1], [0, -3], [-17, 0], [0, 2], [-5, 0], [0, 2], [7, 0], [0, -2], [0, 3], [-1, 0], [0, -4], [-18, 0], [0, 1], [19, 0], [23, 0], [0, -4], [0, -1], [0, 1], [0, 1], [3, 0], [-29, 0], [0, -2], [27, 0], [0, -3], [7, 0], [0, 2], [-1, 0], [0, 1], [-18, 0], [-13, 0], [0, 4], [-5, 0], [0, 4], [0, 0], [-1, 0], [23, 0], [0, -5], [-6, 0], [0, 7], [14, 0], [2, 0], [0, 3], [0, 4], [-34, 0], [0, 0], [0, -3], [-2, 0], [-13, 0], [-29, 0], [-9, 0], [0, 4], [15, 0], [0, -2], [19, 0], [0, -6], [0, 2], [26, 0], [15, 0], [22, 0], [0, 2], [0, 2], [0, -5], [0, -4], [0, -7], [0, 5], [0, -8], [-26, 0], [34, 0], [0, -9], [27, 0], [0, 6], [-17, 0], [0, -2], [-17, 0], [0, 2], [-1, 0], [0, 3], [-17, 0], [0, -4], [-5, 0], [0, -5], [14, 0], [11, 0], [0, 0], [0, -6], [43, 0], [0, 6], [0, -1], [-10, 0], [0, -1], [-29, 0], [11, 0], [-34, 0], [0, 10], [0, 1], [-17, 0], [27, 0], [0, 9], [0, -1], [6, 0], [0, 8], [0, 8], [0, 3], [30, 0], [23, 0], [11, 0], [-41, 0], [0, 3], [23, 0], [23, 0], [0, 0], [11, 0], [0, 0], [0, -1], [-42, 0], [0, 3], [-2, 0], [-25, 0], [0, 6], [0, -5], [35, 0], [10, 0], [0, -6], [0, 3], [0, -4], [0, 1], [0, -7], [-5, 0], [19, 0], [-41, 0], [0, -3], [7, 0], [0, -1], [0, 1], [-22, 0], [0, 4], [0, -7], [-25, 0], [-45, 0], [14, 0], [0, 0], [19, 0], [0, -10], [-45, 0], [0, 3], [-46, 0], [0, 3], [-17, 0], [0, 2], [-5, 0], [0, 10], [0, -11], [39, 0], [2, 0], [0, -2], [-25, 0], [63, 0], [0, 3], [42, 0], [0, -5], [15, 0], [0, -12], [0, -7], [-17, 0], [0, -3], [-38, 0], [-1, 0], [54, 0], [-1, 0], [0, -3], [35, 0], [0, -5], [6, 0], [15, 0], [0, 5], [-10, 0], [0, 9], [0, -2], [34, 0], [0, 9], [-22, 0], [35, 0], [0, 1], [0, -9], [0, 0], [-14, 0], [0, 1], [-61, 0], [0, 6], [-37, 0], [-9, 0], [0, -13], [-29, 0], [0, -14], [0, -12], [-65, 0], [-62, 0], [0, -5], [0, -14], [0, 2], [-33, 0], [0, 9], [0, -9], [-5, 0], [0, -4], [0, 6], [75, 0], [-26, 0], [0, -4], [0, 7], [0, 9], [0, 11], [0, -2], [-26, 0], [-38, 0], [0, 7], [0, -11], [0, 5], [0, 10], [0, -8], [27, 0], [55, 0], [0, -2], [3, 0], [11, 0], [0, 9], [3, 0], [0, 1], [43, 0], [-14, 0], [0, 0], [0, 8], [39, 0], [46, 0], [-46, 0], [0, 12], [-25, 0], [-5, 0], [0, -8], [31, 0], [-25, 0], [0, 1], [18, 0], [0, -8], [39, 0], [0, -10], [0, 0], [-25, 0], [35, 0], [0, 2], [0, 2], [0, -12], [0, -16], [-61, 0], [0, 6], [0, -16], [3, 0], [-2, 0], [0, 9], [-1, 0], [38, 0], [0, -1], [11, 0], [0, 0], [-5, 0], [31, 0], [0, 0], [-49, 0], [0, 5], [0, -10], [-41, 0], [-2, 0], [0, -3], [0, 7], [0, -4], [42, 0], [0, 1], [-42, 0], [0, 16], [83, 0], [0, 11], [62, 0], [-37, 0], [0, -5], [0, -14], [-78, 0], [0, -6], [0, -8], [-9, 0], [58, 0], [0, -18], [-62, 0], [79, 0], [0, 12], [34, 0], [27, 0], [0, 5], [0, -15], [0, 1], [62, 0], [6, 0], [71, 0], [0, 8], [0, -13], [0, 5], [0, 18], [-5, 0], [79, 0], [-30, 0], [0, 14], [-41, 0], [2, 0], [95, 0], [0, 4], [-89, 0], [0, 3], [-69, 0], [0, -4], [0, 4], [-29, 0], [0, 13], [39, 0], [-49, 0], [0, 11], [31, 0], [-5, 0], [0, 14], [0, -11], [58, 0], [0, 1], [-77, 0], [-37, 0], [0, 7], [0, 2], [0, 6], [11, 0], [22, 0], [0, -8], [26, 0], [0, -13], [0, 9], [0, 0], [67, 0], [0, -18], [-33, 0], [0, -11], [0, -2], [-17, 0], [34, 0], [63, 0], [-85, 0], [23, 0], [0, -6], [79, 0], [0, -13], [0, -5], [0, -16], [-93, 0], [0, 9], [0, -8], [-33, 0], [-49, 0], [0, -9], [0, 21], [0, 10], [-57, 0], [0, -5], [71, 0], [0, 5], [-85, 0], [-49, 0], [-22, 0], [0, -1], [-49, 0], [31, 0], [0, -17], [-21, 0], [-101, 0], [0, -11], [0, 12], [-25, 0], [43, 0], [0, 5], [0, 14], [51, 0], [-53, 0], [0, -12], [0, -16], [-13, 0], [0, -18], [-50, 0], [22, 0], [0, 2], [0, -8], [31, 0], [-5, 0], [0, 4], [67, 0], [0, -8], [0, -18]]
