
# q-expansion of newform 882.2.e.m, 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, 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 = 882
weight = 2
poly_data = [4, 0, -2, 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, 0, 0], 1], [[0, 1, 0, 0], 1], [[0, 0, 1, 0], 2], [[0, 0, 0, 1], 2]]

hecke_ring_character_values = [[785, [-1, 0, 1, 0]], [199, [-1, 0, 1, 0]]]
aps_data = [[1, 0, 0, 0], [0, -1, -1, 1], [0, 1, -1, 1], [0, 0, 0, 0], [-2, 0, 2, 0], [0, -2, 0, 4], [0, 0, -2, 0], [-5, 1, 5, -2], [0, 0, 1, 0], [0, 2, 2, 2], [6, 0, 0, 0], [-2, -4, 2, 8], [0, -4, 0, 8], [0, -2, -2, -2], [0, 8, 0, -4], [0, 2, 6, 2], [-2, 0, 0, 0], [9, -2, 0, 1], [-8, -4, 0, 2], [5, -4, 0, 2], [0, 2, 2, 2], [3, -4, 0, 2], [0, 0, -2, 0], [12, 2, -12, -4], [0, -2, 2, -2], [5, 5, -5, -10], [0, 0, -14, 0], [12, 0, -12, 0], [0, -6, 2, -6], [11, -2, -11, 4], [-3, 0, 0, 0], [0, -1, 11, -1], [2, -4, -2, 8], [-7, 1, 7, -2], [0, 0, 6, 0], [5, 0, -5, 0], [-1, -6, 0, 3], [10, 4, -10, -8], [-4, 6, 4, -12], [-8, 4, 0, -2], [0, -6, -6, -6], [-3, -6, 0, 3], [9, -4, 0, 2], [-13, -4, 0, 2], [2, 12, 0, -6], [0, 2, -2, 2], [-8, -2, 8, 4], [0, -2, 16, -2], [3, 1, -3, -2], [0, -5, 11, -5], [7, 0, -7, 0], [3, -4, -3, 8], [4, -2, -4, 4], [15, 2, 0, -1], [0, 4, -18, 4], [-21, -2, 21, 4], [0, 3, 11, 3], [-12, 2, 12, -4], [4, 6, -4, -12], [0, 0, 19, 0], [-23, -2, 0, 1], [-15, 5, 15, -10], [13, 10, 0, -5], [16, 12, 0, -6], [10, -12, 0, 6], [6, 12, 0, -6], [-10, 12, 0, -6], [-6, 12, 6, -24], [0, 16, 0, -8], [0, 2, -16, 2], [6, 0, -6, 0], [-1, 4, 1, -8], [4, 4, -4, -8], [0, 2, 2, 2], [-2, 20, 0, -10], [0, 2, -2, 2], [20, 2, -20, -4], [-24, -6, 24, 12], [0, 2, 15, 2], [-4, -8, 0, 4], [-27, -1, 27, 2], [0, -2, -18, -2], [0, -8, -12, -8], [2, -8, 0, 4], [12, -8, 0, 4], [-10, -4, 0, 2], [1, 16, 0, -8], [7, -20, 0, 10], [0, 3, -13, 3], [11, -6, -11, 12], [-10, 0, 10, 0], [10, -8, -10, 16], [0, 12, 7, 12], [-6, -4, 6, 8], [0, 12, -4, 12], [0, -20, 0, 10], [0, -2, 12, -2], [0, -6, 24, -6], [7, 3, -7, -6], [-6, 10, 6, -20], [0, 8, -12, 8], [0, 2, 8, 2], [-13, -22, 0, 11], [-30, 0, 0, 0], [24, 8, 0, -4], [0, 8, -4, 8], [0, -7, 1, -7], [0, -6, 0, 12], [-24, -8, 0, 4], [0, -6, -2, -6], [0, -6, -6, -6], [0, 6, 0, 6], [-14, 12, 14, -24], [-13, -7, 13, 14], [23, 4, 0, -2], [-17, 10, 17, -20], [10, 12, -10, -24], [0, 6, 36, 6], [0, -4, 0, -4], [0, 6, 10, 6], [7, -2, 0, 1], [0, 8, -11, 8], [0, 12, 0, -6], [0, -4, -42, -4], [29, 18, 0, -9], [22, -24, 0, 12], [8, 16, 0, -8], [0, -4, 0, 8], [0, 10, -16, 10], [0, 1, 15, 1], [0, -8, 6, -8], [-36, 0, 36, 0], [0, 8, -21, 8], [-6, -24, 0, 12], [0, 0, -2, 0], [-10, 18, 10, -36], [0, -11, -7, -11], [18, -24, 0, 12], [-9, 11, 9, -22], [0, -4, 26, -4], [-2, 0, 0, 0], [20, -16, 0, 8], [16, -24, 0, 12], [-12, -20, 0, 10], [0, 6, -16, 6], [0, -2, 40, -2], [0, -13, 7, -13], [30, 2, -30, -4], [0, 0, 10, 0], [-7, -2, 7, 4], [0, -10, 2, -10], [0, 16, 0, -8], [-10, -8, 0, 4], [0, -4, 24, -4], [12, -6, -12, 12], [0, -20, -3, -20], [11, 6, -11, -12], [0, 28, 0, -14], [26, 16, 0, -8], [1, -2, 0, 1], [28, 20, 0, -10], [26, -24, 0, 12], [-5, -8, 5, 16], [27, -11, -27, 22], [-18, -16, 0, 8], [-6, 16, 6, -32], [0, 4, -8, 4], [-35, -7, 35, 14]]
