
# q-expansion of newform 882.2.f.m, downloaded from the LMFDB on 20 February 2026.

# 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 = [3, -12, 19, -15, 10, -3, 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, 0, 0], 1], [[0, 1, 0, 0, 0, 0], 1], [[6, 0, 1, 5, -1, 1], 3], [[0, -3, 2, -5, 1, -1], 3], [[6, -21, 19, -16, 5, -2], 3], [[-9, 33, -22, 19, -5, 2], 3]]

hecke_ring_character_values = [[785, [0, 0, 0, 0, 1, 0]], [199, [1, 0, 0, 0, 0, 0]]]
aps_data = [[-1, 0, 0, 0, -1, 0], [1, -1, -1, 0, 0, 0], [0, -1, 0, 0, -2, 1], [0, 0, 0, 0, 0, 0], [0, -1, 1, -1, 0, -1], [0, -1, 0, 0, -1, 1], [-2, 2, 4, 0, 2, 0], [0, -1, 0, 2, -1, 0], [1, 0, 0, 1, 2, -2], [-1, 3, 2, 3, -1, -2], [1, -3, 0, 1, -6, 1], [1, 2, 0, -4, 2, 0], [1, -2, 0, 1, -5, 0], [5, -2, -3, -2, 5, 3], [-2, -2, -3, -2, -2, 3], [-1, -1, 3, 5, -1, 0], [1, -6, 0, 1, -1, 4], [-2, -5, -2, -5, -2, 2], [-3, 2, 0, -3, 0, 4], [4, -3, -4, 2, -3, 0], [-6, -3, -2, 4, -3, 0], [0, -3, -7, -3, 0, 7], [-3, 1, -1, 1, -3, 1], [-2, -2, -3, 1, -2, 0], [10, 0, 2, 0, 10, -2], [4, -3, 1, -3, 4, -1], [-3, 6, 0, -3, 1, 0], [3, 3, 3, -3, 3, 0], [1, 1, -1, -3, 1, 0], [3, 1, 0, 3, -1, -7], [-9, 5, 1, -9, 5, 0], [-4, 10, 0, -4, 0, -2], [5, 0, -1, 0, 5, 1], [1, -4, 0, 1, -11, 2], [-3, 6, 0, -3, 3, 0], [13, 5, 4, 5, 13, -4], [3, 3, 0, 3, -11, -9], [9, -2, -9, -5, -2, 0], [1, 6, 0, 1, 5, -8], [11, 4, 7, 4, 11, -7], [-3, 3, 0, -6, 3, 0], [-12, -5, -4, 6, -5, 0], [11, 5, 3, 5, 11, -3], [2, -7, 0, 2, -11, 3], [11, -2, -4, 0, -2, 0], [-7, 1, 0, -2, 1, 0], [4, -8, 0, 4, -15, 0], [7, 2, 0, 2, 7, 0], [8, 8, 3, 8, 8, -3], [-2, -4, 0, -2, -2, 8], [-4, -5, -4, 6, -5, 0], [-4, -4, 0, -4, 5, 12], [-5, -4, -12, -4, -5, 12], [19, -2, -3, 1, -2, 0], [8, -13, 0, 8, -4, -3], [10, 1, -3, 1, 10, 3], [-4, 10, 8, -12, 10, 0], [-5, 5, -4, -14, 5, 0], [4, 3, 3, 3, 4, -3], [0, 8, 7, 8, 0, -7], [-5, 8, 0, -5, -2, 2], [-2, 10, 0, -2, 13, -6], [-2, 1, 4, 2, 1, 0], [-2, 9, 0, -2, 5, -5], [-10, -8, -10, -8, -10, 10], [-6, -6, 3, -6, -6, -3], [-11, 6, 3, 6, -11, -3], [2, 6, 0, 2, 15, -10], [-8, 7, 0, -8, -8, 9], [0, -1, 8, -1, 0, -8], [3, -9, -6, -9, 3, 6], [-13, 9, 7, -11, 9, 0], [28, -1, -3, -1, 28, 3], [3, -1, 0, 3, 18, -5], [9, -12, -7, 17, -12, 0], [-3, 1, 0, -3, 14, 5], [11, -3, 5, -3, 11, -5], [-12, -2, -4, 0, -2, 0], [5, -16, 0, 5, -1, 6], [2, -7, 0, 2, -12, 3], [4, -14, 0, 4, -2, 6], [14, 8, 14, 8, 14, -14], [1, -2, 6, 10, -2, 0], [-12, 8, 3, -13, 8, 0], [-11, -5, 1, -5, -11, -1], [14, 1, 1, 1, 14, -1], [-4, 6, 19, 7, 6, 0], [-19, -9, -8, -9, -19, 8], [3, -4, 1, -4, 3, -1], [1, 12, 0, 1, -8, -14], [5, 7, 11, -3, 7, 0], [-16, 3, 8, 3, -16, -8], [5, 3, 12, 6, 3, 0], [-13, 8, 0, -13, 8, 18], [7, -9, 0, 7, -17, -5], [-16, 8, -3, -19, 8, 0], [2, -16, 0, 2, -7, 12], [-3, -15, -15, 15, -15, 0], [16, -6, 3, 15, -6, 0], [7, 9, 16, -2, 9, 0], [12, 15, 17, 15, 12, -17], [-6, 7, -8, -22, 7, 0], [3, 7, 0, 3, 8, -13], [-3, 6, 12, 6, -3, -12], [-2, -7, 0, -2, -7, 11], [6, 6, -5, -17, 6, 0], [-14, 11, 11, 11, -14, -11], [27, -6, -12, 0, -6, 0], [4, 1, 0, 4, 25, -9], [8, 6, 3, 6, 8, -3], [1, 9, 0, 1, -15, -11], [6, -1, 10, 12, -1, 0], [1, -9, 0, 1, -1, 7], [-6, -4, -7, -4, -6, 7], [-15, -5, -15, -5, -5, 0], [7, 12, 13, 12, 7, -13], [3, 6, 0, 3, -11, -12], [30, 3, 3, -3, 3, 0], [-1, 11, 0, -1, 23, -9], [9, 2, 13, 2, 9, -13], [-8, 11, 0, -8, 10, 5], [-9, -4, 0, -4, -9, 0], [-9, 9, 12, 9, -9, -12], [-16, 5, 12, 2, 5, 0], [5, 9, 12, 9, 5, -12], [-29, -1, -8, -6, -1, 0], [8, 4, 9, 4, 8, -9], [19, -2, -3, 1, -2, 0], [-14, -18, -19, -18, -14, 19], [-3, 17, 0, -3, 20, -11], [-2, -6, -23, -11, -6, 0], [0, -6, 0, 0, 9, 6], [-2, 11, 0, -2, 38, -7], [0, 8, 25, 9, 8, 0], [-4, 21, 0, -4, 19, -13], [2, -15, 0, 2, -10, 11], [16, 6, 17, 5, 6, 0], [-5, 3, 0, -5, 18, 7], [-5, 1, 0, -5, -11, 9], [-15, 1, 10, 8, 1, 0], [17, -13, -20, 6, -13, 0], [-10, 4, 1, 4, -10, -1], [13, -6, 0, 13, -5, -20], [-5, 7, 0, -14, 7, 0], [11, -13, -8, 18, -13, 0], [-10, -3, 5, -3, -10, -5], [21, 4, -3, 4, 21, 3], [1, -5, -9, -5, 1, 9], [0, -15, 0, 0, -23, 15], [5, 3, -8, -14, 3, 0], [6, -4, 0, 6, -15, -8], [-37, 5, -3, -13, 5, 0], [0, -14, -24, 4, -14, 0], [1, 1, 0, 1, 31, -3], [16, 7, 14, 7, 16, -14], [-5, -14, -12, -14, -5, 12], [1, -11, -7, 15, -11, 0], [11, -4, -9, -4, 11, 9], [-21, 14, 21, -7, 14, 0], [-6, 2, 0, -6, 16, 10], [5, -12, -22, -12, 5, 22], [8, -8, 2, 18, -8, 0], [10, -11, 0, 10, -9, -9], [17, -4, -6, 2, -4, 0], [-4, 11, 0, -4, 14, -3], [-13, 14, 15, 14, -13, -15], [6, -5, 3, 13, -5, 0], [-30, 13, 21, 13, -30, -21]]
