# q-expansion of newform 7872.2.a.bq, downloaded from the LMFDB on 08 May 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
    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 = 7872
weight = 2
poly_data = [-6, 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, 0], [2, 1], [2, -1], [1, -1], [-2, -1], [1, 1], [-4, 1], [4, -1], [-1, 3], [7, 0], [1, 2], [-1, 0], [-1, 0], [7, 1], [8, -2], [0, -6], [3, -4], [2, -2], [-3, -3], [1, 0], [10, 0], [-10, -1], [-6, 4], [4, 3], [3, 3], [13, 0], [2, 0], [2, 3], [6, 3], [-18, 0], [0, -5], [-1, 1], [10, 4], [10, 0], [0, 8], [-2, -4], [3, -2], [-6, -8], [-10, -1], [1, 3], [-4, 4], [8, 2], [10, 1], [2, -9], [-12, 1], [0, -2], [8, 4], [-21, -1], [12, 1], [6, 0], [-4, 2], [-11, -6], [14, -2], [9, -7], [-5, 5], [8, -2], [-3, 6], [-1, 4], [27, -1], [-5, 2], [-7, 3], [-5, 8], [16, -6], [-2, 3], [-3, -5], [28, 1], [-9, -4], [-21, -3], [-7, 2], [6, -4], [24, 3], [-7, -4], [1, 10], [-2, -12], [21, -7], [6, 5], [18, 7], [12, 0], [-5, 2], [4, -5], [-8, -12], [-20, 7], [1, 10], [-34, 0], [24, 0], [-20, -6], [4, -8], [-6, 12], [-6, -7], [-6, 8], [-17, -5], [-29, 4], [14, 1], [-4, 10], [-3, -7], [23, -5], [15, 1], [-18, 8], [-2, -12], [-8, 2], [-11, -9], [-39, -3], [-24, 1], [-12, 0], [12, 10], [9, 1], [3, 9], [-4, -13], [30, -2], [-10, 2], [4, -16], [-30, 3], [-7, -8], [-17, -2], [-21, 7], [-28, 2], [-12, 9], [-27, -7], [24, 2], [-16, -6], [-14, -10], [-22, 7], [-22, 8], [6, 15], [-22, 2], [-42, -2], [-13, -9], [-12, 7], [37, 6], [-35, 4], [14, 4], [22, 0], [-4, -5], [-28, 10], [-33, -2], [7, 9], [23, 4], [6, 10], [2, 2], [-1, -20], [-16, 1], [-6, 6], [19, -7], [-13, 8], [36, -2], [-12, 0], [0, -2], [-33, -4], [-30, 1], [33, -2], [-12, 0], [0, -15], [21, -3], [22, -4], [-16, -11], [14, -5], [15, 7], [-28, 0], [-34, -4], [-10, -4], [8, 9], [34, 0], [37, 1], [1, 9], [18, -7], [-4, -22], [-14, -3], [-49, -4], [18, 10], [18, 8], [16, 2], [2, 8], [-21, -4], [20, -4], [-45, -1], [-28, 15], [30, -10], [4, 1], [6, -8], [-23, 10], [30, -8], [8, -4], [14, -12], [2, -12], [30, -5], [-4, 0], [-42, -4], [18, -5], [11, -3], [-39, -2], [0, 22], [-14, 2], [-6, -10], [0, -3], [28, -8], [-10, 22], [-42, -7], [-46, 4], [15, -15], [-8, 0], [-1, 6], [32, 5], [14, -16], [-21, -1], [-9, -3], [2, -6], [23, -1], [-24, 15], [9, -4], [-38, 3], [-1, -1], [10, -18], [-44, -2], [21, 19], [35, 6], [-30, -3], [12, -21], [-9, -21], [-58, 6], [-4, 3], [25, -4], [-23, -7], [-4, 7], [30, 11], [-8, 7], [-10, 4], [14, -7], [0, 22], [36, 1], [2, 12], [-26, -7], [-9, 20], [36, -1], [-16, -13], [30, 12], [2, 9], [10, 2], [0, 24], [-39, 9], [24, -2], [-28, 4], [38, -6], [-19, 10], [-24, -8], [14, -1], [38, 10], [-35, 15], [-11, -6], [-34, -10], [7, 14], [-20, -17], [30, 2], [15, -12], [18, -14], [42, 4], [4, -21], [-8, -21], [-27, -3], [-16, 9], [-25, 20], [-9, 19], [-10, -18], [8, -2], [-10, -5], [25, -20], [9, -17], [-30, 2], [6, 0], [9, 15], [24, 0], [1, 16], [35, -6], [41, -6], [-6, -24], [48, -12], [39, -11], [-40, -14], [-20, -1], [27, 5], [45, -1], [-38, -15], [-26, -12], [4, 14], [48, 14], [-7, 31], [40, -9], [14, -2], [42, 7], [27, -13], [27, 19], [-6, -12], [-39, -5], [0, 5], [-16, -25], [-9, -7], [-4, -10], [10, -6], [-28, -16], [-6, 7], [-48, -2], [2, 8], [-31, 20], [-27, 3], [29, 14], [22, 0], [0, 17], [-39, 12], [-42, 0], [-12, 17], [3, 19], [38, 10], [-48, 2], [-29, 18], [-42, -1], [20, -20], [38, -4], [-12, -1], [26, -14], [-3, 3], [26, -10], [64, -6], [26, 3], [30, -5], [20, -3], [-10, 12], [-58, 3], [-2, 27], [-55, -7], [-18, 2], [-28, 18], [54, -12], [40, -8], [0, 14], [43, 8], [-41, 13], [-46, -5], [46, 15], [-10, -9], [-13, 30], [-24, 28], [-44, 16], [88, -2], [-2, -10], [-8, 32], [0, -20], [-60, 10], [-5, 36], [-28, 8], [-8, 10], [26, 9], [-23, 2], [33, 1], [-25, -24], [-8, -2], [4, 26], [54, 11], [-80, 4], [-30, 15], [48, -17], [-35, 22], [6, 12], [15, 27], [-32, 2], [26, -14], [-30, 17], [2, -26], [3, -4], [19, -16], [-16, 0], [-32, 20], [-32, -22], [-33, 23], [-10, 16], [12, 21], [36, 7], [-46, -21], [-32, -18], [-13, -17], [-36, -3], [-51, 5], [-84, 3], [3, -8], [-16, -8], [-38, -6], [-14, 4], [10, 13], [-24, -6], [53, 10], [28, 0], [18, -10], [35, 9], [-48, 10], [4, -16], [40, -8], [-8, -12], [-52, -9], [38, -2], [53, -6], [18, -18], [-43, 2], [-70, 10], [-13, 26], [38, -14], [-72, 11], [-16, -23], [11, 12], [3, 5], [8, 15], [8, 7], [5, -14], [48, -20], [36, 6], [12, -2], [44, -17], [-40, -9], [18, 17], [-48, 3], [-17, 25], [18, 8], [-60, -7], [-10, 32], [-26, 11], [-13, 5], [-29, 0], [-16, -13], [-45, 9], [-43, -25], [52, 5], [-41, 9]]