# q-expansion of newform 6728.2.a.k, downloaded from the LMFDB on 01 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, 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 = 6728
weight = 2
poly_data = [1, -2, -1, 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], 1], [[0, 1, 0], 1], [[-2, 0, 1], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0], [0, -1, 0], [-3, 1, -1], [-2, 1, 0], [0, -2, 3], [-4, 2, -1], [-4, 0, 0], [-6, 1, -2], [-5, -1, 1], [0, 0, 0], [3, -3, 1], [-2, 2, -1], [-6, 0, 2], [-3, -1, -3], [-2, 4, -5], [-1, -3, 1], [-2, -2, 2], [0, 3, 6], [-6, 2, -1], [-10, 6, -9], [-1, -5, 1], [-6, 10, -3], [8, -7, 4], [-1, -3, -7], [4, -1, 8], [-3, -3, -7], [-2, 0, -9], [-4, -2, 5], [0, -11, 2], [-6, -5, 0], [0, 2, -1], [5, 1, 11], [6, 2, -1], [7, -9, -3], [-6, 1, 8], [-11, 7, -5], [4, 4, -10], [2, 4, 5], [0, 1, 12], [-4, -10, 6], [-1, -7, -5], [-4, 13, -2], [-8, -6, 0], [2, -3, 2], [-13, -1, 1], [-8, -5, 4], [4, 16, -7], [-4, 2, 5], [13, -3, 11], [6, -5, -10], [-14, -2, 0], [12, -1, -2], [2, -12, 3], [-14, 17, -18], [-12, -3, 8], [-5, 21, -11], [-10, 14, 5], [-4, 13, -14], [0, -10, 5], [-10, 4, 5], [0, 11, 4], [18, -11, 16], [18, -12, 20], [-10, 2, 5], [17, -9, 21], [-13, 13, -19], [8, 0, 10], [12, 4, -3], [14, -8, 0], [-6, 20, -10], [-15, 9, -15], [-2, 6, 19], [16, -15, 14], [18, -15, 16], [-27, 11, -15], [11, -15, 1], [-14, 20, -8], [-12, 10, -11], [-3, 7, -3], [8, -11, 22], [4, 12, -11], [-25, 11, -7], [-2, -5, 6], [5, -11, 7], [-12, 21, 4], [-2, -10, 1], [-13, -5, -9], [-8, 5, 2], [-10, -19, 6], [-16, 4, -26], [8, -3, -6], [-13, -9, -1], [-21, -7, -3], [13, -5, 5], [17, 1, 15], [4, -5, 6], [-14, 8, -11], [-10, -14, 4], [6, 2, 26], [-24, 15, -18], [-14, 27, -24], [5, -9, -5], [2, 24, -14], [0, -2, 11], [-8, -4, 15], [10, -15, 14], [14, 11, -14], [-26, 16, -7], [-18, -3, -8], [-8, 16, 7], [3, -3, -9], [0, -11, -8], [-2, 4, -15], [-10, -3, -24], [26, -13, 16], [-20, -7, 4], [9, -1, -15], [18, -16, 15], [-21, 27, -21], [-18, 4, -3], [-5, 5, -17], [-1, 21, -1], [34, -19, 34], [2, 10, 7], [15, 11, 1], [-15, 17, -13], [-6, -14, 5], [21, -9, 15], [-11, -7, -19], [24, -6, 11], [15, -11, 9], [-34, 16, -21], [2, 21, -14], [-6, -1, -16], [-32, 5, 4], [7, -1, -1], [-10, 7, 0], [12, -25, 20], [-35, 1, -3], [2, 23, -12], [16, -6, -6], [4, 2, 23], [12, -12, 31], [-19, 1, -5], [36, -22, 34], [13, -27, 21], [-6, -16, 18], [-38, 12, 3], [12, -4, -7], [0, -3, -30], [-16, 17, 12], [-28, 18, -19], [-10, 6, -9], [-14, -32, 14], [-12, -22, -5], [-36, 8, 12], [2, 0, -25], [0, -4, -12], [-24, 20, -21], [-26, -22, 7], [10, -21, 34], [-32, 11, -30], [-6, 37, -10], [-15, -3, -1], [-4, -2, -25], [-2, 11, -14], [19, -25, 29], [-6, -4, -9], [-15, 5, 13], [17, -23, -5], [10, -15, -14], [3, -17, -7], [-6, 10, -3], [34, 4, 3], [-13, 7, 11], [-15, -9, 15], [-10, -1, -14], [-16, 14, -12], [20, -23, 12], [-18, 11, -28], [33, -13, 39], [-26, 2, -1], [36, -26, 43], [-11, 11, 15], [-14, 10, -18], [0, -25, 26], [25, -5, -21], [18, -4, 41], [31, 1, -1], [-10, 30, -7], [-12, -13, -10], [-8, -17, -14], [-2, 2, -27], [-6, 0, 9], [-13, 23, -9], [-36, 23, 6], [-24, 26, 12], [23, 5, 9], [46, 2, 10], [5, -3, 21], [-18, 2, -21], [-38, -10, 5], [14, 13, 20], [-31, 3, 3], [14, -28, -18], [14, -32, 30], [-14, 11, -4], [-24, 11, -2], [-20, 14, -19], [27, -33, 9], [-4, 40, -3], [12, 11, -26], [19, -27, 41], [-7, 15, 35], [-13, -11, -5], [0, 26, -11], [-10, -37, 12], [-5, -11, -11], [-22, 1, -6], [-4, -31, 12], [0, -16, -27], [-18, 19, 8], [-30, 4, 0], [-1, -25, -11], [51, 7, -1], [-12, 4, -9], [-8, 6, -30], [-12, 18, 3], [-38, 35, -34], [10, -2, 12], [16, 17, -6], [2, 4, -31], [0, -22, 21], [-47, 5, -3], [-16, 19, -2], [-46, -6, -13], [50, -21, 16], [11, -21, 17], [28, -16, -1], [-28, 11, -10], [-7, -17, 17], [-17, 47, 3], [5, -43, 33], [-2, 12, 8], [48, -22, 53], [12, 7, -22], [4, 8, 2], [15, -17, -7], [-24, -28, -1], [12, 16, -6], [25, 13, -23], [21, -39, 15], [4, -24, 28], [-5, 5, 5], [-6, 0, 19], [-53, 19, -31], [16, -25, 8], [20, -5, 2], [6, 4, 33], [-30, 5, -32], [18, -11, 8], [-17, 9, 7], [-2, -30, -3], [-8, 6, -3], [-49, 29, -41], [-18, 46, -14], [1, 1, 27], [-22, -27, 6], [-10, -4, -6], [-42, 3, -18], [42, 6, 0], [-4, 1, -28], [12, -26, 17], [-10, -21, -10], [20, -40, 11], [47, -13, -1], [-38, 14, 19], [22, -10, 5], [6, -5, 30], [-24, 20, -5], [-36, 15, 8], [18, -25, -8], [6, -42, -1], [41, -29, 45], [-14, 50, -37], [25, 9, -25], [-24, -6, 4], [0, -20, 11], [31, -5, 39], [42, 3, -6], [-10, 0, 7], [44, 21, 2], [-54, 0, -12], [12, 12, -4], [-6, -25, 6], [5, 1, -15], [2, -42, -5], [8, 16, -12], [-50, -9, -6], [-15, -3, 1], [4, -40, 9], [-30, 19, 10], [-1, -23, 27], [-17, 31, -15], [-4, -37, 6], [-20, 6, -49], [36, -11, 44], [14, -36, 0], [-32, -18, -13], [-9, 17, -29], [24, -25, 34], [16, -29, -14], [28, -9, 28], [-39, -7, 15], [-24, 38, 0], [-38, 22, 2], [44, 0, 11], [13, -37, 45], [66, -5, -12], [12, -20, 6], [-67, 35, -43], [26, 6, 35], [-29, 29, -29], [52, -25, 40], [2, 1, 10], [-1, 23, -33], [10, -1, 18], [-18, 48, -30], [-46, 25, -48], [-58, 36, -71], [-8, -30, -12], [28, -1, 36], [-35, 31, -43], [-2, 1, 20], [20, -62, 11], [-5, 43, -19], [56, -21, 18], [-10, 12, 11], [8, -1, 18], [-20, -5, -8], [-21, 5, -25], [5, 37, -17], [-38, 24, 11], [-20, 56, -13], [32, -10, -7], [26, -31, -26], [0, 2, -17], [5, -35, 17], [5, 21, -33], [-26, 26, -15], [-24, 15, 30], [-8, 4, -38], [-6, -9, 14], [7, 33, -7], [-30, 36, -31], [7, 13, 19], [-14, -32, 17], [-24, 27, 6], [20, 35, -6], [-54, 4, 13], [10, 8, -40], [-73, 19, -35], [34, -12, -1], [33, 35, -25], [-41, 35, -29], [-2, 32, -15], [64, -26, 0], [70, -10, 19], [-7, 43, -35], [-58, 6, -9], [44, -38, -15], [-38, 46, -59], [60, -29, -2], [28, -54, 18], [15, -29, -9], [55, -19, 3], [28, -9, 58], [-36, 34, -22], [-48, 18, -14], [-24, 19, 4], [20, -44, 15], [15, -41, 37], [-36, 48, -31], [18, 32, 1], [-16, -6, 45], [-17, -39, -1], [24, -57, 54], [-66, 30, -47], [11, -21, -7], [52, 5, 6], [72, -14, 15], [-28, -17, 10], [51, 17, 1], [14, -19, 2], [-7, -21, 21], [30, 30, -19], [26, -29, -20], [86, -7, 4], [-5, 31, 25], [15, -21, 39], [-67, 5, -27], [27, 9, 23], [10, -34, 44], [-2, -23, -6], [-5, -9, 5], [30, 45, -6], [22, 34, -7], [-40, 20, 22], [48, 1, -8], [9, 13, -5], [-58, 14, -15], [25, -35, -15], [-4, -8, -30], [2, -42, 23], [61, -31, 47], [20, -13, -48], [50, -24, -5], [-70, 38, -37], [26, 21, -14], [32, -5, 32], [-6, -8, -17], [-16, -22, -10], [-43, -9, -19], [8, -11, 16], [-13, 39, -11], [-8, 2, 38], [-5, -41, -19], [12, -46, 29], [6, -30, 23], [-56, 10, 18]]