# q-expansion of newform 3200.2.a.bk, downloaded from the LMFDB on 02 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 = 3200
weight = 2
poly_data = [-1, -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], 1], [[-1, 2], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0], [1, 1], [0, 0], [-1, 1], [-2, 0], [0, 2], [0, 2], [0, 2], [-7, -1], [2, 0], [-2, 2], [-2, -4], [8, -2], [1, -3], [-5, 1], [-4, -2], [-4, 2], [6, 0], [-1, 3], [-2, -2], [0, 2], [-4, 4], [-3, -3], [-6, 4], [12, -2], [10, 0], [1, -1], [15, -1], [-6, 4], [-6, -4], [-9, 5], [6, -4], [-2, 0], [16, -2], [2, 4], [-10, 6], [14, 4], [5, -3], [-1, -7], [-2, 4], [-16, 2], [-2, 4], [2, 6], [16, 2], [-4, -2], [-8, -4], [-10, -4], [-11, -1], [-1, -5], [6, 4], [-20, -2], [4, -4], [4, -10], [-2, 0], [-6, -4], [-11, 3], [6, -8], [2, -10], [-2, 4], [8, 2], [-15, -3], [-6, 8], [7, 3], [2, -6], [2, -12], [-16, 2], [-2, -8], [-14, -4], [15, -1], [6, -4], [-14, -4], [24, 4], [-25, -3], [14, 4], [20, 2], [17, 3], [6, 8], [-8, 2], [-2, 0], [4, -6], [0, 2], [2, -12], [2, -10], [-4, -2], [-24, -4], [-11, -7], [12, 6], [14, 8], [-6, 16], [-19, -9], [-17, 11], [-4, 4], [15, 1], [-30, -4], [28, 6], [-15, -9], [-10, -12], [2, 8], [9, 13], [-6, 8], [-5, 7], [22, 4], [-11, -3], [-12, 10], [10, 12], [-2, 8], [7, 7], [14, 0], [4, -8], [0, 10], [27, 9], [24, 2], [-12, 6], [16, -2], [30, -2], [-4, -2], [-11, 5], [15, 1], [-36, 6], [8, -6], [-18, 8], [-8, -14], [-8, 10], [-31, -3], [-30, 0], [18, 4], [18, 0], [4, 12], [-5, -3], [34, -8], [0, -22], [-15, 7], [6, 10], [2, -8], [-14, -8], [-6, 4], [-4, -10], [15, -13], [-12, 6], [-30, -8], [-2, 0], [-30, -4], [5, 3], [-5, -5], [38, 0], [-4, 0], [-12, -2], [2, 12], [36, -6], [25, 3], [34, -8], [-20, 2], [21, -3], [-13, -11], [7, -1], [22, -14], [-12, -8], [20, -2], [8, 2], [-18, 12], [-33, -5], [-10, 16], [-17, -7], [6, -8], [8, -14], [1, -17], [22, 2], [-48, -6], [34, -4], [32, -6], [-4, 2], [30, -4], [-30, 2], [12, 22], [0, -24], [2, -4], [-30, -4], [30, -4], [-11, -5], [22, 0], [-17, -19], [30, -4], [16, 2], [48, 2], [21, 15], [2, 8], [-18, 12], [9, -7], [-20, 18], [30, -6], [54, 0], [-15, 13], [-6, 0], [-10, -8], [23, -13], [16, -6], [12, 18], [32, -6], [-18, 16], [41, 7], [-30, -8], [10, 6], [48, -6], [18, -16], [-4, -14], [10, 0], [16, 24], [-31, 17], [-22, -12], [30, -8], [-12, -2], [2, 4], [-39, 7], [-25, -17], [12, 24], [-16, 6], [3, 1], [-34, -16], [19, 13], [12, 14], [-2, -8], [-44, 0], [-54, 4], [1, 11], [-5, 15], [46, 0], [-12, -2], [48, -8], [-13, 5], [6, 24], [28, -2], [-12, 6], [-46, -10], [32, 10], [53, 9], [19, 17], [36, -2], [-8, -6], [-48, 6], [-54, -6], [5, 21], [14, 8], [29, 11], [14, -4], [26, -12], [-32, 4], [43, 1], [30, -4], [-12, -14], [21, -17], [8, 18], [18, 16], [-13, 13], [40, 10], [-30, 8], [-16, -6], [2, -32], [31, 15], [38, -4], [-2, -8], [-27, 7], [-1, -21], [-34, 4], [34, -16], [-12, -2], [52, -2], [14, -12], [-10, 8], [57, -3], [-34, 12], [-18, -12], [-1, -5], [-36, -2], [-48, 8], [14, 8], [-7, -25], [67, 3], [42, 0], [16, 10], [-6, -8], [33, 11], [10, 18], [7, -7], [22, 12], [-45, -13], [6, 26], [-64, 2], [-40, 10], [44, -16], [16, 6], [-6, -28], [43, -17], [44, -10], [-78, 4], [-46, -8], [-38, -4], [14, -6], [-38, -8], [8, -2], [7, 11], [6, -8], [30, 20], [-20, -4], [57, 1], [-10, 0], [32, -6], [19, -5], [-38, -16], [32, -12], [14, -20], [9, 35], [-34, -4], [38, 8], [-19, 29], [-13, -11], [-30, -20], [16, -6], [6, 10], [66, 4], [8, -2], [2, 16], [70, -8], [30, -16], [5, 23], [26, 4], [20, 6], [44, -2], [53, 9], [-13, -15], [-8, -6], [-30, -12], [10, 16], [-12, -4], [33, 9], [-14, -4], [-21, 3], [18, -20], [-2, 8], [-24, -14], [3, -7], [-22, -8], [-66, 8], [34, 12], [14, -10], [-26, -12], [-28, -10], [22, 0], [19, -5], [70, 10], [-28, -10], [2, 32], [26, -20], [-10, 32], [-15, -21], [-74, 4], [-8, -22], [-12, 4], [54, 8], [0, 26], [-19, 3], [16, 26], [-50, -16], [-41, -3], [32, -10], [-41, 3], [-8, 34], [-52, -10], [9, 31], [-24, -18], [18, -8], [12, -6], [53, -17], [-14, -4], [22, 22], [0, -6], [0, 10], [46, -22], [54, 8], [-4, -10], [-38, -12], [-18, 28], [-70, 4], [35, -19], [-2, -32], [36, 6], [41, -25], [18, -2], [6, -12], [9, 29], [-33, 21], [-48, 6], [-18, -12], [16, 22], [11, -9], [86, -2], [30, 16], [-44, 12], [26, 32], [-10, 16], [22, -24], [26, 4], [-36, -26], [59, -7], [54, 8], [38, 16], [62, 14], [-36, 14], [-42, 24], [-27, -27], [-20, 30], [-30, 12], [-82, -4], [-3, -7], [6, -12], [24, 10], [66, -16], [64, 8], [3, 5], [-20, 14], [-71, 7], [14, -24], [24, 26], [-25, -11], [-52, -14], [-18, -8], [38, 12], [37, -19], [48, -14], [-6, -36], [0, -4]]