# q-expansion of newform 7942.2.a.z, downloaded from the LMFDB on 24 October 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 = 7942
weight = 2
poly_data = [-1, -1, 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 = [[1, 0], [0, 1], [2, -3], [0, 2], [1, 0], [2, 2], [0, 4], [0, 0], [2, -3], [4, 4], [2, -5], [7, -1], [2, -8], [0, -2], [-5, 5], [-6, 3], [3, 3], [-6, 8], [-9, 3], [8, -4], [4, -6], [10, 4], [6, 0], [3, -3], [-10, -4], [14, -2], [12, -8], [6, 0], [6, -10], [-3, 11], [10, -12], [2, -12], [9, 3], [-6, 12], [-22, 2], [-10, -4], [1, 9], [8, -3], [22, 2], [12, 2], [-4, -5], [-13, 7], [-5, 1], [2, -16], [-20, 4], [18, -7], [6, -8], [-5, 9], [-10, 20], [-2, -1], [8, -4], [10, 6], [-10, 0], [9, -3], [14, 8], [26, 2], [-5, -9], [-24, 2], [4, 6], [8, 2], [4, 8], [-4, -12], [22, -12], [-3, -17], [10, -4], [-10, 12], [-12, 3], [22, -16], [0, -8], [-8, 16], [-17, -3], [-18, 0], [-27, 11], [24, -14], [-8, 24], [2, 3], [14, -11], [-18, 1], [-20, 21], [-6, -6], [-12, 7], [-7, 25], [-18, 2], [8, -11], [30, -14], [-9, 15], [-9, 9], [-8, 2], [2, -14], [11, -31], [0, 1], [12, -18], [-6, -3], [10, 8], [19, -13], [8, -2], [10, 4], [26, 0], [2, -2], [30, -24], [-18, 2], [-30, -10], [-28, 20], [-20, 8], [24, -24], [24, -27], [-5, -25], [-30, 22], [-13, 13], [-10, 0], [-10, 32], [-36, -2], [-36, 13], [-39, 13], [-10, 21], [28, -21], [-33, 3], [-19, 19], [-45, -1], [-16, 16], [-2, -11], [-4, -16], [14, -24], [-8, 29], [28, -1], [14, 14], [22, -24], [13, -21], [31, -31], [-24, 26], [32, -10], [8, 10], [-18, -7], [-46, 8], [-16, -8], [-12, -20], [35, 7], [-20, 22], [-18, 5], [-36, 8], [-8, 16], [-2, -2], [-26, 3], [-30, -14], [22, -28], [-18, 25], [2, -24], [-6, -30], [-8, 13], [38, -17], [-32, 28], [-18, 0], [32, 1], [8, -6], [28, -9], [18, -1], [6, -34], [-24, 41], [-6, 10], [-26, 12], [-16, -24], [20, -12], [12, -42], [-7, -15], [-7, 3], [-5, -35], [-30, 7], [16, -26], [-28, 22], [-2, -24], [26, -50], [22, -19], [-16, 20], [-38, 8], [25, -45], [2, -8], [28, 4], [31, -5], [6, 16], [-28, -12], [-6, -3], [10, 0], [17, 5], [16, 20], [4, 24], [23, 7], [2, -18], [-5, -13], [24, 22], [-22, 28], [-56, 9], [32, -36], [-20, 24], [14, -15], [14, -10], [-26, 32], [-6, 2], [2, -5], [-24, 42], [-46, -12], [-40, 18], [22, 22], [-26, -9], [2, 34], [4, 23], [30, 8], [22, -53], [0, 20], [-12, 24], [-29, 35], [2, 8], [-37, 39], [0, 12], [-9, -21], [-44, 18], [-16, 15], [36, -46], [-40, -10], [27, -55], [-30, 4], [-6, 14], [-16, 34], [28, -23], [19, -7], [-30, 6], [34, -28], [-16, 9], [44, -8], [52, 10], [-10, 44], [-7, -23], [6, -48], [48, -18], [50, 6], [-20, -1], [-10, 14], [0, 21], [16, -24], [-16, 52], [-26, 37], [-36, 19], [52, -14], [-56, 24], [-26, 9], [-28, 6], [8, -12], [-21, 37], [-4, 47], [-38, 8], [-4, 16], [-6, 32], [-14, -32], [-20, -16], [-29, 37], [16, -20], [40, 12], [-14, 21], [-18, 42], [45, -11], [-22, -32], [38, -8], [46, -12], [-50, 44], [-54, 20], [-29, 17], [-16, -16], [-42, 19], [-32, 5], [-54, -8], [42, -22], [-10, 8], [-5, 31], [4, 25], [14, 34], [-42, -10], [71, 5], [-8, -19], [-2, 30], [4, 18], [4, 12], [36, -6], [-46, 37], [-24, 52], [-2, -12], [24, 30], [-6, 3], [44, -47], [-4, 46], [21, 15], [52, -24], [-15, 13], [8, -21], [-18, 38], [18, -42], [6, -20], [6, -10], [-45, 13], [66, -13], [30, -30], [-34, 30], [38, -28], [-24, -12], [10, -46], [-17, 3], [12, -9], [28, 7], [0, 36], [-70, -3], [34, 17], [-4, -16], [-48, 0], [57, 1], [38, -52], [24, -47], [-18, -10], [10, 40], [28, -51], [-46, -2], [81, -9], [48, -4], [-38, 38], [-56, 21], [70, -6], [-13, 9], [-46, 2], [27, -27], [-3, -35], [16, -32], [16, -40], [6, 4], [54, -8], [-59, -3], [-80, 16], [-4, 24], [-52, -6], [-8, -16], [33, -11], [40, 24], [26, -36], [28, -26], [-30, -3], [-12, 23], [-72, 18], [-3, -49], [6, 3], [-54, 38], [22, -1], [-28, 0], [42, -58], [19, -65], [-64, -10], [62, -36], [-22, -33], [-36, 8], [-40, 30], [68, 12], [20, -24], [-4, 24], [18, -20], [27, 25], [-52, 28], [-10, 18], [-2, -9], [30, -40], [36, -51], [50, 8], [-18, -18], [32, 2], [-58, 6], [-64, -4], [55, -25], [34, -72], [-78, -6], [34, -38], [69, -3], [25, 37], [16, 8], [-34, -24], [-44, 68], [30, 20], [-11, -3], [-26, 16], [8, -2], [2, 14], [-68, 40], [65, -29], [43, 5], [-26, 13], [26, -16], [-41, -27], [-16, -27], [39, 15], [49, -31], [-8, -29], [-62, 7], [22, 30], [-40, 4], [-48, -17], [-31, 65], [0, -40], [20, -4], [-44, 21], [38, -70], [32, 23], [-24, -16], [-32, 48], [-19, 17], [4, -52], [-4, -29], [-93, -1], [0, -2], [40, -76], [65, -23], [-40, -8], [-34, -6], [41, -51], [-80, 32], [25, 39], [48, -9], [52, -10], [23, -69], [14, 14], [-22, 71], [26, -42], [40, -33], [54, 11], [-8, 24], [14, -26], [-28, 7], [-28, 2]]