# q-expansion of newform 7942.2.a.bt, downloaded from the LMFDB on 22 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, 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 = 7942
weight = 2
poly_data = [-3, -90, -351, -240, 486, 396, -285, -198, 90, 41, -15, -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, 0, 0, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-7229, 15183, 78629, -21577, -64398, 16444, 11828, -4066, 1497, -215, -355, 93], 2069], [[18207, -35598, -222939, 17460, 260334, -16782, -115656, 13257, 21924, -3638, -1484, 313], 6207], [[-12234, 165255, 463143, 57987, -501531, -128178, 216255, 35796, -42996, -1027, 3164, -316], 6207], [[744, -32820, 96501, 340233, -57375, -305109, 14529, 97887, -3507, -12383, 388, 487], 6207], [[26592, 179475, 285765, -138084, -368997, 67188, 170301, -30309, -34044, 7315, 2455, -614], 6207], [[27729, 230733, 181131, -421362, -252774, 269892, 117495, -80616, -23010, 11759, 1613, -679], 6207], [[-21936, -272538, -397863, 438822, 513108, -286674, -233151, 93873, 44934, -15397, -3097, 992], 6207], [[35199, 15939, -302556, -413907, 90249, 381630, 57789, -134418, -25935, 21898, 2572, -1379], 6207], [[-20892, -257523, -549474, 192930, 772281, -85401, -400275, 50727, 89769, -16388, -7091, 1642], 6207], [[-404, 91171, 165211, -238978, -269951, 195411, 144983, -72342, -31500, 12742, 2370, -854], 2069]]

hecke_ring_character_values = None
aps_data = [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, -1, -1, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, -1, 1, 2, 0, 0, -1, 0, 0, 0, 0], [0, -1, -2, -1, 0, -1, 0, -2, -1, -1, 0, 0], [0, 1, 1, -1, -1, 1, 0, 1, 1, 2, 3, -1], [4, -2, 1, 2, 1, 0, 0, -2, -2, -1, 0, 0], [2, -1, -1, 0, 1, 2, -3, -1, 2, 2, 1, 1], [3, -1, 0, 2, 1, 1, -2, -3, 0, 2, -1, 2], [-1, -3, 1, 0, -1, 1, 0, 2, -1, -2, 1, 1], [-1, -1, 0, 1, 0, 0, -2, 0, 0, 0, -2, -1], [1, -1, -3, -1, 1, -1, -1, -3, 1, 0, -1, 2], [0, 0, 0, 1, 1, 0, 1, -1, 0, -1, 0, 0], [2, -4, 2, 3, -1, 4, -2, 0, -2, 0, 2, -1], [1, 0, 0, -2, 0, -4, 2, -1, 0, -1, -2, 1], [0, 2, 1, -1, 3, -1, 0, -1, 0, 2, 0, -1], [0, 0, 0, -2, 0, -1, 3, 3, 1, -2, -2, 0], [-6, 1, -1, -2, 1, -1, -1, 2, 0, 0, -2, -2], [-1, 1, -1, -1, 1, 0, 2, 1, 2, 1, 2, -2], [1, -1, 2, 3, -1, 3, 2, 2, 0, 0, -1, 3], [-5, 1, 2, 0, 1, -3, 2, 1, 1, 1, 3, 1], [2, -2, 0, -1, 1, -1, 2, 3, 0, -2, -5, 1], [2, -1, -3, -3, -3, 0, -1, 0, 1, -2, 1, 3], [1, -4, 0, 2, -2, -2, -1, -1, 0, 0, 1, 1], [2, 0, 1, 2, 2, 0, 0, -3, -4, -1, -3, 1], [2, 2, 3, -1, 0, -3, 3, 1, -6, -1, 3, -2], [-1, -3, -8, -2, 0, 1, -4, 0, 4, -3, 0, 2], [-3, 0, 1, 1, 0, 1, 0, -1, -1, 1, 4, 3], [1, -4, -5, -1, -3, -1, -5, 1, 5, -4, -3, 4], [1, 0, 5, 1, -3, 3, 1, -2, -4, -1, 2, -3], [4, 1, 1, 0, 1, -2, 1, -2, -4, 1, 1, -3], [-4, -6, 2, -3, -2, 0, 1, 2, -1, -3, -1, 1], [-2, -2, -1, 4, 1, 7, -1, -1, 4, 1, 0, 1], [-3, 2, 2, 1, 4, 2, 1, 3, 3, 1, 2, 3], [-4, 3, 1, 3, 3, -3, 3, -3, 0, 3, 0, -2], [-1, 2, -1, -1, 1, -5, 1, 2, 0, 3, -4, 2], [-4, 2, 0, -9, -3, -3, 4, 5, 3, -2, 0, -4], [-3, 1, 2, -2, -3, 3, 3, 4, 2, 1, 1, -2], [2, 2, 3, 0, -4, -3, 2, 0, -1, 0, -3, -2], [-6, 0, 2, 1, 1, 0, 2, 1, 0, 2, 0, -4], [-4, -1, 2, -2, -3, -1, -1, 1, 2, 0, 2, -1], [2, 2, 0, -2, 6, 0, 2, 0, 2, 1, -1, 1], [-9, 2, 0, -4, -4, -6, -2, 4, 1, -4, 0, -2], [0, 2, 5, -2, 0, -5, 5, 4, -1, 1, -2, -4], [5, -3, -5, -3, -4, 0, -3, 2, 0, -4, -6, 0], [-10, 1, 2, -1, 6, -4, 3, 2, 4, 6, -1, -2], [-5, 1, -2, 1, 3, 6, -3, -3, 0, 2, 3, 0], [7, -4, 1, 1, -1, 3, -5, -2, 0, -3, -1, 2], [-12, 0, 3, 0, 0, 0, -3, 5, -2, 1, -4, -3], [6, -6, 1, 1, -4, 4, -4, 0, 3, -3, 1, 3], [-6, 2, 1, -3, 0, 4, 5, 4, 2, 1, 2, -1], [-8, -5, -4, 0, 0, -1, -3, -2, 4, 1, 5, 1], [0, 0, 6, 1, 1, 0, 3, 3, -5, 2, -4, 0], [-9, 1, -2, -7, -1, 1, 3, 8, 0, -5, -6, -3], [-8, 5, -4, -6, -3, -8, 1, -1, -1, -6, -10, 2], [-6, 0, 2, 2, 2, -5, 6, 2, 1, 0, -4, -1], [9, 0, -2, 6, 4, -1, -2, -5, -5, 0, -6, 1], [1, -1, 3, 6, 8, 2, 3, 2, 6, 8, 0, 0], [-8, 1, 5, 1, -3, 0, 0, -3, -4, 5, 5, -4], [0, -3, 2, 3, 0, 2, 0, 0, 1, 0, 11, -4], [2, -2, 1, -1, -4, -2, 1, 1, 0, -3, -1, -3], [7, 2, -3, 1, 2, -5, 0, -6, -5, -4, -9, 2], [-10, 5, 2, -6, -2, -6, 3, 1, 0, 5, 4, -4], [5, -1, -3, 8, 2, -1, 0, -7, 1, 4, -3, 3], [-8, 4, -1, -4, 0, 3, 4, 6, -1, 0, 0, -1], [-12, 2, -4, -6, -5, 3, 2, 3, 6, -2, 1, 5], [-3, -5, 4, -3, -3, -9, -2, 4, 0, 3, -6, -7], [2, -4, -2, 0, -6, -6, 2, -3, -7, -6, 1, 3], [-10, -4, 2, -4, -10, 13, -2, 7, 1, -2, 5, -6], [-7, 3, -9, -9, -4, -1, -1, 7, 7, -2, -3, 3], [-4, 3, -10, -5, -1, -1, 0, 3, 5, -1, -1, 2], [-4, 0, -3, 5, 5, -5, -3, -4, -3, 7, 0, 1], [1, -8, -5, 0, 5, -1, -6, -2, -2, -1, -2, 4], [-12, 2, -1, -6, -1, 0, -4, 0, 2, 1, -2, 2], [15, -3, -7, 6, 4, 3, -4, -6, 0, -3, 2, 11], [-12, -3, 6, -7, -3, -1, 1, 6, 0, 1, 3, -5], [-9, -1, 1, -4, 0, -5, 0, 2, -5, -2, -4, -2], [10, 9, -2, 0, -1, -2, -1, -8, 0, 1, 2, -2], [5, 2, 2, 12, 5, 8, -6, -10, 0, 6, 9, 3], [0, -3, 3, -1, -1, 0, 1, 4, 0, 0, 1, -2], [3, -4, -1, 4, 1, -9, 1, -6, -3, 2, -3, 2], [-8, -2, 6, -1, 0, -3, 4, 1, -5, -2, 10, -4], [17, 2, 10, 5, 0, 3, 7, -3, -5, 0, 2, -4], [2, 4, -4, -1, 3, -5, 2, -1, -1, -9, -10, 1], [-1, 0, -3, 3, 3, -2, 0, 2, 1, 1, 5, 8], [-5, -2, 9, 0, -1, 7, 4, 1, -5, 0, 8, -5], [2, -5, 0, 1, -7, 4, -2, 1, 2, 1, -5, 2], [4, 10, -5, 3, 5, 2, -2, -8, 1, 0, -5, 1], [11, -1, -1, 0, 2, 3, -3, 3, 3, 2, 0, 3], [-3, 3, -8, -1, 2, 4, -1, -5, 2, -4, 2, 1], [-15, 5, -6, -7, -1, -7, -5, 1, 3, -1, -7, -5], [-3, 4, 0, 0, -7, 5, 8, 1, 1, -1, 4, -2], [3, 5, 2, 2, 8, 2, 8, 3, 1, 4, 2, 4], [-12, 8, -4, -6, -5, 6, 7, 9, 5, -1, 1, -2], [11, -4, 1, 6, 4, -6, 6, -4, -1, -2, -4, 3], [-10, 4, 0, -3, 2, -4, 5, 0, -3, -4, 0, -2], [-12, 1, -1, -3, 0, -7, 2, 7, 1, 1, 3, -3], [1, 13, 3, 4, 3, -11, 7, -1, -7, 2, 2, -1], [2, 0, -2, 2, 0, 2, -1, 1, 6, 0, -2, -1], [-7, 1, -2, -3, 2, -1, 4, 6, -2, 1, 3, -5], [5, 4, 2, 3, 0, 5, -1, -6, -1, -3, -2, 5], [-17, 6, 4, -4, -4, -3, 8, 5, -6, 0, -9, -3], [-6, 7, 4, 2, -1, 0, 3, 3, 0, 6, 0, -4], [-3, 3, 5, 7, -2, 8, 1, 1, 2, 3, 0, -5], [3, 3, 0, 6, 5, -1, -4, -12, -9, 5, 0, 4], [9, 0, -1, 2, -7, 6, -8, -3, -5, -5, 9, -1], [5, 2, 2, -2, -1, 5, -3, -3, 7, 1, 5, 0], [4, -1, 4, 2, 0, -4, -2, -7, 7, 0, -5, 2], [-3, 1, 7, 6, 9, -2, 4, -8, -8, 2, -7, -6], [-11, -1, 0, 5, 8, 2, -4, -8, 0, 8, 5, -1], [-2, 1, 4, -1, -2, 1, -4, 3, 4, 4, 8, 5], [7, -2, -2, 8, 6, -6, -2, -9, 2, 5, 1, 7], [5, -4, 2, 4, 0, -1, -3, -8, -5, 4, 4, 4], [-12, 10, -2, 1, 5, -6, 4, -6, 2, 2, -3, -4], [-5, -1, 3, -3, -4, 9, 1, -3, -3, -4, 5, -5], [0, -4, -2, 5, 2, 3, -6, 2, 3, -3, -1, 5], [1, 3, 7, -2, -6, 12, -1, 3, 6, 2, 5, -1], [-5, 6, 5, 5, 5, -3, -4, -1, 5, 5, 0, 0], [0, -14, -3, 3, 0, 8, -7, 0, 2, 1, -3, 3], [7, -5, -6, 0, -4, -13, -10, -5, 2, -5, -3, 7], [-14, 11, -3, -3, 2, -5, -2, 1, 7, 4, -8, 1], [-1, 15, 2, -1, 6, -16, 10, -5, -4, 5, -6, -7], [-8, -4, -5, 1, 6, -4, 2, -1, 8, 1, 1, -3], [4, 6, -15, 4, 10, 8, -6, -14, 0, -1, 7, 8], [0, 2, 2, 1, -1, -3, 0, 6, 2, 4, -1, 4], [4, -4, -8, 8, 8, 8, -5, -4, 9, 1, 2, 16], [15, -4, -7, 9, 2, 0, -2, -11, -3, -3, -10, 8], [0, -4, 0, 7, 6, 9, 1, 3, 5, 6, 4, 6], [-13, 13, 3, -7, -6, -4, 10, 6, -2, 0, -6, -10], [-1, 13, 2, 4, 2, 8, 5, 1, -4, -2, 5, 0], [-6, 13, 1, -7, 2, -1, 3, 14, 9, 6, 10, 0], [-28, 12, 3, -12, -6, 7, 6, 9, 3, -1, 3, -9], [8, -1, 2, 3, 0, 9, -6, -6, 2, 6, 15, 11], [-6, 2, -1, -1, 3, 2, -10, -7, 6, 8, 6, 1], [2, -5, 9, 2, -5, -1, 5, 3, -6, -6, 10, -7], [-2, -1, 0, 1, 5, -2, -6, 1, 3, -3, 1, -5], [-4, 2, 11, -3, 1, 11, 12, 4, -4, 0, 14, -8], [7, 3, -3, -2, 11, -1, 2, -6, -4, -3, -1, 0], [-9, 12, -7, -2, 7, -10, 3, -7, 1, 4, -9, -1], [-8, -2, -12, -7, -4, -2, 4, 3, -6, -11, -11, 3], [-12, 6, 6, -1, -7, -5, 3, 1, -4, -5, -11, -3], [3, 4, -2, 0, -1, 1, 1, 4, -5, -4, 16, 3], [-10, 0, 10, -1, -5, -2, 4, -4, -1, -4, -1, -1], [8, -1, 9, 5, -6, 9, -7, -8, -11, -1, -3, -3], [11, -4, -7, 4, 10, -5, -5, -2, -2, -1, -2, 6], [-14, 0, 5, -1, -8, 9, 7, 2, -2, 0, 2, -10], [5, 8, -4, -4, 4, -18, 10, 3, 6, 0, -11, 4], [2, 5, 15, 2, 0, 2, 7, 0, -9, -2, 10, -7], [8, -9, -5, -1, -1, 2, 2, 5, 1, -11, -2, 6], [7, 1, 5, 12, 12, -16, -1, -10, -2, 5, -9, 6], [-5, -4, 5, 17, 6, 5, -2, -6, 0, 10, 3, -1], [5, -1, -2, -14, -15, -9, 6, 10, -6, -9, 1, 0], [-6, -10, -1, -2, -6, 9, -9, 6, 3, -4, -12, 5], [-9, -1, -10, -13, -9, 8, -7, 3, 12, -2, 0, 7], [1, -5, 0, 4, 2, 10, -5, -4, 14, 5, -5, 5], [8, -1, 6, 1, -5, 0, -3, -7, 1, 5, 7, 1], [-6, 0, 2, 1, 1, 7, 4, 8, 6, 2, 16, -2], [-9, 4, 4, -5, -4, 9, 10, 4, 3, 9, 14, -9], [-1, -4, -6, -8, -8, -4, -6, -3, -5, -3, 5, -1], [-18, -3, -9, -16, -13, -8, 2, 8, -11, -16, -2, 1], [7, 7, 4, 1, -8, 8, -6, -1, -9, 1, 5, -4], [9, -4, 9, 0, -3, 8, 7, -2, 0, 3, 11, -8], [17, -7, -4, 10, 10, 0, -7, -8, -2, 6, 7, 5], [10, 4, -4, -1, 1, -9, -7, -10, 0, 4, -8, 0], [-8, 2, 12, 2, -3, 4, 8, 5, -5, 1, 6, -16], [5, 3, 10, 0, 1, -13, 3, 8, -2, 0, -7, -10], [-4, 2, -7, 0, -3, 6, -3, -6, 9, 8, 8, 3], [7, -1, -6, 5, 5, 0, -5, -11, -15, -4, 1, 8], [9, -1, 1, 3, -1, -15, -2, -5, -8, -2, -9, 5], [-20, 6, 6, -10, -14, 2, 3, 9, -1, 5, 7, -14], [-9, 8, 4, -4, -9, 2, 5, 2, -7, -2, 6, -12], [12, -1, -9, -2, -9, 7, 6, -1, -1, -12, 3, 10], [10, -5, 0, 12, -3, -1, 0, -1, -11, -12, -8, 6], [-1, 4, -4, -9, 8, -8, -2, 2, 2, 3, 2, -4], [0, -20, 14, -1, -8, 11, -6, 8, 1, 0, 16, -4], [-13, 1, 12, -11, -12, -7, 13, 7, -12, -5, -4, -15], [-12, 3, 12, -1, -9, -2, -5, 3, 9, 9, 11, -10], [-20, 4, -14, -16, -5, -8, -3, 13, 8, -5, -8, 4], [4, 0, 12, -3, -6, 14, 0, 9, 7, 0, 8, -5], [-19, 5, -6, -1, 4, -15, 2, -7, 9, 4, 10, 6], [12, -4, -4, 6, 6, 6, -7, 3, 10, 3, -4, 12], [-15, 12, -8, 1, 11, -20, -3, -4, 15, 6, -16, 5], [22, 5, 5, -3, 4, -15, 9, 5, -6, 2, -4, 3], [11, -4, 10, 0, 3, 14, 6, 5, -5, 0, 1, -7], [2, -5, -10, -4, -5, 4, -7, 0, 12, -5, 10, -4], [1, -9, -4, 2, 8, -10, -6, -7, 4, 13, 10, 1], [-23, 11, 2, -9, -2, -9, 6, 2, -6, 1, 8, -8], [1, -20, -11, 0, 1, 0, -4, 1, 5, -6, -9, 12], [-24, 8, 8, 0, 6, 5, 5, 7, -7, 5, 9, -6], [-1, 2, 8, -4, -6, -6, 3, -10, -5, -2, 0, -11], [14, 2, -8, -2, 4, 5, -3, -1, -4, -4, 6, 6], [6, 4, -5, -6, 7, 4, -3, 1, 8, 2, -2, 7], [-12, -1, -3, -2, 6, -4, -4, -3, 4, 3, -3, 3], [-16, -4, -3, -9, -15, 11, 8, 13, 3, -5, 5, -1], [-2, -9, 7, 2, -6, 5, -1, 7, -5, -1, 0, -7], [-7, -1, 10, -2, -6, 14, -7, 6, 3, 4, 0, -9], [1, -12, 9, -2, -2, 1, 6, 10, -9, -3, -3, -1], [-12, -3, 0, 1, -7, 0, -1, 14, -1, -3, 4, -1], [-10, 4, -3, -11, 5, 4, 1, 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