# q-expansion of newform 2940.2.q.p, downloaded from the LMFDB on 30 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 = 2940
weight = 2
poly_data = [4, 0, 2, 0, 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], 1], [[0, 1, 0, 0], 1], [[0, 0, 1, 0], 2], [[0, 0, 0, 1], 2]]

hecke_ring_character_values = [[1471, [1, 0, 0, 0]], [1961, [1, 0, 0, 0]], [1177, [1, 0, 0, 0]], [1081, [-1, 0, -1, 0]]]
aps_data = [[0, 0, 0, 0], [0, 0, 1, 0], [1, 0, 1, 0], [0, 0, 0, 0], [0, 0, -2, 0], [0, 0, 0, -2], [0, -2, -4, -2], [-4, 1, -4, 0], [6, -1, 6, 0], [-2, 0, 0, 4], [0, -3, 0, -3], [2, -6, 2, 0], [-2, 0, 0, 0], [-4, 0, 0, 4], [2, -6, 2, 0], [0, -1, -6, -1], [0, -2, 0, -2], [0, -7, 0, 0], [0, 2, -4, 2], [-2, 0, 0, -6], [0, -6, -4, -6], [10, 0, 10, 0], [-2, 0, 0, -2], [2, -2, 2, 0], [4, 0, 0, 0], [0, -4, 6, -4], [0, 2, 0, 0], [2, 5, 2, 0], [0, -4, -10, -4], [-2, 0, 0, -3], [0, 0, 0, 10], [0, 8, 0, 0], [0, 7, 2, 7], [8, 0, 0, -11], [14, -4, 14, 0], [0, -8, -6, -8], [0, -10, -4, -10], [4, -10, 4, 0], [-16, 0, 0, 0], [0, -10, 0, 0], [0, 6, -2, 6], [20, 0, 0, 3], [2, 12, 2, 0], [0, -2, 10, -2], [6, 0, 0, 5], [0, -3, 4, -3], [-12, 0, 0, -4], [8, 0, 0, -2], [0, 0, -4, 0], [4, -1, 4, 0], [10, 9, 10, 0], [-14, 0, 0, -4], [0, -1, 28, -1], [20, 0, 0, -6], [10, -8, 10, 0], [0, 5, -10, 5], [0, -2, -18, -2], [-16, 3, -16, 0], [0, -8, -6, -8], [14, 0, 0, -12], [0, 6, 0, 6], [2, 0, 0, -4], [-4, 0, 0, 4], [0, 8, 20, 8], [-20, 6, -20, 0], [-10, -7, -10, 0], [12, -16, 12, 0], [14, 0, 0, 14], [0, 1, 14, 1], [8, 0, 0, 15], [0, -20, 2, -20], [-10, 12, -10, 0], [0, 2, -28, 2], [-14, -8, -14, 0], [-2, 0, 0, 24], [18, 10, 18, 0], [0, 2, 10, 2], [0, -20, 0, 0], [-30, 6, -30, 0], [0, 9, -12, 9], [-16, 0, 0, -12], [-4, 0, 0, -20], [0, -6, 30, -6], [16, 0, 0, -2], [-12, 9, -12, 0], [22, 1, 22, 0], [22, 0, 0, 0], [-18, -6, -18, 0], [-14, 0, 0, 18], [-20, 0, 0, 10], [4, 24, 4, 0], [0, 24, 0, 24], [0, 20, 8, 20], [2, 0, 0, -22], [2, -20, 2, 0], [-30, 0, 0, -2], [30, -6, 30, 0], [0, 22, 10, 22], [16, 2, 16, 0], [-8, 20, -8, 0], [-4, 0, 0, 22], [0, 21, -2, 21], [0, 6, -30, 6], [-26, 6, -26, 0], [0, -20, -8, -20], [0, 2, 0, 2], [14, 0, 0, 2], [-14, -12, -14, 0], [0, 0, -14, 0], [24, 0, 0, -3], [4, 18, 4, 0], [0, -8, 26, -8], [6, 0, 0, 21], [0, -7, -24, -7], [18, 0, 0, -8], [0, 12, 26, 12], [4, 0, 0, 20], [0, 8, -24, 8], [-18, -1, -18, 0], [-10, 0, 0, 14], [0, 3, 40, 3], [38, 0, 0, 2], [-26, 12, -26, 0], [0, -11, -18, -11], [32, 9, 32, 0], [-14, 0, 0, -12], [-20, -8, -20, 0], [-4, 6, -4, 0], [-4, 0, 0, -28], [-16, 6, -16, 0], [0, 16, 16, 16], [6, 0, 0, -17], [-8, 4, -8, 0], [-2, 0, 0, 2], [2, 6, 2, 0], [8, 0, 0, -15], [0, -18, 12, -18], [0, 22, -8, 22], [-16, 0, 0, 22], [0, 6, 14, 6], [8, 0, 0, -7], [2, 30, 2, 0], [0, -18, 0, -18], [-10, 0, 0, -3], [0, -5, -4, -5], [24, 0, 0, 6], [12, 0, 0, 20], [0, 18, 0, 18], [-20, -7, -20, 0], [26, -7, 26, 0], [30, -12, 30, 0], [46, 0, 0, 0], [0, 0, 0, -12], [30, 2, 30, 0], [0, -4, 28, -4], [-14, 0, 0, -24], [0, 20, 0, 0], [-22, -2, -22, 0], [-12, 0, 0, -4], [0, 32, -6, 32], [-30, -5, -30, 0], [22, 0, 0, 17], [-40, 0, 0, 14], [-12, 6, -12, 0], [0, -17, 6, -17], [0, -28, 16, -28], [0, 8, 0, 8], [0, -8, -8, -8], [18, 0, 0, -32], [22, -16, 22, 0], [0, 28, 6, 28], [8, 0, 0, 17], [-18, -6, -18, 0], [0, 32, 2, 32], [0, 13, 4, 13], [-10, 0, 0, 10], [18, 0, 0, -12], [0, 2, -22, 2], [-28, 0, 0, 4], [24, -7, 24, 0], [28, 2, 28, 0], [-4, 0, 0, 2], [-38, 0, 0, -4], [32, 22, 32, 0], [0, -35, -10, -35], [0, 8, -30, 8], [0, 2, -54, 2], [0, 2, -36, 2], [8, 20, 8, 0], [0, 2, 20, 2], [20, -4, 20, 0], [-30, 0, 0, 3], [10, 4, 10, 0], [-10, 2, -10, 0], [0, 23, 2, 23], [0, -36, -10, -36], [0, 4, 0, 4], [14, 24, 14, 0], [-26, 0, 0, -28], [24, -24, 24, 0], [0, -38, -14, -38], [-20, 0, 0, -25], [-28, -16, -28, 0], [0, -7, 12, -7], [0, 0, 0, -2], [0, 22, 0, 22], [12, -9, 12, 0], [54, -5, 54, 0], [14, 0, 0, 2], [0, -35, -20, -35], [14, 36, 14, 0], [14, 0, 0, 18], [4, 0, 0, 20], [-34, -22, -34, 0], [0, -18, -40, -18], [-8, -25, -8, 0], [0, 8, -12, 8], [0, 0, 18, 0], [42, 11, 42, 0], [-10, 0, 0, 13], [12, 16, 12, 0], [-48, 0, 0, 9], [34, -26, 34, 0], [-28, -16, -28, 0], [4, 0, 0, 8], [-24, 0, 0, -24], [-50, -4, -50, 0], [0, 6, 22, 6], [28, -10, 28, 0], [10, 10, 10, 0], [0, 30, -6, 30], [0, -23, 4, -23], [-38, 0, 0, 16], [0, -18, 34, -18], [-44, 0, 0, -4], [0, -12, 16, -12], [44, 11, 44, 0], [34, -3, 34, 0], [-42, 0, 0, 10], [0, 0, 0, 2], [0, -7, -46, -7], [24, -11, 24, 0], [0, -8, -20, -8], [34, -4, 34, 0], [-24, 0, 0, -38], [-44, -6, -44, 0], [8, 0, 0, 20], [0, 20, -32, 20], [0, 8, -54, 8], [-6, 0, 0, -21], [-18, 0, 0, -22], [-42, 24, -42, 0], [0, -23, -6, -23], [-36, 0, 0, 5], [0, -26, -4, -26], [34, 24, 34, 0], [0, 12, -60, 12], [0, -18, -20, -18], [36, 0, 0, -10], [12, -6, 12, 0], [0, 14, 12, 14], [14, 0, 0, 27], [0, 3, 20, 3], [8, 0, 0, 14], [0, -4, -38, -4], [-8, -41, -8, 0], [-14, 0, 0, 22], [2, 0, 0, 6], [48, 0, 0, -14], [0, -13, -26, -13], [32, 7, 32, 0], [0, -20, -52, -20], [0, 16, 24, 16], [8, 20, 8, 0], [16, 0, 0, -12], [-16, -12, -16, 0], [-42, -21, -42, 0], [0, -16, -32, -16], [60, -8, 60, 0], [-8, 30, -8, 0], [0, -16, -16, -16], [0, 32, 2, 32], [56, 0, 0, 16], [-40, 0, 0, 11], [52, -22, 52, 0], [-66, -8, -66, 0], [0, -30, 30, -30], [-22, 0, 0, -19], [0, -13, 8, -13], [2, 0, 0, -6], [0, 38, 14, 38], [0, 18, 34, 18], [26, -31, 26, 0], [0, 0, 0, -20], [-22, 0, 0, -16], [0, 38, 2, 38], [-20, -3, -20, 0], [-2, 0, 0, 12], [-36, 10, -36, 0], [-36, 0, 0, -2], [-24, 20, -24, 0], [34, -27, 34, 0], [0, 44, 12, 44]]