  It is known that a massive charged scalar field can trigger a superradiant
instability in the background of a Kerr-Newman black hole. In this paper, we
present a numerical study of such an instability by using the continued
fraction method. It is shown that for given a black hole, the unstable scalar
mode with a specific azimuthal index $m$ only occurs in a finite region in the
parameter space of the scalar field. The maximum mass of the scalar cloud is
exactly the upper bound of the mass of the unstable modes. We show that due to
the electromagnetic interaction between the scalar field and the Kerr-Newman
black hole, the growth rate of the instability can be $15.7\%$ larger than that
of a scalar field in Kerr spacetime of the same rotation parameter. In
addition, we find a maximum value of the growth rate $\tau^{-1}=1.788\times
10^{-7}M^{-1}$, which is about $4\%$ larger than that in the Kerr case.
