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

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... nstable 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 τ^-1=1.788× 10^-7M^-1, which is about 4% larger than that in the Kerr case.