Does only filter coefficients can represent it's own filter characteristic?
You can compute the filter's frequency response to check the behavior of the filter:

$$H(\omega)=\sum_{n=0}^{N-1}h[n]e^{-jn\omega}\tag{1}$$

where $h[n]$ are the filter coefficients, and $N$ is the filter length (i.e., the number of coefficients). The frequency response $H(\omega)$ is a complex function. You can compute its magnitude and phase to evaluate the filter's magnitude and phase characteristics.

You can compute (and plot) the frequency response using Matlab's freqz command.