The discrete-time Fourier transform (DTFT) of a length $N$ "discrete box" sequence starting at $n=0$ is
$$\sum_{n=0}^{N-1}e^{-jn\omega}=e^{-j(N-1)/2}\frac{\sin\left(\frac{N\omega}{2}\right)}{\sin\left(\frac{\omega}{2}\right)}\tag{1}$$
The numerator of $(1)$ has zeros at frequencies
$$\omega_{0,k}=\frac{2k\pi}{N}\tag{2}$$
You can interpret the convolution as filtering the signal $x[n]=e^{-jn\pi /2}$ with a filter with its frequency response given by $(1)$. With $N=4$ you get zeros of the frequency response at $\omega_{0,k}=k\pi /2$, so the complex exponential with frequency $\omega=-\pi/2$ is completely suppressed because the filter's frequency response has a zero at exactly that frequency.