Sampling period

The signal $x(t)=e^{-t^2}text{sinc}(t)$ was sampled at interval $T$. It was then found that the discrete time Fourier transform of the sampled signal is: $X(e^{jomega})=1$. What is the minimum $T$ for which such a result is possible? If this is impossible for any $T$, explain why.

I started it but didn’t how to continue , any help? My steps so far:

begin{align}
x[n] &= e^{-n^2}text{sinc}[n] \
&= e^{-n^2}frac{text{sin}(pi n)}{pi n} \
&= frac{e^{-n^2}}{2j pi n}big(e^{jpi n} – e^{-jpi n}big) \
&= frac{1}{2j pi n}big( e^{jpi n – n^2} – e^{-jpi n – n^2} big) \
end{align}

begin{align}
X(e^{jomega}) &= 1 \
&= sum_{n=0}^{infty} x[n]e^{-jomega n} \
&= sum_{n=0}^{infty} frac{1}{j2pi n} big( e^{jpi n – n^2} – e^{-jpi n – n^2} big)
end{align}