  We propose and analyze a novel symmetric Gautschi-type exponential wave
integrator (sEWI) for the nonlinear Schr\"odinger equation (NLSE) with low
regularity potential and typical power-type nonlinearity of the form $
|\psi|^{2\sigma}\psi $ with $ \psi $ being the wave function and $ \sigma > 0 $
being the exponent of the nonlinearity. The sEWI is explicit and stable under a
time step size restriction independent of the mesh size. We rigorously
establish error estimates of the sEWI under various regularity assumptions on
potential and nonlinearity. For ``good" potential and nonlinearity
($H^2$-potential and $\sigma \geq 1$), we establish an optimal second-order
error bound in the $L^2$-norm. For low regularity potential and nonlinearity
($L^\infty$-potential and $\sigma > 0$), we obtain a first-order $L^2$-norm
error bound accompanied with a uniform $H^2$-norm bound of the numerical
solution. Moreover, adopting a new technique of \textit{regularity compensation
oscillation} (RCO) to analyze error cancellation, for some non-resonant time
steps, the optimal second-order $L^2$-norm error bound is proved under a weaker
assumption on the nonlinearity: $\sigma \geq 1/2$. For all the cases, we also
present corresponding fractional order error bounds in the $H^1$-norm, which is
the natural norm in terms of energy. Extensive numerical results are reported
to confirm our error estimates and to demonstrate the superiority of the sEWI,
including much weaker regularity requirements on potential and nonlinearity,
and excellent long-time behavior with near-conservation of mass and energy.
