  We present a new filtered low-regularity Fourier integrator for the cubic
nonlinear Schr\"odinger equation based on recent time discretization and
filtering techniques. For this new scheme, we perform a rigorous error analysis
and establish better convergence rates at low regularity than known for
classical schemes in the literature so far. In our error estimates, we combine
the better local error properties of the new scheme with a stability analysis
based on general discrete Strichartz-type estimates. The latter allow us to
handle a much rougher class of solutions as the error analysis can be carried
out directly at the level of $L^2$ compared to classical results \black in
dimension $d$, \black which are limited to higher-order (sufficiently smooth)
Sobolev spaces $H^s$ with $s>d/2$. In particular, we are able to establish a
global error estimate in $L^2$ for $H^1$ solutions which is roughly of order
$\tau^{ {1\over 2} + { 5-d \over 12} }$ in dimension $d \leq 3$ ($\tau$
denoting the time discretization parameter). This breaks the "natural order
barrier" of $\tau^{1/2}$ for $H^1$ solutions which holds for classical
numerical schemes (even in combination with suitable filter functions).
