  In this paper, we develop an ultra-weak discontinuous Galerkin (DG) method to
solve the one-dimensional nonlinear Schr\"odinger equation. Stability
conditions and error estimates are derived for the scheme with a general class
of numerical fluxes. The error estimates are based on detailed analysis of the
projection operator associated with each individual flux choice. Depending on
the parameters, we find out that in some cases, the projection can be defined
element-wise, facilitating analysis. In most cases, the projection is global,
and its analysis depends on the resulting $2\times2$ block-circulant matrix
structures. For a large class of parameter choices, optimal $\textit{a priori}$
$L^2$ error estimates can be obtained. Numerical examples are provided
verifying theoretical results.
