  A detailed study of complex-space singularities of the two-dimensional
incompressible Euler equation is performed in the short-time asymptotic
r\'egime when such singularities are very far from the real domain; this allows
an exact recursive determination of arbitrarily many spatial Fourier
coefficients. Using high-precision arithmetic we find that the Fourier
coefficients of the stream function are given over more than two decades of
wavenumbers by $\hat F(\k) = C(\theta) k^{-\alpha} \ue ^ {-k \delta(\theta)}$,
where $\k = k(\cos \theta, \sin \theta)$. The prefactor exponent $\alpha$,
typically between 5/2 and 8/3, is determined with an accuracy better than 0.01.
It depends on the initial condition but not on $\theta$. The vorticity diverges
as $s^{-\beta}$, where $\alpha+\beta= 7/2$ and $s$ is the distance to the
(complex) singular manifold. This new type of non-universal singularity is
permitted by the strong reduction of nonlinearity (depletion) which is
associated to incompressibility. Spectral calculations show that the scaling
reported above persists well beyond the time of validity of the short-time
asymptotics. A simple model in which the vorticity is treated as a passive
scalar is shown analytically to have universal singularities with exponent
$\alpha =5/2$.
