  The Schur limit of the superconformal index of a four-dimensional N = 2
superconformal field theory encodes rich physical information about the
protected spectrum of the theory. For a Lagrangian model, this limit of the
index can be computed by a contour integral of a multivariate elliptic
function. However, surprisingly, so far it has eluded exact evaluation in
closed, analytical form. In this paper we propose an elementary approach to
bring to heel a large class of these integrals by exploiting the ellipticity of
their integrand. Our results take the form of a finite sum of (products of) the
well-studied flavored Eisenstein series. In particular, we derive a compact
formula for the fully flavored Schur index of all theories of class S of type
a1, we put forward a conjecture for the unflavored Schur indices of all N=4
super Yang-Mills theories with gauge group SU(N), and we present closed-form
expressions for the index of various other gauge theories of low ranks. We also
discuss applications to non-Lagrangian theories, modular properties, and defect
indices.
