  We give new necessary and sufficient conditions for the numerical range
$W(T)$ of an operator $T \in \mathcal{B}(\mathcal{H})$ to be a subset of the
closed elliptical set $K_\delta \subseteq \mathbb{C}$ given by \[ K_\delta
{\stackrel{\rm def}{=}}
  \left\{x+iy: \frac{x^2}{(1+\delta)^2} + \frac{y^2}{(1-\delta)^2} \leq
1\right\}, \] where $0 < \delta < 1$. Here $\mathcal{B}(\mathcal{H})$ denotes
the collection of bounded linear operators on a Hilbert space $\mathcal{H}$.
Central to our efforts is a direct generalization of Berger's well-known
criterion for an operator to have numerical radius at most one, his so-called
strange dilation theorem.
  We next generalize the lemma of Sarason that describes power dilations in
terms of semi-invariant subspaces to operators $T$ that satisfy appropriate
dilation properties. This generalization yields a characterization of the
operators $T\in \mathcal{B}(\mathcal{H})$ such that $W(T)$ is contained in
$K_\delta$ in terms of certain structured contractions that act on $\mathcal{H}
\oplus \mathcal{H}$.
  As a corollary of our results we extend Ando's parametrization of operators
having numerical range in a disc to those $T$ such that $W(T)\subseteq
K_\delta$. We prove that, if $T$ acts on a finite-dimensional Hilbert space
$\mathcal{H}$, then $W(T)\subseteq K_\delta$ if and only if there exist a pair
of contractions $A,B \in \mathcal{B}(\mathcal{H})$ such that $A$ is
self-adjoint and \[ T=2\sqrt\delta A + (1-\delta)\sqrt{{1+A}}\ B\sqrt{{1-A}}.
\] We also obtain a formula for the B. and F. Delyon calcular norm of an
analytic function on the inside of an ellipse in terms of the extremal
$H^\infty$-extension problem for analytic functions defined on a slice of the
symmetrized bidisc.
