  A new concept of a deformed numerical range $W^\rho(T)$ is introduced. Here
$T$ is a bounded linear operator or a matrix and $ \rho \in[1,+\infty)$ is a
parameter. Each $W^\rho(T)$ is a closed convex set that contains the spectrum
of $T$. Furthermore, $W^\rho(T)$ is decreasing with respect to $ \rho $ and
$W^2(T)$ coincides with the numerical range. It is also shown that $W^\rho(T)$
is contained in the closed unit disc if and only if $T$ has a $\rho$ unitary
dilation in the sense of N\'agy-Foia\c s. The spectral constants of $W^\rho(T)$
are investigated, it is shown that it is monotone and continuous with respect
to the parameter $ \rho $.
