Dilating a Lattice Polygon

Below is a drawing of the lattice polygon $P$. It has 16 integer points (i.e., points with integer coordinates) on its boundary (indicated by circles), and none in its interior (shaded light gray above). The polygon $26P$ is obtained by dilating $P$ by a factor of 26. That is, it consists of the points $(26x, 26y)$ wherein $(x,y)$ is a point in the interior or on the boundary of $P$. How many integer points are in the interior of $26P$?