Answer
$\gamma = 1.0019569$
Work Step by Step
We can find $\gamma$:
$K = (\gamma-1)~mc^2$
$\gamma-1 = \frac{K}{mc^2}$
$\gamma = 1+\frac{K}{mc^2}$
$\gamma = 1+\frac{(1.0000000\times 10^3~eV)(1.602 176 462\times 10^{-19}~J/eV)}{(9.109 381 88\times 10^{-31}~kg)(2.998\times 10^8~m/s)^2}$
$\gamma = 1.0019569$
