The number of representations is calculated by N. Bagis and M.L Glasser in On the Number of Representations of Integers by various Quadratic and Higher Forms (2014):
http://ilorentz.org/beenakker/MO/sumofsquares_2.png
where $r(n)$ is Jacobi's formula for the number of representations of $n=x^2+y^2$.