Let $S_2^{+}(p)$ be the space of newforms of level $p$ that have Aitkin-Lehner eigenvalue +1, and $S_2^{-}(p)$ be the space that have Aitkin-Lehner eigenvalue $-1$. What is known about the asymptotics of $S_2^{+}(p)/S_2(p)$ and $S^{-}(p)/S_2(p)$ as $p$ goes to infinity?

I did a computation involving the mass formula that gave me a surprising answer to this question, but since it involves the mass formula I am not sure I did it correctly.