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Let $S_2^{+}(p)$ be the space of newforms of level $p$ that have Atkin-Lehner eigenvalue +1, and $S_2^{-}(p)$ be the space that have Atkin-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.

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    $\begingroup$ Both ratios are $1/2$ in the limit. The difference $S_2^- - S_2^+$ is basically a class number which is $O(p^{1/2 + \epsilon})$. [BTW it's Atkin, not Aitkin.] $\endgroup$
    – Noam D. Elkies
    Commented Mar 16, 2018 at 21:44

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