On the norms of $p$-stabilized elliptic newforms (with an appendix by Keith Conrad)

Let $f \in S_κ(Γ_0(N))$ be a Hecke eigenform at $p$ with eigenvalue $λ_f(p)$ for a prime $p$ not dividing $N$. Let $α_p$ and $β_p$ be complex numbers satisfying $α_p + β_p = λ_f(p)$ and $α_p β_p = p^{κ-1}$. We calculate the norm of $f_{p}^{α_p}(z) = f(z) - β_{p} f(pz)$ as well as the norm of $U_p f$, both classically and adelically. We use these results along with some convergence properties of the Euler product defining the symmetric square L-function of $f$ to give a `local' factorization of the Petersson norm of $f$.

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