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

Abstract

Let $f\in S_{\kappa }({\Gamma }_0(N))$ be a Hecke eigenform at $p$ with eigenvalue ${\lambda }_f(p)$ for a prime $p$ not dividing $N$. Let ${\alpha }_p$ and ${\beta }_p$ be complex numbers satisfying ${\alpha }_p+{\beta }_p={\lambda }_f(p)$ and ${\alpha }_p{\beta }_p=p^{\kappa -1}$. We calculate the norm of $f_p^{{\alpha }_p}(z)=f(z)-{\beta }_{p}f(pz)$ 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 $f$.

16 pages, with appendix by Keith Conrad. Simplified section 3 and corrected some errors. Reorganized some material in sections 3 and 4

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