Central $L$-values of newforms and local polynomials

In this paper, we characterize the vanishing of twisted central $L$-values attached to newforms of square-free level in terms of so-called local polynomials and the action of finitely many Hecke operators thereon. Such polynomials are the ``local part'' of certain locally harmonic Maass forms constructed by Bringmann, Kane and Kohnen in $2015$. We offer a second perspective on this characterization for weights greater than $4$ by adapting results of Zagier to higher level. To be more precise, we establish that a twisted central $L$-value attached to a newform vanishes if and only if a certain explicitly computable polynomial is constant. We conclude by proving an identity between these constants and generalized Hurwitz class numbers, which were introduced by Pei and Wang in $2003$. We provide numerical examples in weight $4$ and levels $7$, $15$, $22$, and offer some questions for future work.