arXiv subjects
Publications and source records attributed to Maya Wassercug.
In the recent paper ``Murmurations", Zubrilina showed that the Fourier coefficients of modular forms follow a certain murmuration pattern. The proof given in ``Murmurations" proceeded by estimating the terms of the Skoruppa-Zagier trace formula. However, a few technical issues arise in the proof, including an omission of certain Chebyshev factors that arise in summations from the Skoruppa-Zagier trace formula. In this work, we address these issues to solidify the proof.
Let $S_k^σ(N)$ denote the space of cusp forms of level $N$, weight $k$, and Atkin-Lehner sign pattern $σ$, and $S_k^{\mathrm{new},σ}$ denote its new subspace. In this paper, we study the asymptotic behavior of the coefficients of the $m$-th Hecke polynomial over $S_k^σ(N)$ and $S_k^{\mathrm{new},σ}(N)$. In particular, we show that in certain settings, all but finitely many of these coefficients take a particular sign. We also study settings in which the coefficients do not tend to any particular sign.