Bounds on the multiplicity of the Hecke eigenvalues

Abstract

Fix an integer $N$ and a prime $p\nmid N$ where $p\ge 5$. We show that the number of newforms $f$ (up to a scalar multiple) of level $N$ and even weight $k$ such that ${\mathcal{T}}_p(f)=0$ is bounded independently of $k$, where ${\mathcal{T}}_p(f)$ is the Hecke operator.

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