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arXiv · 2104.10055

Divisors of Fourier coefficients of two newforms

Arvind Kumar·Moni Kumari
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Abstract

For a pair of distinct non-CM newforms of weights at least 2, having rational integral Fourier coefficients $a_{1}(n)$ and $a_{2}(n)$, under GRH, we obtain an estimate for the set of primes $p$ such that $$ \omega(a_1(p)-a_2(p)) \le [ 7k+{1}/{2}+k^{1/5}],$$ where $\omega(n)$ denotes the number of distinct prime divisors of an integer $n$ and $k$ is the maximum of their weights. As an application, under GRH, we show that the number of primes giving congruences between two such newforms is bounded by $[7k+{1}/{2}+k^{1/5} ]$. We also obtain a multiplicity one result for newforms via congruences.

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Arvind Kumar, Moni Kumari. 2021-04-20. Divisors of Fourier coefficients of two newforms. https://doi.org/10.2140/pjm.2023.326.85

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