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arXiv · 1504.07381

Non-vanishing Fourier coefficients of modular forms

Peng Tian·Hourong Qin
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Abstract

In this paper, we generalize D. H. Lehmer's result to give a sufficient condition for level one cusp forms $f$ with integral Fourier coefficients such that the smallest $n$ for which the coefficients $a_n(f)=0$ must be a prime. Then we describe a method to compute a bound $B$ of $n$ such that $a_n(f)\ne0$ for all $n<B$. As examples, we achieve the explicit bounds $B_k$ for the unique cusp form $\Delta_{k}$ of level one and weight k with $k=16, 18, 20, 22, 26$ such that $a_n(\Delta_k)\ne0$ for all $n<B_k$.

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Peng Tian, Hourong Qin. 2015-04-28. Non-vanishing Fourier coefficients of modular forms. https://arxiv.org/abs/1504.07381

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