On the first sign change of Fourier coefficients of cusp forms at sum of two squares
Since Fourier coefficients of modular forms carry a lot of arithmetic information, several problems about them are being studied. One of them is certainly the sign change problem. There are many papers about sign change problem with different settings. In the paper under review, the authors consider the problem of first sign change of Fourier coefficients of a primitive cusp form at a sparse sequence, where elements of the sparse sequence can be written as a sum of two squares. As a main result, they give an effective upper bound on elements of such a sparse sequence to have first sign change. The main tool is coming from \textit{W. Kohnen} and \textit{J. Sengupta}'s work [Math. Z. 254, No. 1, 173--184 (2006; Zbl 1220.11060)] based on analytic number theory calculations.
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