{ "id": "2208.14786", "version": "v1", "published": "2022-08-31T11:51:07.000Z", "updated": "2022-08-31T11:51:07.000Z", "title": "A modular analogue of a problem of Vinogradov", "authors": [ "Ratnadeep Acharya", "Sary Drappeau", "Satadal Ganguly et Olivier Ramaré" ], "comment": "14 pages, to appear at Ramanujan J", "categories": [ "math.NT" ], "abstract": "Given a primitive, non-CM, holomorphic cusp form $f$ with normalized Fourier coefficients $a(n)$ and given an interval $I\\subset [-2, 2]$, we study the least prime $p$ such that $a(p)\\in I$ . This can be viewed as a modular form analogue of Vinogradov's problem on the least quadratic non-residue. We obtain strong explicit bounds on $p$, depending on the analytic conductor of $f$ for some specific choices of $I$.", "revisions": [ { "version": "v1", "updated": "2022-08-31T11:51:07.000Z" } ], "analyses": { "subjects": [ "11M06", "11N56", "11N80" ], "keywords": [ "modular analogue", "holomorphic cusp form", "modular form analogue", "strong explicit bounds", "normalized fourier coefficients" ], "note": { "typesetting": "TeX", "pages": 14, "language": "en", "license": "arXiv", "status": "editable" } } }