Abstract
We prove that given any \(\epsilon >0\), a non-zero adelic Hilbert cusp form \({\mathbf {f}}\) of weight \(k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n\) and square-free level \(\mathfrak {n}\) with Fourier coefficients \(C_{{\mathbf {f}}}(\mathfrak {m})\), there exists a square-free integral ideal \(\mathfrak {m}\) with \(N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }\) such that \(C_{{\mathbf {f}}}(\mathfrak {m})\ne 0\). The implied constant depends on \(\epsilon , F\).
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Acknowledgements
The authors are indebted to Professor B. Ramakrishnan for his encouragement and for many fruitful suggestions. The authors are grateful to anonymous referee for his/her valuable suggestions and remarks which improved the exposition of the paper. The authors acknowledge Harish-Chandra Research Institute for fantastic facilities and for the serene ambience that it facilitates.
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Agnihotri, R., Chakraborty, K. On the Fourier coefficients of certain Hilbert modular forms. Ramanujan J 58, 167–182 (2022). https://doi.org/10.1007/s11139-021-00401-2
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DOI: https://doi.org/10.1007/s11139-021-00401-2