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 \(\epsilon , F\).

我们证明了给定的任何\（\小量> 0 \） ，非零adelic希尔伯特尖形式\（{\ mathbf {F}} \）重量的\（K =（K_1，K_2，\ ldots，k_n）\ in（{\ mathbb {Z}} _ +）^ n \）和无平方水平\（\ mathfrak {n} \）中具有傅立叶系数\（C _ {{\ mathbf {f}}}（\ mathfrak {m }）\），存在一个无平方积分理想\（\ mathfrak {m} \），其中\（N（\ mathfrak {m}）\ ll k_0 ^ {3n + \ epsilon} N（\ mathfrak {n}） ^ {\ frac {6n ^ 2 + 1} {2} + \ epsilon} \）使得\（C _ {{\ mathbf {f}}}（\ mathfrak {m}）\ ne 0 \）。隐含常数取决于\（\ epsilon，F \）。