In this work, the authors prove that Hermitian cusp forms of weight \(k\) for he Hermitian modular group of degree \(2\) are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square-free. They prove the following main theorem:
Theorem 1. Let \(F\in S_{k}(O_{k})\) be non-zero. Then
(a) \(a(F,T)\neq 0\) for infinitely many matrices \(T\) such that \( |D_{K}|\det (T)\) is of the form \(p_{K}^{\alpha }n\), where \(n\) is square-free with \((n,p_{K})=1 \) and \(0\leq \alpha \leq 2\) if \(D_{K}\neq -8\) and \(0\leq \alpha \leq 3\) if \(D_{K}=-8.\)
(b) For and \(\varepsilon >0\), \(\#\{0<n<X,\ n\text{ square-free, }(n,p_{K})=1,a(F,T)\neq 0,p_{K}^{\alpha }n=|D_{K}|\det (T)\}\gg _{F,\varepsilon }X^{1-\varepsilon }.\)