Consider the set of Siegel modular forms \(f\) of genus \(n\), weight \(r\) and level \(q\) which do not vanish at all zero-dimensional cusps. It is known that if such an \(f\) is an eigenform of some power \(T(p)^m\) \((m\geq 1)\) of the Hecke operator \(T(p)\) for at least one prime \(p\equiv \pm 1\bmod q\) and if \(r>n+1\), then \(f\) is uniquely determined by its values at the zero-dimensional cusps (see \textit{E. Freitag} [Abh. Math. Semin. Univ. Hamb. 66, 229-247 (1996; Zbl 0870.11028)]). One of the aims of the paper under review is to replace the condition ``\(p\equiv\pm 1\bmod q\)'' in this theorem by the more natural condition ``\((p,q)=1\)''. In addition, the author gives an estimate on the exponent \(m\). The proof is based on a study of the Hecke operators for \(\Delta_n[q] \supset \Gamma_n[q]\) and on an investigation of various Eisenstein series connected with the problem at hand. In fact, the functions \(f\) under consideration are expressed in terms of suitable Eisenstein series. There is also a precise description of the characters \(\chi\) for which the corresponding Eisenstein series \(E_{\chi,r}\) can be written as a linear combination of theta series.