Let \(G\) be a subgroup of finite index of the full modular group \(\Gamma\), \(G_0\) be the subgroup of linear translations lying in \(G\). Denoting a generator of the group \(G_0\) by \(T^q=\begin{pmatrix} 1 & q \\ 0 & 1\end{pmatrix}\) and letting \(\mathcal K\) be a set of representatives of left cosets of \(G\) modulo \(G_0\), in the upper complex half plane \(H\) we define the Poincaré series of weight \(k\) and character \(m\) with respect to the subgroup \(G\): \[ \phi_m(\tau,k;G) = \sum_{M\in\mathcal K}\frac{\exp \{\frac1{q}2\pi imM(\tau)\}}{\mu_M(\tau)}. \] Here \(\tau =x+iy\) is a complex variable, \(M=\begin{pmatrix} a & b \\ c & d\end{pmatrix}\), \(\mu_M(\tau)=(c\tau +d)^k\), \(k\) is an even positive integer, \(m\) is a natural number.] The author proves the following theorem. There exist positive constants \(k_0\) and \(B\), where \(B>4 \log 2\), such that for all \(k\geq k_0\) and all positive integers \(m\leq k^2 \exp \{-B \log k/\log \log k\}\), the Poincaré series \(\phi_{qm}(\tau,k;G)\) is identically different from zero. Moreover, the series \(\phi_q(\tau,k;G)\), \(\phi_{2q}(\tau,k;G),\ldots,\phi_{dq}(\tau,k;G)\), where \(d=\dim S_k(\Gamma)\), are linearly independent. (Here \(S_k(\Gamma)\) is the space of cusp forms of weight \(k\) on \(\Gamma\).) This theorem generalizes results of \textit{H. Petersson} [Jahresber. Dtsch. Math.-Ver. 49, 49--75 (1939; Zbl 0021.02502)] and \textit{R. A. Rankin} [Proc. Edinb. Math. Soc., II. Ser. 23, 151--161 (1980; Zbl 0454.10013)] for the case \(q=1\).