Let \(f(z)=\sum_{n\geq 1} a_f(n) n^{(k-1)/2} e(nz)\) be a holomorphic eigenform form of level 1 and weight \(k\), and let \(\chi\) be a Dirichlet character \(\pmod q\). The corresponding twisted \(L\)-function is defined for \(\text{Re}(s)> 1\) as follows \[ L(f\times \chi,s)= \sum^\infty_{n=1} {a_f(n)\chi(n)\over n^s}= \prod_p \Biggl(1-{a_f(p) \chi(p)\over p^s}+ {\chi^2(p)\over p^{2s}}\Biggr)^{-1} \] and by the analytic continuation elsewhere. The authors study square mean values of these \(L\)-functions at the central point of the critical line and prove the following result. (Theorem 1.1) For integers \(q\) satisfying \[ \sum_{p|q,p> x} {1\over p}\leq(\log\log q)^{-10}, \] where \(x=\exp((\log\log q)/(200\log\log\log q))\), we have \[ \sum_{\substack{\chi\pmod q\\ \chi\text{-primitive}}}|L(f\times\chi, 1/2)|^2= KP_q(1) \psi(q) q\log q(1+ O((\log\log q)^{-1})), \] where \(K= 6(4\pi)^k\| f\|^2/(\pi\Gamma(k))\) (\(\|\cdot\|\) being the Petersson norm), \[ \begin{aligned} P_q(1) &= \prod_{p|q} \Biggl(1-{1\over p}\Biggr)^{-1}\,\Biggl(1- {a_f(p)^2- 2\over p}+ {1\over p^2}\Biggr),\\ \psi(q) &= \prod_{p||q} \Biggl(1-{2\over p}\Biggr) \prod_{p^2|q} \Biggl(1-{1\over p}\Biggr)^2\end{aligned} \] and the implied constant depends on \(f\). This result should be compared with [T. Stefanicki, J. Reine Angew. Math. 474, 1–24 (1996; Zbl 0848.11023)], where a similar asymptotic formula was obtained with a better remainder term but in a more restricted range for \(q\).