  We study Cohen-Macaulay non-Gorenstein local rings $(R,\mathfrak{m},k)$
admitting certain totally reflexive modules. More precisely, we give a
description of the Poincar\'{e} series of $k$ by using the Poincar\'{e} series
of a non-zero totally reflexive module with minimal multiplicity. Our results
generalize a result of Yoshino to higher-dimensional Cohen-Macaulay local
rings. Moreover, from a quasi-Gorenstein ideal satisfying some conditions, we
construct a family of non-isomorphic indecomposable totally reflexive modules
having an arbitrarily large minimal number of generators.
