Special factors of holomorphic eta quotients

The levels of the factors of a holomorphic eta quotient f are bounded above with respect to the weight and the level of f. Unfortunately, this bound remains implicit due to the ineffectiveness of Mersmann's finiteness theorem. On the other hand, for checking whether f is irredicble, it is essential to know at least an explicit upper bound for the minimum m_f among the levels of the proper factors of f. Recently, such an explicit upper bound has been established. But this bound is far larger than the conjectured minimum. Here, by constructing a special factor of f, we show that the least upper bound for m_f is indeed equal to the level of f if the level of f is a prime power. Also, under suitable conditions we generalize the construction of the special factors for holomorphic eta quotients of arbitrary levels.