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.

的全纯ETA商的因子水平˚F在上面相对于所述重量和水平边界˚F。不幸的是，由于 Mersmann 有限性定理的无效性，这个界限仍然是隐含的。另一方面，为了检查f是否不可减，必须至少知道最小值的明确上限${米}_F$在f的适当因素的水平之间。最近，已经建立了这样一个明确的上限。但是这个界限远大于推测的最小值。在这里，通过构造f 的一个特殊因子，我们证明了${米}_F$的确等于水平˚F如果水平˚F是一个主要动力。此外，在合适的条件下，我们推广了任意水平的全纯 eta 商的特殊因子的构造。