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 irreducible, 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. In the case where the level of f is a prime power, the least upper bound for \(m_f\) has been recently determined via construction of a special factor. However, this construction does not generalize unconditionally to arbitrary levels. Here, we provide an explicit upper bound \(M_N\) for the minimum of the levels of the proper factors of a holomorphic eta quotient f of level N.

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是否不可约，必须知道f的适当因子中的最小值\（m_f \）的至少一个明确的上限。在f的级别为素数幂的情况下，\(m_f\)的最小上界最近通过构建一个特殊的因素来确定。但是，这种构造不会无条件地推广到任意级别。在这里，我们为N级 的全纯 eta 商f的适当因子的最小值提供了一个明确的上限\(M_N\)。