= Eta-quotient
An eta-quotient is a finite product of integer powers of <Dedekind eta functions> at rescaled arguments:
$$
f(\tau)=\prod_{\delta\mid N}\eta(\delta\tau)^{r_\delta},\qquad r_\delta\in\mathbb Z.
$$
Negative exponents place factors in the denominator; its formal modular weight is $\tfrac12\sum_\delta r_\delta$. A product of this form is not automatically a holomorphic <modular form>: its transformation law and behavior at cusps require additional conditions.
In string and <conformal field theory> calculations, inverse eta powers package oscillator-state counting. For example, $\eta(\tau)^{-24}=q^{-1}\prod_{n\geq1}(1-q^n)^{-24}$ displays the vacuum offset and the degeneracies of 24 independent chiral bosonic oscillators. Checking its leading power and modular weight provides a useful consistency check before combining it with zero-mode or ghost contributions.
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