  We give an explicit formula for dimensions of spaces of rational-weight
modular forms whose multiplier systems are induced by eta-quotients of
fractional exponents. As the first application, we give series expressions of
Fourier coefficients of the $n$-th root of certain infinite $q$-products. As
the second application, we extend Yves Martin's list of multiplicative
holomorphic eta-quotients of integral weights by first extending the meaning of
multiplicativity, then identifying one-dimensional spaces, and finally applying
Wohlfahrt's extension of Hecke operators. A table containing $2277$ of such
eta-quotients is presented. As a related result, we completely classify the
multiplier systems induced by eta-quotients of integral exponents. For
instance, there are totally $384$ such multiplier systems on $\Gamma_0(4)$ for
any fixed weight. We also provide SageMath programs on checking the theorems
and generating the tables.
