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We show that modular forms of fractional weights on principal congruencesubgroups of odd levels, which are found by T. Ibukiyama, naturally appear ascharacters being multiplied $\eta^{c_{\text{eff}}}$ of the so-called minimalmodels of type $(2, p)$, where $c_{\text{eff}}$ is the effective central chargeof a minimal model. Using this fact and modular invariance property of thespace of characters, we give a different proof of the result showned byIbukiyama that is the explicit formula representing$\mathrm{SL}_{2}(\mathbb{Z})$ on the space of the ibukiyama modular forms. Wealso find several pairs of spaces of the Ibukiyama modular forms on $\Gamma(m)$and $\Gamma(n)$ with $m|n$ having the property that the former are included inthe latter. Finally, we construct vector-valued modular forms of weight$k\in\frac{1}{5}\mathbb{Z}_{>0}$ starting from the Ibukiyama modular forms ofweight $k\in\frac{1}{5}\mathbb{Z}_{>0}$ with some multiplier system by asymmetric tensor product of this representation and show that the componentsfunctions coincides with the solution space of some monic modular lineardifferential equation of weight $k$ and the order $1+5k$.

Minimal models, modular linear differential equations and modular forms of fractional weights