  We show that modular forms of fractional weights on principal congruence
subgroups of odd levels, which are found by T. Ibukiyama, naturally appear as
characters being multiplied $\eta^{c_{\text{eff}}}$ of the so-called minimal
models of type $(2, p)$, where $c_{\text{eff}}$ is the effective central charge
of a minimal model. Using this fact and modular invariance property of the
space of characters, we give a different proof of the result showned by
Ibukiyama that is the explicit formula representing
$\mathrm{SL}_{2}(\mathbb{Z})$ on the space of the ibukiyama modular forms. We
also 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 in
the latter. Finally, we construct vector-valued modular forms of weight
$k\in\frac{1}{5}\mathbb{Z}_{>0}$ starting from the Ibukiyama modular forms of
weight $k\in\frac{1}{5}\mathbb{Z}_{>0}$ with some multiplier system by a
symmetric tensor product of this representation and show that the components
functions coincides with the solution space of some monic modular linear
differential equation of weight $k$ and the order $1+5k$.
