Modular forms of integral weight are prominent in number theory. Furthermore, there are $\theta$-functions and the $\eta$-function, having weight 1/2, which also have a rich theory.

But I have never seen a modular form of weight e.g. 1/3.

I have been wondering about this for a long time. Are there examples of modular forms of fractional weights other than multiples of 1/2? And if yes, is there are reason why they are poorly studied?