Some applications of eta-quotients

We show that every modular form on ${\mathrm{\Gamma }}_0(2^n)$ ($n\ge 2$) can be expressed as a sum of eta-quotients. Furthermore, we construct a primitive generator of the ring class field of the order of conductor $4N$ ($N\ge 1$) in an imaginary quadratic field in view of the special value of certain eta-quotient.