Some expressions for binary theta series by eta-quotients and their applications

In this paper, we show that some single theta series of positive definite binary quadratic forms can be expressed in terms of eta-quotients. We also give two applications of our theorem as follows; (1) to prove that a certain modular form, which is a product of theta series and eta-product, has complex multiplication, (2) an explicit description for the image of η ( τ ) 2 η ( N τ ) 2 under the Hecke operator T 2 .