On the representation of binary quadratic forms by quadratic forms

Let $Q$ and $T$ denote two matrices that represent positive integral quadratic forms. We are investigating the solubility of $X^T Q X = T$ for an integer matrix $X$. Therefore, we consider the Siegel theta series of degree two and derive an asymptotic formula for its Fourier coefficients in the level aspect. This involves computing inner-products of the cuspidal part of theta series and evaluating Fourier coefficients of Eisenstein-Klingen series.