Newforms of squarefree level and theta series

We show that the cuspidal part of the span of the theta series associated to maximal integral lattices on a definite quaternion algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {A}}$$\end{document} over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Q}}$$\end{document} is precisely the space of newforms. Furthermore, using the Eichler commutation relation and an idea coming out of the Yoshida lifting, we show how to express each newform as an explicit linear combination of such theta series. The coefficients of these linear combinations come from cuspidal eigenvectors of Brandt matrices.