Modular forms for the $A_{1}$-tower

In the 1960's Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct $\chi_{5}$, the cusp form of lowest weight for the group $\operatorname{Sp}(2;Z)$. In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identifed with the orthogonal group of signature $(2,3)$ for the lattice $A_{1}$ and Igusa's form $\chi_{5}$ appears as the roof of this tower. We use this interpretation to construct a framework for this tower which uses three different types of constructions for modular forms. It turns out that our method produces simple coordinates for the graded rings of modular forms.