This is a continuation of my RTG colloquium lecture on November 8. In this lecture, we study the method of Birch in more detail, to see how it can be used to compute essentially arbitrary spaces of classical modular forms. This involves relating Birch's construction to orthogonal modular forms and Clifford algebras, and applying a form of the Jacquet-Langlands correspondence. We also report on some limited computational evidence that this method can also be applied to GSp(4) Siegel modular forms. A short computer demonstration using Sage may be included if time permits. Note: this is a report on the PhD thesis of Jeffery Hein, written under John Voight at Dartmouth in consultation with Gonzalo Tornaria.