In recent years, our understanding of stable homotopy groups of spheres at $p=2$ increased drastically due to work of Isaksen, Wang, and Xu. A primary method they used is the "cofiber-of-tau philosophy" in the stable infinity category of 2-complete $\mathbb{C}$-motivic spectra. To a sufficiently nice spectrum $E$, Pstragowski produced an infinity-categorical deformation of spectra called "$E$-synthetic spectra," which exhibits and generalizes the cofiber-of-tau phenomena seen in $\mathbb{C}$-motivic spectra. $E$-synthetic spectra are closely related to the $E$-Adams spectral sequence and this relation has had many applications in recent years for Adams spectral sequence calculations. 

In this talk, we discuss some of the basic calculational features of synthetic spectra in the case of $E=H\mathbb{F}_2$, including how to compute bigraded synthetic homotopy groups and their applications to classical Adams spectral sequence calculations for $p=2$. In particular, we discuss our computation of the bigraded synthetic homotopy groups of 2-complete tmf, the connective topological modular forms spectrum.