Given an arithmetic variety $\mathscr{X}$ and a hermitian line bundle $\overline{\mathscr{L}}$, the arithmetic Hilbert-Samuel theorem describes the asymptotic behavior of the co-volumes of the lattices $H^0(\mathscr{X}, \mathscr{L}^{\otimes k})$ in the normed spaces $H^0(\mathscr{X}, \mathscr{L}^{\otimes k}) \otimes \mathbb{R}$ as $k \to \infty$. Using his work on quasi-filtered graded algebras, Chen proved a variant of the arithmetic Hilbert-Samuel theorem which studies the asymptotic behavior of the successive minima of the lattices above. Chen's theorem, however, requires that the metric on $\overline{\mathscr{L}}$ is continuous, and hence does not apply to automorphic vector bundles for which the natural metrics are often singular. In this thesis, we discuss a version of Chen's theorem for the line bundle of modular forms for a finite index subgroup $\Gamma \subseteq \text{PSL}_2(\mathbb{Z})$ endowed with the logarithmically singular Petersson metric. This generalizes work of Chinburg, Guignard, and Soul'{e} addressing the case $\Gamma = \text{PSL}_2(\mathbb{Z})$.