This question is about automorphic forms for the group $\mathrm{GL}_2$, over a rational function field. Let's say $\mathbf{F}_q$ is a finite field, and $X=\mathbf{P}^1_{\mathbf{F}_q}$ is the projective line, with function field $K=\mathbf{F}_q(T)$. For an effective divisor $D\subset X$ we have the space $S(\Gamma_0(D))$ of cusp forms with $\Gamma_0(D)$ structure. By this I mean: complex-valued smooth functions on $\mathrm{GL}_2(\mathbf{A}_K)$, which are left $\mathrm{GL}_2(K)$-invariant, which have trivial central character, which are right-invariant under the appropriate compact open subgroup (integral matrices which are upper-triangular modulo $D$), and which satisfy the cuspidality condition. This space is finite-dimensional.

What is the dimension of $S(\Gamma_0(D))$?

I'm happy to have an answer in the case that $D$ is a sum of distinct degree 1 points. As I understand it, there should be no cusp forms when $D$ is empty. I expect to get cusp forms when $D$ has degree at least 4, because then there are non-constant elliptic curves over $\mathbf{F}_q(T)$ with four places of multiplicative reduction; then Langlands' theory predicts cusp forms of this level. What if $D$ has degree $\leq 3$?