  Given a projective smooth curve $X$ over any field $k$, we discuss two
notions of global $\mathbb{A}^1$ degree of a finite morphism of smooth curves
$f: X \to \mathbb{P}^1_k$ satisfying certain conditions. One originates from
computing the Euler number of the pullback of the line bundle
$\mathscr{O}_{\mathbb{P}^1}(1)$ as a generalization of Kass and Wickelgren's
construction of Euler numbers. The other originates from the construction of
global $\mathbb{A}^1$ degree of morphisms of projective curves by Kass, Levine,
Solomon, and Wickelgren as a generalization of Morel's construction of
$\mathbb{A}^1$-Brouwer degree of a morphism $f: \mathbb{P}^1_k \to
\mathbb{P}^1_k$. We prove that under certain conditions on $N$, both notions of
global $\mathbb{A}^1$ degrees of covering maps between modular curves $X_0(N)
\to X(1)$, $X_1(N) \to X(1)$, and $X(N) \to X(1)$ agree to be equal to sums of
hyperbolic elements $\langle 1 \rangle + \langle -1 \rangle$ in the
Grothendieck-Witt ring $\mathrm{GW}(k)$ for any field $k$ whose characteristic
is coprime to $N$ and the pullback of $\mathscr{O}_{\mathbb{P}^1}(1)$ is
relatively oriented.
