Let $N \in \mathbb{N}$, and let $X_0(N)$ denote the compact complex modular curve associated to the congruence subgroup $\Gamma_0(N) \subset SL_2(\mathbb{Z})$. Let $H^1_B(X_0(N), \mathbb{C})$ denote the first Betti cohomology of the underlying compact Riemann surface with complex coefficients, and let $H^0(X_0(N), \Omega^1)$ denote the complex [vector space](/page/Vector%20Space) of holomorphic differentials on $X_0(N)$.

Under the de Rham comparison map, the assignment $\omega \mapsto [\omega]$ embeds $H^0(X_0(N), \Omega^1)$ into $H^1_B(X_0(N), \mathbb{C})$, and complex conjugation embeds the conjugate space $\overline{H^0(X_0(N), \Omega^1)}$ by $\omega \mapsto [\overline{\omega}]$. There is a canonical direct-sum decomposition

\begin{align*} H^1_B(X_0(N), \mathbb{C}) = H^0(X_0(N), \Omega^1) \oplus \overline{H^0(X_0(N), \Omega^1)}. \end{align*}

Moreover, under the standard isomorphism

\begin{align*} S_2(\Gamma_0(N)) &\longrightarrow H^0(X_0(N), \Omega^1) \\ f &\longmapsto f(z)\,dz, \end{align*}

this decomposition becomes

\begin{align*} H^1_B(X_0(N), \mathbb{C}) \cong S_2(\Gamma_0(N)) \oplus \overline{S_2(\Gamma_0(N))}. \end{align*}