Let $N \in \mathbb{N}$, let $X_0(N)$ be the compact modular curve attached to $\Gamma_0(N)$ over $\mathbb{C}$, and let $H^1_B(X_0(N), \mathbb{C})$ denote its first Betti cohomology with complex coefficients. For each $n \in \mathbb{N}$, let $T_n$ denote both the classical Hecke operator on $S_2(\Gamma_0(N))$ and the Hecke operator on $H^1_B(X_0(N), \mathbb{C})$ induced by the usual Hecke correspondence on $X_0(N)$.

There is a natural Hecke-equivariant complex-linear isomorphism

\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*}

More precisely, under this isomorphism the first summand corresponds to holomorphic differentials on $X_0(N)$ and the second summand corresponds to antiholomorphic differentials. If $f \in S_2(\Gamma_0(N))$ is a nonzero simultaneous Hecke eigenform satisfying

\begin{align*} T_n f = a_n(f) f \end{align*}

for every $n \in \mathbb{N}$, then the holomorphic summand contains a Hecke eigenline on which $T_n$ acts by $a_n(f)$, and the antiholomorphic summand contains a Hecke eigenline on which $T_n$ acts by $\overline{a_n(f)}$.