Let $k \ge 2$ and $N \ge 1$ be integers. Let $Y_1(N)(\mathbb C)$ be the complex modular curve of level $\Gamma_1(N)$, and let $\mathbb V_{k-2,\mathbb Q}$ be the rational local system on $Y_1(N)(\mathbb C)$ associated to the algebraic representation

\begin{align*} \operatorname{Sym}^{k-2}\mathbb Q^2 \end{align*}

of $GL_2$. Then there is a Hecke-equivariant isomorphism of complex vector spaces

\begin{align*} H^1_{\mathrm{par}}(Y_1(N)(\mathbb C), \mathbb V_{k-2,\mathbb Q}) \otimes_{\mathbb Q} \mathbb C \cong S_k(\Gamma_1(N)) \oplus \overline{S_k(\Gamma_1(N))}, \end{align*}

where $H^1_{\mathrm{par}}$ denotes parabolic cohomology, $S_k(\Gamma_1(N))$ denotes the space of holomorphic cusp forms of weight $k$ and level $\Gamma_1(N)$, and $\overline{S_k(\Gamma_1(N))}$ denotes the complex-conjugate antiholomorphic cusp-form space. The isomorphism is compatible with all Hecke operators $T_\ell$ for primes $\ell \nmid N$.