Let $f$ be a cusp modular form of weight $k$ (assume it has rational coefficients to get things simpler). To $f$ one associates some periods $\Omega^{\pm}(f)$ by looking at the subspace of the cohomology $$ H^1(X(N), \mathrm{Sym}^{k-2} \mathcal{L}) $$ cut by the Hecke operators. Here $X(N)$ is the modular curve, $\mathcal{E} \stackrel{\pi}{\to} X(N)$ is the universal elliptic curve and $\mathcal{L}$ is the rank two local system $R^1\pi_\ast \mathbb{Q}_{\mathcal{E}}$.
Question: Are there some explicit formulae known for $\Omega^{\pm}(f)$, at least in some cases (say when $f$ has complex multiplication)?
