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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)?

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In the CM case, the two periods are both algebraic multiples of the "CM period" attached to the imaginary quadratic field that $f$ has CM by. One reference for the relation between these periods is the paper by Bouganis and Venjakob in Asian J Math. –  David Loeffler Feb 25 '13 at 8:11
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