The product of the periods associated to $f$ ($\omega^{\pm}$ in your notation) is equal modulo $\mathbf{Q}(f)^\times$ to $i \langle f, f\rangle$ times some explicit powers of $\pi$ depending on the weight. So the result follows from the rationality theorems of Manin (and Shimura), exactly as you suggest.
David Loeffler
- 37k
- 3
- 89
- 194