For \(S_k\) , the space of cusp forms of weight k for the full modular group, we first introduce periods on \(S_k\) associated to symmetric square L-functions. We then prove that for a fixed natural number n, if k is sufficiently large relative to n, then any n such periods are linearly independent. With some extra assumption, we also prove that for \(k\ge e^{12}\) , we can always pick up to \(\frac{\log k}{4}\) arbitrary linearly independent periods.
