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Let $k$ be a positive integer divisible by 4, $p\>k$ a prime, $f$ an elliptic cuspidal eigenform (ordinary at $p$) of weight $k-1$, level 4 and non-trivial character. In this paper we provide evidence for the Bloch-Kato conjecture for the motives ${\text{ad}}^{0}M(-1)$ and ${\text{ad}}^{0}M\left(2\right)$, where $M$ is the motif attached to $f$. More precisely, we prove that under certain conditions the $p$-adic valuation of the algebraic part of the symmetric square $L$-function of $f$ evaluated at $k$ provides a lower bound for the $p$-adic valuation of the order of the Pontryagin...