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Find all solutions to the equation $2n^3 + 6n^2 + 102n + 98 \; = \; m^3 \hspace{2cm} m,n \in \mathbb{Z} \;.$ If the answers are $(m_1,n_1)\,,\,(m_2,n_2)\,,\,\ldots\,,\,(m_N,n_N)$, evaluate $S \; = \; \sum_{j=1}^N (m_j^2 + n_j^2) \;.$

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