For a simple graph $G$ with vertex set $V(G)$ and edge set $E(G)$, let $\deg(u)$ be the degree of the vertex $u \in V(G)$. The forgotten index of $G$ and its coindex are defined as $F(G)=\sum_{v\in V(G)}\deg^3(v)$ and $\overline{F}(G) = \sum_{uv\not\in E(G)}\big[\deg^2(u)+\deg^2(v)\big]$. New bonds for the first Zagreb index $M_1(G)=\sum_{v \in V(G)}\deg(v)^2$, forgotten index, and its coindex are obtained.