The green paramagnetic species formed by heating KMnO _4 at 513 K is

When potassium permanganate ($\text{KMnO}_4$) is heated at $513\text{ K}$, it decomposes to form potassium manganate, manganese dioxide, and oxygen gas.

$2\text{KMnO}_4 \xrightarrow{513\text{ K}} \text{K}_2\text{MnO}_4 + \text{MnO}_2 + \text{O}_2$

The species $\text{K}_2\text{MnO}_4$ (potassium manganate) is green in color.

In $\text{K}_2\text{MnO}_4$, the oxidation state of $\text{Mn}$ is $+6$. The electronic configuration of $\text{Mn}^{6+}$ is $[\text{Ar}] 3d^1$.

Since it contains one unpaired electron in the $d$-orbital, it is paramagnetic in nature.

Thus, the green paramagnetic species formed is $\text{K}_2\text{MnO}_4$.

Answer: $\text{K}_2\text{MnO}_4$