A new graph, called the symplectic inner product graph $Spi\big(2ν,q\big)$, over a finite field $\mathbb{F}_q$ is introduced. We show that $Spi\big(2ν,q\big)$ is connected with diameter $4$ if and only if $ν\geq2$ and the automorphism group of $Spi\big(2ν,q\big)$ is determined. Two necessary and sufficient conditions for two vertices of $Spi\big(2ν,q\big)$ and two edges of $Spi\big(2ν,q\big)$ respectively are in the same orbit under the action of the automorphism group of $Spi\big(2ν,q\big)$ are obtained.