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We construct the quasi probability distribution $W(p,q)$ on even dimensionalvector space with marginality and invariance under the transformation inducedby projective representation of the group ${\rm Sp}(2,\mathbb{Z})$ whoseelements correspond to linear canonical transformation.  On even dimensional vector space, non-existence of such a quasi probabilitydistribution whose arguments take physical values was shown in our previouspaper(Phys.Rev.A{\bf 65} 032105(2002)). For this reason we study a quasiprobability distribution $W(p,q)$ whose arguments $q$ and $p$ take not only $N$physical values but also $N$ unphysical values, where $N$ is dimension ofvector space. It is shown that there are two quasi probability distributions oneven dimensional vector space. The one is equivalent to the Wigner functionproposed by Leonhardt, and the other is a new one.

Sp(2,$\mathbb{Z}$) invariant Wigner function on even dimensional vector space