cases, one obtains the following formula $$\begin{gathered}
 \label{eq:ind2d} 
 m( g, z) = k + \tfrac{1}{2} ( 1 - ( -1) ^{k + \epsilon})\end{gathered}$$ where $k$ and $\epsilon$ are determined by $$\arg (z) \in \bigl[  k
\tfrac{\pi}{2}, (k+1) \tfrac{\pi}{2} \bigr[ , \qquad \epsilon =
\begin{cases} 0 \text{ if } \operatorname{tr} ( g) > 2 \\ 1 \text{
    otherwise}.
\end{cases}$$

The formula does not depend on the parametrization. Unfortunately such a simple description doesn’t generalize in higher dimension. Nevertheless we can characterize the index in any dimension by considering product.

Let $S_1$ and $S_2$ be two symplectic vector spaces with positive polarizations $E_1$ and $E_2$. Then $E= E_1 \times E_2$ is a positive polarization of $S = S_1 \times S_2$. We have a morphism from ${\operatorname{Mp}}(S_1,E_1) \times {\operatorname{Mp}}( S_2, E_2) $ to ${\operatorname{Mp}}( S,E)$ sending $((g_1, z_1),$ $(g_2, z_2))$ into $( g_1 \times g_2 , z_1
  z_2)$. Furthermore, $$\begin{gathered}
 \label{eq:indprod}
 m (  g_1 \times g_2 , z_1
  z_2) = m ( g_1, z_1) + m ( g_2 , z_2). \end{gathered}$$ So the image of ${\operatorname{Mp}}_* (S_1,E_1) \times {\operatorname{Mp}}_* ( S_2,E_2)$ meets each connected component of ${\operatorname{Mp}}_* (S,E)$. This gives the following characterization.

The collection $(m_{S,
  E}: {\operatorname{Mp}}(S, E) \rightarrow {{\mathbb{Z}}}/4 {{\mathbb{Z}}})$, where 