  We study the scaling behavior of $M$-furcation $(M\!=\!2, 3, 4,\dots)$
sequences of $M^n$-period $(n=1,2,\dots)$ orbits in two coupled one-dimensional
(1D) maps. Using a renormalization method, how the scaling behavior depends on
$M$ is particularly investigated in the zero-coupling case in which the two 1D
maps become uncoupled. The zero-coupling fixed map of the $M$-furcation
renormalization transformation is found to have three relevant eigenvalues
$\delta$, $\alpha$, and $M$ ($\delta$ and $\alpha$ are the parameter and
orbital scaling factors of 1D maps, respectively). Here the second and third
ones, $\alpha$ and $M$, called the ``coupling eigenvalues'', govern the scaling
behavior associated with coupling, while the first one $\delta$ governs the
scaling behavior of the nonlinearity parameter like the case of 1D maps. The
renormalization results are also confirmed by a direct numerical method.
