Calibrate k, Then Find the Fastest Mode

Three sensors sit in a line and leak signal into their neighbours. Their coupling matrix is symmetric, and the middle sensor's self-coupling $k$ was never recorded:

$M = \begin{bmatrix} 4 & 2 & 0 \\ 2 & k & 2 \\ 0 & 2 & 4 \end{bmatrix}$

The calibration sheet that survived records only two things about $M$:

The signal in this network evolves as $\mathbf{s}_{t+1} = M\mathbf{s}_t$, so the largest eigenvalue of $M$ is the growth rate of the mode that eventually dominates everything else.

Recover $k$ from the determinant, then report the largest eigenvalue of $M$.

The answer is an exact integer — enter it with no rounding.