If the characteristic polynomial of a $3 \times 3$ matrix $M$ over $\mathbb{R}$ (the set of real numbers) is $\lambda^3 - 4\lambda^2 + a\lambda + 30$, $a \in \mathbb{R}$, and one eigenvalue of $M$ is 2, then the largest among the absolute values of the eigenvalues of $M$ is _________.

Given the characteristic equation:

Since $\lambda = 2$ is an eigenvalue, it satisfies the characteristic equation:

Substituting $a = -11$ into the polynomial:

Factoring out $(\lambda - 2)$:

Thus, the eigenvalues of $M$ are $\lambda_1 = 2$, $\lambda_2 = 5$, and $\lambda_3 = -3$.

The absolute values of the eigenvalues are $|2| = 2$, $|5| = 5$, and $|-3| = 3$.

The largest among the absolute values is 5.