  We prove that Klein quartic maximizes the multiplicity of the first positive
eigenvalue of the Laplacian among all closed hyperbolic surfaces of genus $3$,
with multiplicity equal to $8$. We also obtain partial results in genus $2$,
where we find that the maximum multiplicity is between $3$ and $6$. Along the
way, we show that for every $g\geq 2$, there exists some $\delta_g>0$ such that
the multiplicity of any eigenvalue of the Laplacian on a closed hyperbolic
surface of genus $g$ in the interval $[0,1/4+\delta_g]$ is at most $2g-1$
despite the fact that this interval can contain arbitrarily many eigenvalues.
This extends a result of Otal to a larger interval but with a weaker bound,
which nevertheless improves upon the general upper bound of S\'evennec.
