  We develop a numerical method based on canonical conformal variables to study
two eigenvalue problems for operators fundamental to finding a Stokes wave and
its stability in a 2D ideal fluid with a free surface in infinite depth. We
determine the spectrum of the linearization operator of the quasiperiodic
Babenko equation, and provide new results for eigenvalues and eigenvectors near
the limiting Stokes wave identifying new bifurcation points via the
Fourier-Floquet-Hill (FFH) method. We conjecture that infinitely many secondary
bifurcation points exist as the limiting Stokes wave is approached. The
eigenvalue problem for stability of Stokes waves is also considered. The new
technique is extended to allow finding of quasiperiodic eigenfunctions by
introduction of FFH approach to the canonical conformal variables based method.
Our findings agree and extend existing results for the Benjamin-Feir,
high-frequency and localized instabilities. For both problems the numerical
methods are based on Krylov subspaces and do not require forming of operator
matrices. Application of each operator is pseudospectral employing the fast
Fourier transform (FFT), thus enjoying the benefits of spectral accuracy and
$O(N \log N)$ numerical complexity. Extension to nonuniform grid spacing is
possible via introducing auxiliary conformal maps.
