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We developed a method to calculate the eigenvalues and eigenfunctions of thesecond derivative (Hessian) of action at choreographic three-body solutionsthat have the same symmetries as the figure-eight solution. A choreographicthree-body solution is a periodic solution to equal mass planar three-bodyproblem under potential function $\sum_{i<j} U(r_{ij})$, in which three masseschase each other on a single closed loop with equal time delay.  We treat choreographic solutions that have the same symmetries as thefigure-eight, namely, symmetry for choreography, for time reversal, and fortime shift of half period. The function space of periodic functions aredecomposed into five subspaces by these symmetries. Namely, one subspace oftrivial oscillators with eigenvalue $4\pi^2/T^2\times k^2$, $k=0,1,2,\dots$,four subspaces of choreographic functions, and four subspaces of"zero-choreographic" functions. Therefore, the matrix representation of theHessian is also decomposed into nine corresponding blocks. Explicit expressionsof base functions and the matrix representation of the Hessian for eachsubspaces are given.  The trivial eigenvalues with $k\ne 0$ are quadruply degenerated, while with$k=0$ are doubly degenerated that correspond to the conservation of linearmomentum in $x$ and $y$ direction. The eigenvalues in choreographic subspacehave no degeneracy in general. In "zero-choreographic" subspace, everyeigenvalues are doubly degenerated.

Decomposition of the Hessian matrix for action at choreographic three-body solutions with figure-eight symmetry