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The transfer operator for $\Gamma_0(N)$ and trivial character $\chi_0$possesses a finite group of symmetries generated by permutation matrices $P$with $P^2=id$. Every such symmetry leads to a factorization of the Selberg zetafunction in terms of Fredholm determinants of a reduced transfer operator.These symmetries are related to the group of automorphisms in$GL(2,\mathbb{Z})$ of the Maass wave forms of $\Gamma_0(N)$ . For the group$\Gamma_0(4)$ and Selberg's character $\chi_\alpha$ there exists just onenon-trivial symmetry operator $P$. The eigenfunctions of the correspondingreduced transfer operator with eigenvalue $\lambda=\pm 1$ are related to Maassforms even respectively odd under a corresponding automorphism. It then followsfrom a result of Sarnak and Phillips that the zeros of the Selberg functiondetermined by the eigenvalues $\lambda=-1$ of the reduced transfer operatorstay on the critical line under the deformation of the character. Fromnumerical results we expect that on the other hand all the zeros correspondingto the eigenvalue $\lambda=+1$ leave this line for $\alpha$ turning away fromzero.

Symmetries of the transfer operator for Γ0(N) and a character deformation of the Selberg zeta function for Γ0(4)

Symmetries of the transfer operator for Γ₀(N) and a character deformation of the Selberg zeta function for Γ₀(4)