Let $f:[0,1]\times [0,1] \to \mathbb C$ be a double periodic function (periodic in both variables) that depends real-analytically on its argument. We can thus write $f$ as $$ f(x) = \sum_{n \in \mathbb Z} a(x_1,n) e^{2\pi i nx_2}.$$
Now assume that there is $q \in (0,1)$ such that $a(x_1+q,n) = c_n a(x_1,n)$ for all $n$ and a constant $c_n$ independent of $x_1.$ Does this condition impose any easy to check restrictions on the function $f$? I am trying to understand the implications of this condition on functions $f$. It seems like it does not allow me to directly relate $f(x_1+q,x_2)$ to $f(x_1,x_2)$ though.
One small observation: If $q=p/r$ is rational, then $a(x_1,n)=a(x_1 + r q,n) = c_n^r a(x_1,n)$, so $c_n^r = 1.$
