A function of a complex variable P(z) which is holomorphic around z=0 has a power series expansion P(z)=sum a_n z^n. Suppose that the a_n are all integers: what restrictions does that place on the function P(z)? We explore the relationship between this problem to questions in complex analysis, number theory, and to Klein’s famous observation that not all finite index subgroups of SL_2(Z) are determined by congruence conditions.