Double-coset operator on modular forms ($[A\alpha B]f={\sum }_{j}f[{\alpha }_j{]}_k$)

Decompose $A\alpha B={\coprod }_{j}A{\alpha }_j$ for finite-index congruence subgroups $A,B$. The sum is independent of representative choices because $f$ is invariant under $A$. Right multiplication by $B$ permutes these left cosets, so the sum is invariant under $B$. Each summand is holomorphic in the half-plane; cusp holomorphy under rational slash operators proves holomorphy at every new cusp. The same reasoning preserves vanishing at cusps. With the Hecke-normalized rational slash operator, no extra scalar is needed in the defining sum.