I have a question on http://www.personal.psu.edu/rcv4/567c10.pdf
I do not understand the proof of Theorem 10.6. I get that from Theorem 10.5, we get that Tn(f) satisfies the weakly modular equation. However, in the proof of Theorem 10.6, how c(m) and $\gamma{(m)}$ having similar support show Tnf is holomorphic everywhere including infinity.
Being holomorphic in the upper half-plane follows from the fact that sum and product and composition of holomorphic functions are holomorphic.
Holomorphicity at infinity comes from the fact that Fourier series of $T_nf$ at infinity starts at 0. i.e. : $$ T_nf = \sum_{n=0}^{\infty} a_n q^n $$ for some $a_n$'s and this (by definition) means that $f$ is holomorphic at infinity.