I am currently reading Milne's book on modular forms. Towards the end of chapter 4, letting $P \neq i\infty$ be a cusp for $\Gamma(n)$ and letting $\sigma\in \Gamma(1)$ such that $\sigma(P) = i\infty$, lemma 4.35 states:
The function $\varphi(z)$ is a modular form of weight $2k$ for $\Gamma(n)$; moreover $\varphi$ takes the value $1$ at P, and it is $0$ at every other cusp.
where $\varphi$ is defined as $\varphi(z) = j_\sigma(z)^k\varphi_0(\sigma z)$, $\varphi_0$ being the poincare series of weight $2k$ and character $0$ for $\Gamma(n)$, and $j$ is the automorphy factor $j_\gamma(z) = 1/(cz + d)^2$ for $\gamma = \begin{pmatrix}a & b\\ c & d \end{pmatrix}$. I understand the proof of the first part, but for the second part he states:
The second statement is a consequence of the definition of $\varphi$ and properties of $\varphi_0$.
I can't quite follow this. For $z = P$, we know that $\varphi(P) = j_\sigma(P)\varphi_0(i\infty)$ and $\varphi_0$ takes value $1$ at $i\infty$, so $\varphi(P) = j_\sigma(P)$. But I don't see the rest of the argument. Why should $j_\sigma(P)$ be $1$?