Operators on modular forms. My question is a lemma in the following paper.

Serre J P, Stark H M. Modular forms of weight 1/2 Modular functions of one variable VI. Springer Berlin Heidelberg, 1977: 27-67.

Let $\chi:(\mathbb{Z}/N\mathbb{Z})^*\longrightarrow\mathbb{C}^*$ be a character ($\mod N$),
and $\kappa$ be a positive
odd integer.
A function $f$ on upper half plane $H$ is called a modular form of type $(\kappa,\chi)$ on $\Gamma_0(N)$ if


*

*$f(\gamma z) = \chi(d) j(\gamma,z)^\kappa f (z)$ for every $\gamma=\left(
                 \begin{array}{cc}
                   a & b \\
                   c & d \\
                 \end{array}
               \right)$
           in $\Gamma_0$
this makes sense since $4\mid N$; 

*f is holomorphic, both on H and at the cusps.


One then calls $\kappa/2$ the weight of $f$, and $\chi$ its character. 
The space
of such forms will be denoted by $M_0(N,\kappa/2,\chi).$
Besides, operator $V(m)$ is defined by
\begin{align}
[f\mid V(m)](z):=f(mz).
\end{align}
Equivalently, if $f(z)=\sum_{n=0}^{\infty}a_nq^n$, then $f|V(m)=\sum_{n=0}^{\infty}a_nq^{mn}$.
With the notation above, they claim in their Lemma 2 in page 40 that

The operator $V(m)$ takes $M_0(N,\kappa/2,\chi)$ to $M_0(Nm,\kappa/2,\chi\chi_m)$.

I do not understand why $f|V(m)$ is in $M_0(Nm,\kappa/2,\chi\chi_m)$, where $\chi_m$ is the Kronecker character for $\mathbb{Q}(\sqrt{m})$.  Any help will be appreciated. :)  
 A: Some notations first, (please skip to the answer if familiar):
It will be easier if we work with the $slash$ notation, $\big\vert_{k}$. Since the slash notation is not defined now, let us first do it. Let
$$ G= \left\{ (\alpha,\phi(z)): \alpha = \left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in GL^+_2(\mathbb{Q}) \right\},$$ where $\phi(z)$ is a holomorphic function on $\mathbb{H}$ such that $\phi^2(z)=t\frac{cz+d}{\det(\alpha)}$, where $|t|=1$. Then,
$$ f\big\vert_{k}(\alpha,\phi(z)) = (\phi(z))^{-k}f(\alpha z). $$
Also, $G$ forms a group under the group law:
$$ (\alpha,\phi(z))(\beta,\psi(z)):= (\alpha\beta, \phi(\beta z)\psi(z)). $$
The $V(m)$ operator is given by
$$ f|V(m) =  m^{-k/4}f\big\vert_{k}\left(\begin{pmatrix}m&0\\0&1\end{pmatrix}, m^{-1/4} \right)  = m^{-k/4}(m^{-1/4})^{-k}f\left(\begin{pmatrix}m&0\\0&1\end{pmatrix}z \right) = f(mz).$$
(These are given in the mentioned paper by Serre-Stark).
Answer:
I will use the notation $M_{k/2}(\Gamma_{0}(N),\chi)$ for $M_0(N,k/2,\chi)$.
Let $f\in M_{k/2}(\Gamma_{0}(N),\chi)$. To prove that
$$ g = f|V(m) \in M_{k/2}(\Gamma_{0}(mN),\chi\chi_m);$$
i.e., for $\gamma=\begin{pmatrix}a&b\\mNc&d\end{pmatrix}\in \Gamma_0(mN)$, we need to show that $ g|_k(\gamma,j(\gamma,z)) = \chi\chi_m(d)g. $
\begin{align}
g\big\vert_k (\gamma,j(\gamma,z)) 
&= m^{-k/4}f\big\vert_k \left[\begin{pmatrix}m&0\\0&1\end{pmatrix}, m^{-1/4} \right]\left[ \begin{pmatrix}a&b\\mNc&d\end{pmatrix},\left(\frac{mNc}{d}\right)\left(\frac{-4}{d}\right)^{-1/2}(mNcz+d)^{1/2} \right] \\
&= m^{-k/4}f\big\vert_k \left[\begin{pmatrix}m&0\\0&1\end{pmatrix}\begin{pmatrix}a&b\\mNc&d\end{pmatrix}, m^{-1/4} \left(\frac{mNc}{d}\right)\left(\frac{-4}{d}\right)^{-1/2}(mNcz+d)^{1/2} \right]  \\
&= m^{-k/4}f\big\vert_k \left[\begin{pmatrix}a&mb\\Nc&d\end{pmatrix}\begin{pmatrix}m&0\\0&1\end{pmatrix},  \left(\frac{m}{d}\right)\left(\frac{Nc}{d}\right)\left(\frac{-4}{d}\right)^{-1/2}(mNcz+d)^{1/2}m^{-1/4} \right]  \\
&= m^{-k/4}f\big\vert_k \left[\begin{pmatrix}a&mb\\Nc&d\end{pmatrix}, \left(\frac{m}{d}\right)\left(\frac{Nc}{d}\right)\left(\frac{-4}{d}\right)^{-1/2}(Ncz+d)^{1/2} \right] \left[\begin{pmatrix}m&0\\0&1\end{pmatrix}, m^{-1/4}  \right] \\
&= m^{-k/4}f\big\vert_k \left[\begin{pmatrix}a&mb\\Nc&d\end{pmatrix}, \left(\frac{Nc}{d}\right)\left(\frac{-4}{d}\right)^{-1/2}(Ncz+d)^{1/2} \right]  \\
&\hspace{50mm}\left[\begin{pmatrix}1&0\\0&1\end{pmatrix}, \left(\frac{m}{d}\right) \right]\left[\begin{pmatrix}m&0\\0&1\end{pmatrix}, m^{-1/4}  \right] \\
&= m^{-k/4}\chi(d)f\big\vert_k \left[\begin{pmatrix}1&0\\0&1\end{pmatrix}, \left(\frac{m}{d}\right) \right]\left[\begin{pmatrix}m&0\\0&1\end{pmatrix}, m^{-1/4}  \right]  \\
&= m^{-k/4}\chi(d) \left(\frac{m}{d}\right)^{-k}f\big\vert_k \left[\begin{pmatrix}m&0\\0&1\end{pmatrix}, m^{-1/4}  \right]  \\
&= \chi(d) \left(\frac{m}{d}\right) m^{-k/4} f\big\vert_k \left[\begin{pmatrix}m&0\\0&1\end{pmatrix}, m^{-1/4}  \right]  \\
&= \chi(d) \chi_m(d) f|V(m)\\
&= \chi\chi_m(d) g,  
\end{align}
as required; this completes the checking of transformation law.
I am not quite clear about checking holomorphicity of $f|V(m)$; but if $f$ is holomorphic(at $\mathbb{H}$ and at cusps), I think it should be true for $f|V(m)$ also. I don't know how to prove that part.
