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\begin{document}
\title{\bf{Fourier Coefficients in Vector Valued Modular Forms of positive dimension}}
\author{Pepe Gimenez}

\date{\today}
\maketitle

\noindent
{\setlength{\baselineskip}%
{1.5\baselineskip} Knopp and Mason \cite {[3]} obtained growth
conditions of the Fourier coefficients of vector-valued modular
forms of negative dimension. In \cite{[4]} they developed a
general theory of vector-valued modular forms.

Let $F(\tau)=(F_1(\tau),\ldots,F_p(\tau))^t$ be a $p$-tuple of
functions holomorphic in the complex upper half-plane $
\mathcal{H}$ and $\rho:\Gamma \longrightarrow GL(p,\mathbb{C})$ a
$p$-dimensional complex representation.$(F,\rho)$, or simply $F$,
is a vector-valued form of real dimension $r$ on the modular group
$\Gamma=SL(2,\mathbb{Z})$ if

\begin{enumerate}
    \item For all $V=\left(%
\begin{array}{cc}
  a & b \\
  c & d \\
\end{array}%
\right) \in \Gamma$ we have

    \begin{equation} \label{eq:ftau}
    (F_1(\tau),\ldots,F_p(\tau))\mid_r
    V(\tau)=\rho(V)(F_1(\tau),\ldots,F_p(\tau))
    \end{equation}
    \item Each component function $F_j(\tau)$ has a convergent
    $x$-expansion meromorphic at infinity:
    \begin{equation} \label{eq:fourier}
    F_j(\tau)=x^{m_j}\sum_{\nu\geq \mu_j} a_\nu(j)x^{\nu}
    \end{equation}
    with $0\leq m_j<1$ a positive rational number, $\mu_j$ an integer and $x=e^{2\pi i \tau}$. The
    Slash operator $\mid_r V$ in (\ref {eq:ftau}) is defined by:
        \begin{equation} \label{eq:slash}
   F\mid_r V(\tau)=F\mid ^\varepsilon _r V(\tau)=\varepsilon(V)^{-1}\left(c\tau +
   d \right)^{r}F(V\tau).
    \end{equation}
\end{enumerate}
In the rest of the paper we will assume that  $- \pi \leq \arg
\omega < \pi$ for $ \omega \in \mathbb{C}$. Also, only be
interested in the elements of $\Gamma$ of the form
\begin{equation}
V=\left(%
\begin{array}{cc}
  h' & -\frac{hh'+1}{k} \\
  k & -h \\
\end{array}%
\right), \quad k> 0, \quad 0 \leq h,h'<k
\end{equation}
 Note that since $V \in \Gamma$, we have that $hh'\equiv -1
\pmod {k}$, and therefore $-\frac{hh'+1}{k}$ will be an integer,
and there always exists an $h'$ such that $0 \leq h'<k$, and it is
unique and therefore $V$ is uniquely determined by $h$ and $k$,
i.e. $V=V_{k,h}$. To simplify the computations later it will be
convenient to change the multiplier $\varepsilon$ for a new
quantity $v$ such that

\begin{equation}\label{Eq:vep}
F\mid ^v _r V_{k,h}(\tau)=v(V_{k,h})^{-1}\left(-i(k\tau -
   h) \right)^{r}F(V_{k,h}\tau),
\end{equation}
but first we have to make sure that we can do that. Since $k>0$,
we have that
\begin{equation}\label{Eq:argtau}
0<\arg (k\tau -h) < \pi, \quad-\frac{\pi}{2} < \arg -i(k\tau -
   h)<\frac{\pi}{2}
\end{equation}
and since
\begin{equation}
arg (k\tau -h)-\frac{\pi}{2} \equiv arg -i(k\tau - h) \pmod{2 \pi}
\end{equation}
we get
\begin{equation}
arg (k\tau -h)-\frac{\pi}{2}=arg -i(k\tau - h)
\end{equation}
and therefore
  \begin{eqnarray}
\left( - i (k \tau - h)^r\right)&=& |k \tau - h|^r e^{-ir
\arg\left(
- i (k \tau - h)\right)}\\
&=&|k \tau - h|^r e^{-ir \arg\left( (k \tau - h) -
\frac{\pi}{2}\right)}\\
&=& (-i)^r(k \tau -h)^r)
  \end{eqnarray}
and we can define $v(V)= \varepsilon(V)(-i)^{-r}$

 Also given a vector-valued modular form
$(F,\rho)$, we can define
\begin{equation} \label{eq:fx}
f_j(x)=x^{-m_j}F_j(\tau)=\sum_{\nu\geq
\mu_j}^{\infty}a_{\nu}(j)x^{\nu}, \quad 1\leq j\leq p,
\end{equation}
which are analytic in the unit circle and have a pole at $x=0$ of
order $-\mu_j$, provided that $\mu_j$ is negative. We will use
this function to get the coefficients of the vector-valued modular
form $a_\nu(j)$ by using Cauchy's formula. But to do so we will
need later in the paper a bound for $\rho (V)$.

To bound $\rho (V)$ we introduce the Eichler length of $V$
\cite{[10]} with respect to the generators $S=\left(%
\begin{array}{cc}
  0 & -1 \\
  1 & 0 \\
\end{array}%
\right)$, $T=\left(%
\begin{array}{cc}
  1 & 1 \\
  0 & 1 \\
\end{array}%
\right)$ of $\Gamma$. Namely we write $V$ as a product $V=\pm V_1\ldots
V_L$, where each $V_L$ is equal to either $S$ or $T^{n_j}$, for
some integer $n_j$, no two consecutive $V_j$ are both equal to $S$
or a power of $T$, and where $L$ is minimal. Eichler proved that
\begin{equation}
L(V)\leq n_1\log \mu (V)+n_2,
\end{equation}
where $\mu (V)={h'}^2+\left(\frac {hh'+1}{k} \right)^2+k^2+h^2$,
and $n_1$, $n_2$ are constants independent of $V$. Also Knopp and
Mason \cite{[3]} showed that
\begin{equation}
\mid\rho_{jm} (V)\mid \leq p^{L(V)-1}K_1^{L(V)}, \quad 1\leq m,j
\leq p
\end{equation}
where $K_1$ is a constant that satisfies $\mid\rho_{jm} (S)\mid
\leq K_1$ for all $1\leq m,j \leq p$. Therefore if we use the
usual norm
\begin{equation}
\mid\rho (V)\mid = \sqrt{\sum_{1\leq m,j \leq p} \mid\rho_{jm}
(V)\mid^2}
\end{equation}
we see that
\begin{equation}\label{eq:alpha}
\mid\rho (V)\mid \leq K_2\mu(V)^{\alpha}, \quad \alpha =n_1\log
pK_1
\end{equation}
where $K_2$ is independent of $V$. Note that since the length of
$V^{-1}$ is the same as the length of $V$, we can use the same
bound for $\mid\rho (V)^{-1}\mid = \mid\rho (V^{-1})\mid$
\newtheorem{thm}{Theorem}
\begin{thm}\label{th:1} Let $(F,\rho)$ be  a vector-valued modular form of
dimension $r>2\alpha$, where
\begin{equation}\label{Eq:def1}
F_j(\tau)=x^{m_j}\sum_{\nu\geq \mu_j}^{\infty}a_{\nu}(j)x^{\nu},
\end{equation}
where $\alpha$ is given by the Eichler estimate(\ref {eq:alpha}),
then the coefficients $a_{\nu} (j)$ are independent of $a_0(j)$,
$a_1(j)$, $\ldots$.
\end{thm}

To prove the theorem we will use the circle method as applied by
Rademacher and  Zuckerman \cite{[11]}, but now in the context of
 vector-valued modular forms.

\newtheorem{lem}[thm]{Lemma}
\begin{lem}\label{...}Let $(F,\rho)$ be  a vector-valued modular form of
dimension $r$ and $V=\left(%
\begin{array}{cc}
  h' & -\frac{hh'+1}{k} \\
  k & -h \\
\end{array}%
\right) \in \Gamma$, then if  $z=-i(k\tau -h)$, we have
\begin{equation}\label{Eqtrans}
f(e^{-2\pi\frac{z-ih}{k}})=\Omega_{h,k}\Psi_k(z)f(e^{2\pi i
\frac{h'}{k}}e^{-2\pi\frac{1}{kz}})
\end{equation}
where $f(x)=\left( f_1(x),\ldots,f_p(x)\right)^t$,

\begin{equation}\label{Eq:psik}
\Psi_k(z)=z^{r} \left(%
\begin{array}{ccc}
  e^{2\pi m_1\frac{z-1/z}{k}} & \cdots & 0 \\
  \vdots & \ddots & \vdots\\
  0 & \cdots & e^{2\pi m_p\frac{z-1/z}{k}} \\
\end{array}%
\right)
\end{equation}

\begin{equation}
\Omega_{h,k}=v^{-1}(V)\left(\rho(V)^{-1} \right)^t \left(%
\begin{array}{ccc}
  e^{2\pi i m_1\frac{h'-h}{k}} & \cdots  & 0 \\
  \vdots & \ddots & \vdots\\
  0  & \cdots  & e^{2\pi i m_p\frac{h'-h}{k}} \\
\end{array}%
\right)
\end{equation}

\end{lem}
\begin{proof}First note that if we let $k>0$ and choose the unique $h'$ such that $0 \leq h'<k$, then $V \in \Gamma$ is determined by $h$ and $k$, and $\Omega_{h,k}$ depends only on $h$ and $k$.
From the definition of vector-valued modular form (\ref {eq:ftau}
and \ref{Eq:vep}), we get that
\begin{equation}
F(\tau)=v(V)^{-1}z^{r} \rho(V)^{-1}F(V\tau)
\end{equation}
\begin{align}\label{Eq:relfvtauftau}
 f(x) & = \left(%
\begin{array}{c}
  f_1(x) \\
  \vdots \\
  f_p(x) \\
\end{array}%
\right)\\
& = \left(%
\begin{array}{ccc}
  e^{-2\pi i m_1\tau} & \cdots &o \\
  \vdots & \ddots &\vdots\\
  o & \cdots & e^{-2\pi i m_p\tau} \\
\end{array}%
\right)\left(%
\begin{array}{c}
  F_1(\tau) \\
  \vdots \\
  F_p(\tau) \\
\end{array}%
\right)\\
& =\left(%
\begin{array}{ccc}
  e^{-2\pi i m_1\tau} & \cdots &o \\
  \vdots & \ddots &\vdots\\
  o & \cdots & e^{-2\pi i m_p\tau} \\
\end{array}%
\right)v^{-1}(V)z^{r}\rho(V)^{-1} F(V\tau)\\
& =v^{-1}(V)z^{r}\left( \rho(V)^{-1} \right)^t \left(%
\begin{array}{ccc}
  e^{-2\pi i m_1\tau} & \cdots &o \\
  \vdots & \ddots &\vdots\\
  o & \cdots & e^{-2\pi i m_p\tau} \\
\end{array}%
\right)\left(%
\begin{array}{ccc}
  e^{2\pi i m_1 V\tau} & \cdots &o \\
  \vdots & \ddots &\vdots\\
  o & \cdots & e^{2\pi i m_p V\tau} \\
\end{array}%
\right)f(e^{2 \pi i V\tau})\label{Eq:relfvtauftau}
\end{align}
Now since $-i(k\tau - h)=z$, and since $hh'\equiv -1 \pmod {k}$,
\begin{equation}\label{Eqdefz}
\tau=\frac{i z}{k}+\frac{h}{k},\quad
V\tau=\frac{i}{kz}+\frac{h'}{k}
\end{equation}
Note that since $k>0$ then $\Re(z)>0$. Now it only remains to
apply (\ref{Eqdefz}) into (\ref{Eq:relfvtauftau}) to prove the
lemma
\end{proof}
\begin{lem}\label{..1.}Let $(F,\rho)$ be  a vector-valued modular form of
dimension $r$, then the Fourier coefficients $a_m=(a_m(1),\ldots ,
a_m(p))^t$,are given by the following formula
\begin{equation}\begin{split}
a_m=e^{2\pi N^{-2}m}\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}\Omega_{h,k}e^{-2\pi  i h\frac{m}{k}}\int_{-\theta_{h,k}^{'}}^{\theta_{h,k}^{''}}\Psi_k(k(N^{-2}-i
\varphi))\\
f\left(e^{\frac{2\pi}{k}\left(ih'-k^{-1}(N^{-2}-i
\varphi)^{-1}\right)}\right) e^{-2\pi i m \varphi}d\varphi
\end{split}\end{equation}
for any $N$ positive integer, where $\theta_{h,k}^{'}$ and
$\theta_{h,k}^{''}$ are given by the Farey dissection of the
circle $|x|=e^{2\pi N^{-2}}$, using Farey series of order $N$.
\end{lem}
\begin{proof}Since the functions $f_j(x)$ are analytic inside the unit circle
except possibly at zero where there could be a pole, we can use
the Cauchy formula to get
\begin{equation}
a_m(j)=\frac{1}{2\pi i}\int_C \frac {f_j(x)} {x^{m+1}}dx
\end{equation}
where C is the circle $|x|=e^{2\pi N^{-2}}$, for $N$ a positive
integer. We can change the path of integration by making the usual
dissection of the circle $C$ into arcs $\xi_{h,k}$, using the
Farey series of order $N$. thus we have
\begin{equation}
a_m(j)=\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}\frac{1}{2\pi i}\int_{\xi_{h,k}}\frac {f_j(x)} {x^{m+1}}dx
\end{equation}
We can make the change of variable
\begin{equation}\label{EqCV}
x=e^{-2\pi N^{-2}+ 2\pi i\frac{h}{k} + 2\pi i \varphi}, \quad
-\theta_{h,k}^{'}\leq \varphi \leq \theta_{h,k}^{''}
\end{equation}

\begin{equation}\label{Eqamj}
a_m(j)=e^{-2\pi N^{-2}m}\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}e^{-2\pi i m
\frac{h}{k}}\int_{-\theta_{h,k}^{'}}^{\theta_{h,k}^{''}}f_j(e^{2\pi
i
 \frac{h}{k}-2\pi(N^{-2}-i \varphi)})e^{-2\pi i m
 \varphi}d\varphi
\end{equation}
and therefore if we write the same expression in column vectors,
we get
\begin{equation}\label{Eqamjvectors}
a_m=e^{-2\pi N^{-2}m}\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}e^{-2\pi i m
\frac{h}{k}}\int_{-\theta_{h,k}^{'}}^{\theta_{h,k}^{''}}f(e^{2\pi
i
 \frac{h}{k}-2\pi(N^{-2}-i \varphi)})e^{-2\pi i m
 \varphi}d\varphi
\end{equation}
now we can apply the result of the previous lemma (\ref{Eqtrans})
on the column vector $f(e^{2\pi i
 \frac{h}{k}-2\pi(N^{-2}-i \varphi)})$, where $z=k(N^{-2}-i \varphi)$ and the lemma is
 proven.
 \end{proof}
\begin{proof}[Proof of the Theorem]We will show now that $f(x)$ in the neighborhood of $x=0$ is
dominated by the principal part $P(x)$, where $P(x)$ is the column
vector with components

\begin{align}\label{Eq:defPx}
P_j(x)=\sum_{\nu= 1}^{\mu_j}a_{- \nu}(j)x^{- \nu}, &&1\leq j \leq
p,
\end{align}
provided that $\mu_j < 0$, otherwise $P_j(x)=0$. For that purpose
we split the formula for $a_m$ in two parts
\begin{align}\label{Eq:am}
a_m&= Q_m(N) + R_m(N)
\end{align}

where

\begin{equation}\label{Eq:defQm}\begin{split}
Q_m(N)=e^{2\pi N^{-2}m}\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}\Omega_{h,k}e^{-2\pi i h\frac{m}{k}}\int_{-\theta_{h,k}^{'}}^{\theta_{h,k}^{''}}\Psi_k(k(N^{-2}-i
\varphi))\\
P\left(e^{\frac{2\pi}{k}\left(ih'-k^{-1}(N^{-2}-i
\varphi)^{-1}\right)}\right) e^{-2\pi i m \varphi}d\varphi
\end{split}\end{equation}


\begin{equation}\begin{split}
R_m(N) =e^{2\pi N^{-2}m}\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}\Omega_{h,k}e^{-2\pi  i h\frac{m}{k}}\int_{-\theta_{h,k}^{'}}^{\theta_{h,k}^{''}}\Psi_k(k(N^{-2}-i
\varphi))\\
D\left(e^{\frac{2\pi}{k}\left(ih'-k^{-1}(N^{-2}-i
\varphi)^{-1}\right)}\right) e^{-2\pi i m \varphi}d\varphi
\end{split}\end{equation}
To prove the theorem we will show that $\lim_{N\rightarrow
\infty}R_m(N)=0$. From the theory of Farey fractions we have
\begin{align}
\frac{1}{2kN} \leq \theta_{h,k}^{'} \leq \frac{1}{kN}, &&
\frac{1}{2kN} \leq \theta_{h,k}^{''} \leq \frac{1}{kN}
\end{align}
and therefore since $k \leq N$, we find for $-\theta_{h,k}^{'}
\leq \varphi \leq \theta_{h,k}^{''}$
\begin{equation}
\Re\left(k(N^{-2}-i \varphi) \right)=kN^{-2},
\end{equation}

\begin{equation}
\Re\left(\frac{1}{k(N^{-2}-i \varphi)
}\right)=\frac{N^{-2}}{k(N^{-4}+ \varphi^2)}\geq
\frac{N^{-2}}{k(N^{-4}+
k^{-2}N^{-2})}=\frac{k}{k^2N^{-2}+1}\geq\frac{k}{2},
\end{equation}


\begin{equation}
|k(N^{-2}-i \varphi)|=k(N^{-4}+ \varphi^2)^{\frac{1}{2}}\leq (k^2
N^{-4}+ N^{-2})^{\frac{1}{2}} \leq 2^{\frac{1}{2}}N^{-1}
\end{equation}


\begin{equation}
|e^{\frac{2\pi m_j}{k}\left(k(N^{-2}-i \varphi))-1/(k(N^{-2}-i
\varphi)\right)}|\leq e^{\frac{2\pi m_j}{k}\left( kN^{-2}
-\frac{k}{2}\right)}=e^{2\pi m_j N^{-2}}e^{-\pi m_j}
\end{equation}

Therefore
  \begin{eqnarray*}
  \lefteqn{|\Psi_k(z_0)| } \\
  & & \leq |(z_0)|^{r}\sqrt{|e^{\frac{2\pi m_1}{k}z_0-1/z_0}|^2+\ldots +
|e^{\frac{2\pi m_p}{k}z_0-1/z_0}|^2} \\
& & \leq 2^{\frac{r}{2}}N^{-r}\sqrt{p}e^{2\pi m_{max}
N^{-2}}e^{-\pi m_{min} }
  \end{eqnarray*}
  where  $z_0=k(N^{-2}-i \varphi)$, $m_{max}= max\{m_1, \ldots, m_p\}$ and $m_{min}=min \{m_1, \ldots, m_p\}$
  Also, using the fact that if $a,b>0$, then $(a+b)^2=a^2+2ab+b^2
  \geq a^2 +b^2$



  \begin{eqnarray*}
\lefteqn{\mid D\left(e^{\frac{2\pi}{k}\left(ih'-k^{-1}(N^{-2}-i
\varphi )^{-1}\right)}\right) \mid }\\
&& \leq \sum_{\nu=0}^\infty |a_\nu|e^{\frac {-2\pi \nu}{k}
\Re\left(k^{-1}(N^{-2}-i
\varphi)^{-1} \right)}\\
&& \leq \sum_{\nu=0}^\infty |a_\nu|e^{-\pi \nu}\\
&& = \sum_{\nu=0}^\infty\sqrt{|a_\nu(1)|^2+ \ldots +|a_\nu(p)|^2}e^{-\pi \nu}\\
&& \leq \sum_{\nu=0}^\infty\left(|a_\nu(1)|+ \ldots +|a_\nu(p)|
\right)e^{-\pi
\nu}\\
&& = \sum_{\nu=0}^\infty|a_\nu(1)|e^{-\pi \nu}+ \ldots
+\sum_{\nu=0}^\infty|a_\nu(p)|e^{-\pi \nu}
  \end{eqnarray*}

  the last equality holds since all the series  converge absolutely because $|e^{-\pi}|\leq
  1$

Using these results we have

\begin{eqnarray*}
\lefteqn{|\Psi_k(k(N^{-2}-i
\varphi))D\left(e^{\frac{2\pi}{k}\left(ih'-k^{-1}(N^{-2}-i
\varphi)^{-1}\right)}\right)|}\\
&& \leq 2^{\frac{r}{2}}N^{-r}\sqrt{p}e^{2\pi m_{max}
N^{-2}}e^{-\pi m_{min} }\left(\sum_{\nu=0}^\infty|a_\nu(1)|e^{-\pi
\nu}+\sum_{\nu=0}^\infty|a_\nu(2)|e^{-\pi \nu}\right)\\
&& =CN^{-r}e^{2\pi m_{max} N^{-2}}
\end{eqnarray*}
where
\begin{equation}
C=2^{\frac{r}{2}}\sqrt{p}e^{-\pi m_{min}
}\left(\sum_{\nu=0}^\infty|a_\nu(1)|e^{-\pi \nu}+ \ldots
+\sum_{\nu=0}^\infty|a_\nu(p)|e^{-\pi \nu}\right)
\end{equation}

Which is finite since $|e^{-\pi}|\leq
  1$ and both series are convergent inside the unit circle

Now in order to bound $|\Omega_{h,k}|$, we need to use the Eichler
estimate discussed before (\ref{eq:alpha}), and therefore we have
that

\begin{eqnarray}
\lefteqn{|\rho(V)|}\\
&&\leq K_2 \left({h'}^2+\left(
\frac{hh'+1}{k}\right)^2+k^2+h^2\right)^\alpha\\
&&\leq K_3 k^{2\alpha} \label{Eq:kalpha}\\
&&\leq K_3N^{2\alpha}
\end{eqnarray}

Where $\alpha= n_1\log(pK_1)$, and $K_2$, $K_3$ are a constants
independent of $V$. Note that (\ref{Eq:kalpha}) holds since in our
case we have that $0 \leq h,h'<k$ for all $V$'s

 Therefore

  \begin{eqnarray}
\lefteqn{|\Omega_{h,k}|} \\
&& =\mid v^{-1}(V) \left( \rho(V)^{-1} \right)^t\left(%
\begin{array}{ccc}
  e^{2\pi i m_1\frac{h'-h}{k}} & \cdots & 0 \\
  \vdots & \ddots & \vdots\\
  0 & \cdots &  e^{2\pi i m_2\frac{h'-h}{k}} \\
\end{array}%
\right)\mid\\
&& \leq K_3N^{2\alpha}\sqrt{p}\\
&&\label{Eq:boundomega}=K_4N^{2\alpha}
\end{eqnarray}



  thus we have,

\begin{eqnarray}
\lefteqn{|R_m(N)|}\\
&& \leq e^{2\pi N^{-2}m}\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}K_4N^{2\alpha}\int_{-\theta_{h,k}^{'}}^{\theta_{h,k}^{''}}CN^{-r}e^{2\pi \alpha
N^{-2}}d\varphi\\
&&\leq K_5e^{2\pi
N^{-2}(m+\alpha)}N^{-r+2\alpha}\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}\int_{-\theta_{h,k}^{'}}^{\theta_{h,k}^{''}}d\varphi\\
&&\label{EQ:rmbounded} =K_5e^{2\pi N^{-2}(m+\alpha)}N^{-r+2\alpha}
\end{eqnarray}
where $K_5$ is a constant independent of $V$, and since $N$ can be
very large, we can conclude that $|R_m|=0$ for $r>2\alpha$.
\end{proof}
\begin{thm}Let $F(\tau)$ be a vector-valued modular form of dimension $r > 2 \alpha > 0$, then the
coefficients in (\ref{eq:fourier}) are given by the formula
\begin{equation}
a_m= 2 \pi \sum_{k=1}^{\infty}\frac{1}{k}
\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}

- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)A_{k,\nu,m, \rho, \varepsilon}\ B_{k, \nu,m, \rho,
\varepsilon, r}
\end{equation}where,
\begin{equation}\label{Eq:defam}
A_{k,\nu,m, \rho, \varepsilon}=\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \\
  (h,k)=1 \\
\end{array}}}\Omega_{h,k}e^{-2\pi  i \frac{hm+h'\nu}{k}}
\end{equation}
where $B_{k, \nu,m, \rho, \varepsilon, r}$ is a column vector in
which the $jth$ component is given by
\begin{eqnarray}\label{Eq:defBjknum}
\lefteqn{B_{k, \nu,m, \rho, \varepsilon, r}^j}\\
&&=\left( \frac{\nu -
m_j}{m+m_j}\right)^{\frac{r+1}{2}}I_{r+1}\left(\frac{4 \pi}{k}(\nu
- m_j)^{\frac{1}{2}}(m+m_j)^{\frac{1}{2}}\right)\quad if \quad
m+m_j
>
0\\
&&= \frac{2 \pi}{(r+1)!} \left(\frac{2 \pi \nu}{k^2} \right)^{r+1}
\quad if \quad m=m_j=0
\end{eqnarray}
where
\begin{eqnarray}\label{EQ:defIrho}
\lefteqn{I_\rho(z)}\\
&&=\frac{(\frac{z}{2})^{\rho}}{2\pi i}\int_{- \infty}^{(0+)}t^{-
\rho-1}e^{t+\frac{z^2}{4t}}dt , \quad  z \in
\mathbb{R}\\
&&\label{Eq:powerseriesof Iro}= \sum_{n=o}^{\infty}
\frac{\left(\frac{z}{2}\right)^{2n + \rho}}{n!(n+ \rho)!}, \quad
z \in \mathbb{R}
\end{eqnarray}
\end{thm}

\begin{lem}\label{Lem:convabs}
The following series converges absolutely for $r>2 \alpha$
\begin{equation}\label{Eq:sum AknumBknum}
\sum_{k=1}^{\infty}\frac {1}{k}A_{k,\nu,m, \rho, \varepsilon}B_{k,
\nu,m, \rho, \varepsilon, r}
\end{equation}
where $A_{k,\nu,m, \rho, \varepsilon}$ is given by
(\ref{Eq:defam}) and $B_{k, \nu,m, \rho, \varepsilon, r}$ is given
by (\ref{Eq:defBjknum})
\end{lem}
\begin{proof}
From (\ref{Eq:kalpha})  and (\ref{Eq:defam}) we have that
\begin{equation}
\left|\frac{1}{k}A_{k,\nu,m, \rho, \varepsilon}\right| \leq K_4
k^{2 \alpha}
\end{equation}
On the other hand we have that
\begin{eqnarray}
\lefteqn{ \left( \frac{\nu -
m_j}{m+m_j}\right)^{\frac{r+1}{2}}I_{r+1}\left(\frac{4 \pi}{k}(\nu
- m_j)^{\frac{1}{2}}(m+m_j)^{\frac{1}{2}}\right)}\\
&&=\frac{2 \pi (\nu - m_j)^{r+1}}{k^{r+1}}
\sum_{l=0}^{\infty}\frac{(\frac{2 \pi}{k})^{2l}(\nu - m_j)^{l}(\nu
+ m_j)^{l}}{l! \Gamma (r+l+2)}
\end{eqnarray}
which is the jth component of $B_{k, \nu,m, \rho, \varepsilon,
r}$.  Therefore the series (\ref{Eq:sum AknumBknum}) converges
absolutely for $r>2 \alpha$.
\end{proof}

Note that since (\ref{Eq:sum AknumBknum}) converges absolutely for
$r> 2 \alpha$, then

\begin{equation}\label{Eq:convdelscollonets} \begin{split}
2 \pi \sum_{k=1}^{\infty}\frac{1}{k}
\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}
- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)A_{k,\nu,m, \rho, \varepsilon}\ B_{k, \nu,m, \rho, \varepsilon, r}\\
=2 \pi \sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}
- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)\sum_{k=1}^{\infty}\frac{1}{k} A_{k,\nu,m, \rho,
\varepsilon}\ B_{k, \nu,m, \rho, \varepsilon, r}\end{split}
\end{equation}

\begin{lem}\label{lemma:formulaam}Let $F(\tau)$ be a vector-valued modular form of
dimension $r > 2 \alpha >0$. Then if $m+min\{m_1, \ldots ,
m_p\}>0$ the Fourier coefficients in (\ref{eq:fourier}) are given
by the formula
\begin{equation}
a_m= \sum_{k=1}^{\infty}k^r
\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}

- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}\left(%
\begin{array}{ccc}
  a_{-\nu(1)} & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu(p)} \\
\end{array}%
\right)A_{k,\nu,m, \rho, \varepsilon}\left( L_{k, \nu,m, \rho,
\varepsilon, r}+H_{k, \nu,m, \rho, \varepsilon, r} \right)
\end{equation}
where $A_{k,\nu,m, \rho, \varepsilon}$  is given by
(\ref{Eq:defam})

\begin{equation}
L_{k, \nu,m, \rho, \varepsilon, r}=\frac{1}{i}
\int_{-\infty}^{(0+)}f(\omega,k,\nu)d\omega, \quad H_{k, \nu,m,
\rho, \varepsilon, r}=2\sin \pi r
\int_{0}^{\infty}f(\omega,k,\nu)d\omega
\end{equation}
\begin{equation}
f(\omega,k,\nu)= \left(%
\begin{array}{c}
  \omega^{r}e^\frac{2\pi
(\nu-m_1)}{k^2\omega}e^{2\pi  (m+m_1) \omega} \\
  \vdots \\
  \omega^{r}e^\frac{2\pi
(\nu-m_p)}{k^2\omega}e^{2\pi  (m+m_p) \omega} \\
\end{array}%
\right)
\end{equation}

\end{lem}


\begin{proof}
We now evaluate $Q_m(N)$, under the above conditions.

Now if we make the substitution $\omega=N^{-2}-i\varphi$ in
(\ref{Eq:defQm}) we have,
\begin{equation}\begin{split}
Q_m(N)=\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}\Omega_{h,k}e^{-2\pi  i h\frac{m}{k}}\frac{1}{i}\int_{N^{-2}-\theta_{h,k}^{''}}^{N^{-2}+\theta_{h,k}^{'}}\Psi_k(k\omega)
P\left(e^{\frac{2\pi}{k}\left(ih'-k^{-1}\omega^{-1}\right)}\right)
e^{2\pi  m \omega}d\omega
\end{split}
\end{equation}
Therefore by (\ref{Eq:psik}) and (\ref{Eq:defPx}), we have
\begin{eqnarray}\label{Eq:QmwithIk}
\lefteqn{Q_m(N)}\\
&&\begin{split} =\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}\Omega_{h,k}e^{-2\pi  i h\frac{m}{k}}\frac{1}{i}\int_{N^{-2}-\theta_{h,k}^{''}}^{N^{-2}+\theta_{h,k}^{'}}k^{r} \omega^{r} \left(%
\begin{array}{ccc}
  e^{2\pi m_1\omega}e^{-\frac{2\pi m_1}{k^2\omega}} &  \cdots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \cdots & e^{2\pi m_p\omega}e^{-\frac{2\pi m_p}{k^2\omega}} \\
\end{array}%
\right)\\
\left(%
\begin{array}{c}
  \sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}
- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}a_{-\nu} (1)e^-\frac{2\pi i h' \nu}{k}e^\frac{2\pi  \nu}{k^2\omega} \\
\vdots \\
  \sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}
- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}a_{-\nu} (p)e^-\frac{2\pi i h' \nu}{k}e^\frac{2\pi  \nu}{k^2\omega} \\
\end{array}%
\right) e^{2\pi  m \omega}d\omega
\end{split}\\
&&=\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}\Omega_{h,k}e^{-2\pi  i h\frac{m}{k}}k^{r}\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}

- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}e^-\frac{2\pi i h'
\nu}{k}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)I_{k,m,\nu}
\end{eqnarray}

where


\begin{eqnarray*}
\lefteqn{I_{k,m,\nu}}\\
&&=\left(%
\begin{array}{c}
  \frac{1}{i}\int_{N^{-2}-i\theta_{h,k}^{''}}^{N^{-2}+i\theta_{h,k}^{'}} \omega^{r}e^\frac{2\pi  (\nu-m_1)}{k^2\omega}e^{2\pi  (m+m_1) \omega} d\omega \\
  \vdots \\
  \frac{1}{i}\int_{N^{-2}-i\theta_{h,k}^{''}}^{N^{-2}+i\theta_{h,k}^{'}} \omega^{r}e^\frac{2\pi  (\nu-m_p)}{k^2\omega}e^{2\pi  (m+m_p) \omega} d\omega \\
\end{array}%
\right)\\
&&=\frac{1}{i}\int_{N^{-2}-i\theta_{h,k}^{''}}^{N^{-2}+i\theta_{h,k}^{'}}f(\omega,k,\nu)d\omega\\
&&=\left(%
\begin{array}{c}
  I_{k,m,\nu}^1 \\
  \vdots \\
  I_{k,m,\nu}^p \\
\end{array}%
\right)
\end{eqnarray*}

Now we cut the complex plane from $0$ to $-\infty$ along the
negative real axis, and consider the path shown in the figure
below

\includegraphics[scale=.7]{path.eps}
Then we can write

\begin{eqnarray}\label{EQ:ikmnuj}
\lefteqn{I_{k,m,\nu}^j}\\
&&=\frac{1}{i}\int_{-\infty}^{(0+)}-\frac{1}{i}\int_{-\infty}^{-\varepsilon}-\frac{1}{i}\int_{-\varepsilon}^{-\varepsilon-i\theta_{h,k}^{''}}-\frac{1}{i}\int_{-\varepsilon-i\theta_{h,k}^{''}}^{N^{-2}-i\theta_{h,k}^{''}}-\frac{1}{i}\int_{N^{-2}+i\theta_{h,k}^{'}}^{-\varepsilon+i\theta_{h,k}^{'}}-\frac{1}{i}\int_{-\varepsilon+i\theta_{h,k}^{'}}^{-\varepsilon}-\frac{1}{i}\int_{-\varepsilon}^{-\infty}\\
&&=L_{k,m,\nu}^j-J_1^j(N)-J_2^j(N)-J_3^j(N)-J_4^j(N)-J_5^j(N)-J_6^j(N)
\end{eqnarray}

Where the integrand in all the integrals is

\begin{equation}\label{Eq:integrandomega}
 \omega^{r}e^\frac{2\pi  (\nu-m_j)}{k^2\omega}e^{2\pi  (m+m_j) \omega}
\end{equation}


We will also assume that $0<\varepsilon<N^{-2}$. Now in the
integral $J_2^j(N)$ we have

\begin{equation}
\begin{array}{cc}
  \omega=-\varepsilon+i\upsilon, & 0\geq\upsilon\geq -\theta_{h,k}^{''},\\
  \Re(\omega)=-\varepsilon, & \Re\left( \frac{1}{\omega}\right)=\frac{-\varepsilon}{\varepsilon^2+\upsilon^2}<0, \\
  |\omega|=\left(\varepsilon^2+\upsilon^2\right)^{\frac{1}{2}}\leq \left(N{-4}+k^{-2}N{-2}\right)^{\frac{1}{2}}\leq2^{\frac{1}{2}}k^{-1}N^{1}  \\
\end{array}
\end{equation}
and therefore
\begin{equation}
|J_2^j(N)| \leq
\theta_{h,k}^{''}2^{\frac{r}{2}}k^{-r}N^{-r}e^{-2\pi(m+m_j)\varepsilon}<2^{\frac{r}{2}}k^{-r-1}N^{-r-1}
\end{equation}
Similarly we have
\begin{equation}
|J_5^j(N)| <2^{\frac{r}{2}}k^{-r-1}N^{-r-1}
\end{equation}
In the integral $J_3^j(N)$, we have

\begin{equation}
\begin{array}{cc}
  \omega=-u-i\theta_{h,k}^{''}, & -N^{-2}\leq -\varepsilon\leq u \leq N^{-2},\\
  \Re(\omega)=u \leq N^{-2}, & \Re\left( \frac{1}{\omega}\right)=\frac{u}{u^2+ \theta_{h,k}^{''2}} \leq \frac{N^{-2}}{\theta_{h,k}^{''2}} \leq 4k^2, \\
  |\omega|=\left(u^2+\theta_{h,k}^{''2}\right)^{\frac{1}{2}}\leq \left(N{-4}+k^{-2}N{-2}\right)^{\frac{1}{2}}\leq2^{\frac{1}{2}}k^{-1}N^{1}  \\
\end{array}
\end{equation}

and therefore,

\begin{equation}
|J_3^j(N)| \leq \left( N^{-2}+ \varepsilon\right)2^{\frac
{r}{2}}k^{-r}N^{-r}e^{2\pi(m+m_j)N^{-2}+ 8 \pi (\nu-m_j)} \leq
2^{1+ \frac {r}{2}}k^{-r-1}N^{-r-1}e^{2\pi(m+m_j)N^{-2}+ 8 \pi
(\nu-m_j)}
\end{equation}
Similarly,

\begin{equation}
|J_4^j(N)|  \leq 2^{1+ \frac
{r}{2}}k^{-r-1}N^{-r-1}e^{2\pi(m+m_j)N^{-2}+ 8 \pi (\nu-m_j)}
\end{equation}

Finally we have
\begin{equation}
J_1^j(N) +J_6^j(N)=\frac{e^{- \pi i r}}{i}
\int_{-\infty}^{-\varepsilon}+\frac{e^{\pi i r}}{i}
\int_{-\varepsilon}^{-\infty}
\end{equation}
Where the integrand is given by (\ref{Eq:integrandomega}), and
therefore

\begin{equation}
J_1^j(N) +J_6^j(N)=-2\sin \pi r
\int_{\varepsilon}^{\infty}t^{r}e^\frac{2\pi (m_j-
\nu)}{k^2t}e^{-2\pi (m+m_j) t}dt
\end{equation}
Now by (\ref{Eq:QmwithIk}) , (\ref{Eq:boundomega}) and using the
fact that $\sum_1^N k^{-1} \leq N$

\begin{eqnarray}
\lefteqn{Q_m(N)}\\
&&\begin{split} =\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}\Omega_{h,k}e^{-2\pi  i h\frac{m}{k}}k^{r}\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}
- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}e^{-\frac{2\pi i h'
\nu}{k}}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)(L_{k, \nu,m, \rho, \varepsilon, r}+H_{k, \nu,m, \rho, \varepsilon, r})\\
+\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}\Omega_{h,k}e^{-2\pi  i h\frac{m}{k}}k^{r}\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}
- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}e^{-\frac{2\pi i h'
\nu}{k}}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)(\xi_1(N)k^{-r-1})
\end{split}\\
&&\label{Eq:qmdelscollons}\begin{split}=\sum_{\tiny{\begin{array}{c}
  h,k \\
  0 \leq h < k \leq N\\
  (h,k)=1 \\
\end{array}}}\Omega_{h,k}e^{-2\pi  i h\frac{m}{k}}k^{r}\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}
- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}e^{-\frac{2\pi i h'
\nu}{k}}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)(L_{k, \nu,m, \rho, \varepsilon, r}+H_{k, \nu,m, \rho, \varepsilon, r})\\
+ \xi_2(N,\alpha)
\end{split}
\end{eqnarray}
where $\xi_1(N)$ is a column vector of components that are
$O(N^{-r-1}e^{2\pi(m+m_j)N^{-2}})$, and $\xi_2(N,\alpha)$ is a
column vector of components that are $O(N^{-r+ 2
\alpha}e^{2\pi(m+m_j)N^{-2}})$.

And thus by (\ref{Eq:qmdelscollons}),  (\ref{Eq:am}) and
(\ref{EQ:rmbounded}), we have
\begin{equation}
\begin{split}a_m=\sum_{k=1}^N k^r \sum_{\tiny{\begin{array}{c}
  0 \leq h < k\\
  (h,k)=1 \\
\end{array}}}\Omega_{h,k}e^{-2\pi  i h\frac{m}{k}}\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}
- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}e^{-\frac{2\pi i h'
\nu}{k}}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)(L_{k, \nu,m, \rho, \varepsilon, r}+H_{k, \nu,m, \rho, \varepsilon, r})\\
+\xi_2(N,\alpha) \end{split}
\end{equation}
Now if we let $N$ go to infinity and since $r>0$, and by
(\ref{Eq:convdelscollonets}) we have that the series converges
absolutely and therefore
\begin{equation}\label{Eq:amintermsofLandH}
a_m= \sum_{k=1}^{\infty}k^r
\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}

- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)A_{k,\nu,m, \rho, \varepsilon}\left( L_{k, \nu,m, \rho,
\varepsilon, r}+H_{k, \nu,m, \rho, \varepsilon, r} \right)
\end{equation}
where $A_{k,\nu,m, \rho, \varepsilon}$ is given by
(\ref{Eq:defam})
\end{proof}
Now from the theory of Bessel function we have that
\begin{equation}
L_{k, \nu,m, \rho, \varepsilon, r}^j+H_{k, \nu,m, \rho,
\varepsilon, r}^j=\frac{2 \pi}{k^{r+1}} \left( \frac{\nu -
m_j}{m+m_j}\right)^{\frac{r+1}{2}}I_{r+1}\left(\frac{4 \pi}{k}(\nu
- m_j)^{\frac{1}{2}}(m+m_j)^{\frac{1}{2}}\right)
\end{equation}
which makes sense since $m+m_j>0$,  where $I_\rho(z)$ is given by
(\ref{EQ:defIrho}), this reduces (\ref{Eq:amintermsofLandH}) to
\begin{equation}
a_m= 2 \pi \sum_{k=1}^{\infty}\frac{1}{k}
\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}

- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)A_{k,\nu,m, \rho, \varepsilon}\ B_{k, \nu,m, \rho,
\varepsilon, r}
\end{equation}
where $B_{k, \nu,m, \rho, \varepsilon, r}$ is given by
(\ref{Eq:defBjknum})

\begin{lem}Let $F(\tau)$ be a vector-valued modular form of
dimension r, then
\begin{equation}
6m_j+\frac{r}{2}
\end{equation}
is always an integer, where $0 \leq m_j <1$, $m_j \in \mathbb{Q}$,
such that (\ref{eq:fourier}) holds.
\end{lem}
\begin{proof}From (\ref{eq:ftau}), (\ref{eq:slash}) and
(\ref{Eq:vep}), we can see that
\begin{equation}
F\left( \frac{-1}{\tau} \right)= v\left(%
\begin{array}{cc}
  0 & -1 \\
  1 & 0 \\
\end{array}%
\right)\rho \left(%
\begin{array}{cc}
  0 & -1 \\
  1 & 0 \\
\end{array}%
\right) \left(-i \tau \right)^{-r}F\left( \tau \right)
\end{equation}
If we replace $\tau$ by $\frac{-1}{\tau}$ and applying the above
result, we get
\begin{eqnarray}
F\left( \tau \right)&&= v\left(S \right)^2\rho \left(S \right)^2
\left(\frac{i}{ \tau} \right)^{-r} \left(-i \tau
\right)^{-r}F\left( \tau \right)\\
&&= v\left(S \right)^2\rho \left(S \right)^2 \left|\frac{1}{ \tau}
\right|^{-r} \left| \tau \right|^{-r}e^{-ir\left( \arg
\frac{i}{\tau} +\arg (-i \tau )\right)}F\left( \tau \right)
\end{eqnarray}
From (\ref{Eq:argtau}), we infer
\begin{equation}
\begin{array}{cc}
  -\frac{\pi}{2} < \arg \frac{i}{\tau} < \frac{\pi}{2}, & -\frac{\pi}{2} < \arg (-i \tau) < \frac{\pi}{2} \\
\end{array}
\end{equation}
and hence
\begin{equation}\label{Eq:argtau2}
- \pi < \arg \frac{i}{\tau} + \arg (-i \tau) < \pi
\end{equation}
but we have
\begin{equation}\label{Eq:argtau3}
\arg \frac{i}{\tau} + \arg (-i \tau) \equiv \arg \left(
\frac{i}{\tau}(-i \tau)\right)\equiv 0 \pmod {2 \pi}
\end{equation}
and therefore by (\ref{Eq:argtau2}) and (\ref{Eq:argtau3}) we get
\begin{equation}
\arg \frac{i}{\tau} + \arg (-i \tau)=0
\end{equation}
hence
\begin{equation}
F \left( \tau \right) = v\left(S \right)^2\rho \left(S
\right)^2F\left( \tau \right)
\end{equation}
Therefore
\begin{equation}\label{Eq:id1}
v\left(S \right)^2\rho \left(S \right)^2=I
\end{equation}
As discussed by Knopp and Mason in \cite {[11]}, we can assume
that
\begin{equation}
v\left(T \right)\rho \left(T \right)=\left(%
\begin{array}{ccc}
  e^{2\pi i m_1} & \cdots & 0 \\
  \vdots & \ddots & \vdots\\
  0 & \cdots & e^{2\pi i m_p} \\
\end{array}%
\right)
\end{equation}
and therefore
\begin{equation}
F \left( 1- \frac{1}{\tau} \right)=\left(%
\begin{array}{ccc}
  e^{2\pi i m_1} & \cdots & 0 \\
  \vdots & \ddots & \vdots\\
  0 & \cdots & e^{2\pi i m_p} \\
\end{array}%
\right)v\left(S \right)\rho \left(S \right) \left(-i \tau
\right)^{-r}F\left( \tau \right)
\end{equation}
if we replace $\tau$ by $1-\frac{1}{\tau}$ twice we get
\begin{eqnarray}
F(\tau)&&=\left(%
\begin{array}{ccc}
  e^{2\pi i m_1} & \cdots & 0 \\
  \vdots & \ddots & \vdots\\
  0 & \cdots & e^{2\pi i m_p} \\
\end{array}%
\right)^3  v\left(S \right)^3\rho \left(S \right)^3 \left(
\frac{i}{\tau - 1}\right)^{-r}\left( -i \left( 1 - \frac{1}{\tau}
\right)\right)^{-r}(-i \tau)^{-r}F(\tau)\\
&&\label{Eq:foneminusoneovertau}\begin{split}=\left(%
\begin{array}{ccc}
  e^{6\pi i m_1} & \cdots & 0 \\
  \vdots & \ddots & \vdots\\
  0 & \cdots & e^{6\pi i m_p} \\
\end{array}%
\right)v\left(S \right)\rho \left(S \right)\left| \frac{1}{\tau -
1}\right|^{-r}\left| 1 - \frac{1}{\tau}
\right|^{-r}|\tau|^{-r}\\
e^{-ir\left( \arg \left( \frac{i}{\tau - 1}\right)+ \arg \left( -i
\left( 1 - \frac{1}{\tau} \right)\right)+ \arg(-i
\tau)\right)}F(\tau) \end{split}
\end{eqnarray}
From (\ref{Eq:argtau}), we infer
\begin{equation}
 -\frac{3 \pi}{2} \leq \arg \left( \frac{i}{\tau - 1}\right)+ \arg \left( -i
\left( 1 - \frac{1}{\tau} \right)\right)+ \arg(-i \tau) \leq
\frac{3 \pi}{2}
\end{equation}
On the other hand we have that
\begin{eqnarray}
\arg \left( \frac{i}{\tau - 1}\right)+ \arg \left( -i \left( 1 -
\frac{1}{\tau} \right)\right)+ \arg(-i \tau) &\equiv& \arg \left(
\frac{i}{\tau - 1}\right) \left( -i \left( 1 - \frac{1}{\tau}
\right)\right)(-i \tau)\\
&&\equiv \arg (-i)\\
&&\equiv -\frac{\pi}{2} \pmod{ 2 \pi}
\end{eqnarray}
hence
\begin{equation}
\arg \left( \frac{i}{\tau - 1}\right)+ \arg \left( -i \left( 1 -
\frac{1}{\tau} \right)\right)+ \arg(-i \tau)= -\frac{\pi}{2}
\end{equation}
and therefore (\ref{Eq:foneminusoneovertau}) becomes
\begin{equation}
F(\tau)=\left(%
\begin{array}{ccc}
  e^{6\pi i m_1} & \cdots & 0 \\
  \vdots & \ddots & \vdots\\
  0 & \cdots & e^{6\pi i m_p} \\
\end{array}%
\right)v\left(S \right)\rho \left(S \right)e^{\frac{\pi i
r}{2}}F(\tau)
\end{equation}
and therefore
\begin{equation}
\left(%
\begin{array}{ccc}
  e^{6\pi i m_1} & \cdots & 0 \\
  \vdots & \ddots & \vdots\\
  0 & \cdots & e^{6\pi i m_p} \\
\end{array}%
\right)v\left(S \right)\rho \left(S \right)e^{\frac{\pi i r}{2}}=I
\end{equation}
and using (\ref{Eq:id1}), we get
\begin{equation}
v\left(S \right)\rho \left(S \right)=\left(%
\begin{array}{ccc}
  e^{6\pi i m_1+\frac{\pi i r}{2}} & \cdots & 0 \\
  \vdots & \ddots & \vdots\\
  0 & \cdots & e^{6\pi i m_p + \frac{\pi i r}{2}} \\
\end{array}%
\right)
\end{equation}
and since
\begin{equation}
\left(v\left(S \right)\rho \left(S \right) \right)^2=I
\end{equation}
we have that
\begin{equation}
e^{6\pi i m_j + \frac{\pi i r}{2}}=\pm 1, \quad 1 \leq j \leq p
\end{equation}
and therefore,
\begin{equation}
6m_j+\frac{r}{2}
\end{equation}
is an integer as claimed, moreover we can see that if one of the
$m_j$'s is zero then $r$ has to be an even integer.
\end{proof}
\begin{lem}Let $F(\tau)$ be a vector-valued modular form of
dimension $r>2 \alpha > 0$. Then if $min\{m_1, \ldots , m_p\}=0$,
then
\begin{equation}
a_0= 2 \pi \sum_{k=1}^{\infty}\frac{1}{k}
\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}

- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)A_{k,\nu,0, \rho, \varepsilon}\ B_{k, \nu,0, \rho,
\varepsilon, r}^*
\end{equation}
where $A_{k,\nu,0, \rho, \varepsilon}$ is given by
(\ref{Eq:defam}) when $m=0$, and
\begin{equation}\label{Eq:defbkinustar}
B_{k, \nu,0, \rho, \varepsilon, r}^{j*}=\begin{array}{cc}
  \frac{1}{(r+1)!} \left(\frac{2 \pi \nu}{k} \right)^{r+1}& if m_j=0 \\
  B_{k, \nu,0, \rho,
\varepsilon, r}^{j} & if m_j\neq 0 \\
\end{array}
\end{equation}
\end{lem}
\begin{proof}Since $min\{m_1, \ldots , m_p\}=0$ and by the
previous lemma, we have that $r$ is an integer, and therefore the
integrand
\begin{equation}\label{Eq:integrandomega}
 \omega^{r}e^\frac{2\pi  (\nu-m_j)}{k^2\omega}e^{2\pi  (m+m_j) \omega}
\end{equation}
is unique in the plane, and there is no need to cut the plane
along the negative axis. So if $m_j=0$ we can use the path of
integration along a rectangle $R$ with the vertices
\begin{equation}
-\varepsilon -i \theta '', \quad N^{-2} - i \theta'', \quad N^{-2}
+ i \theta', \quad -\varepsilon + i \theta'
\end{equation}
and get
\begin{eqnarray}
I_{k,0,\nu}^j &=& \frac{1}{i} \int_{R} - \frac{1}{i} \int_{N^{-2}
+ i \theta'}^{-\varepsilon + i \theta'} - \frac{1}{i}
\int_{-\varepsilon + i \theta'}^{-\varepsilon -i \theta ''} -
\frac{1}{i} \int_{-\varepsilon -i \theta ''}^{N^{-2} - i
\theta''}\\
&&= L_{k,\nu,0, \rho, \varepsilon, r}^{j*} - J_1^* - J_2^* - J_3^*
\end{eqnarray}
Here $J_1^*$ corresponds to $J_4$, $J_3^*$ to $J_3$ and $J_2^*$ to
$J_2 + J_5$ in (\ref{EQ:ikmnuj}). The estimations are the same as
before. The difference now is that there are no improper integrals
and there is no need for the convergence condition $m+m_j>0$. and
instead of (\ref{Eq:amintermsofLandH}) we get
\begin{equation}\
a_0= \sum_{k=1}^{\infty}k^r
\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}

- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)A_{k,\nu,0, \rho, \varepsilon} L_{k, \nu,0, \rho,
\varepsilon, r}^*
\end{equation}
where $L_{k, \nu,0, \rho, \varepsilon, r}^*$ is a column vector in
which the jth component is
\begin{equation}
L_{k, \nu,0, \rho, \varepsilon, r}^{j*}=\begin{array}{cc}
   L_{k, \nu,0, \rho, \varepsilon, r}+H_{k, \nu,0, \rho, \varepsilon, r} & if m_j \neq 0 \\
  \frac{1}{i} \int^{(0+)}\omega^re^{\frac{2 \pi \nu}{k^2\omega}}d\omega & if m_j=0 \\
\end{array}
\end{equation}
Now since
\begin{equation}
 \frac{1}{i} \int^{(0+)}\omega^re^{\frac{2 \pi
 \nu}{k^2\omega}}d\omega=\frac{2 \pi}{(r+1)!} \left(\frac{2 \pi \nu}{k^2} \right)^{r+1}
\end{equation}
Therefore, after interchanging the summations we get
\begin{equation}
a_0= 2 \pi \sum_{k=1}^{\infty}\frac{1}{k}
\sum_{\tiny{\nu=1}}^{\tiny{\begin{array}{c}

- min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}\left(%
\begin{array}{ccc}
  a_{-\nu}(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_{-\nu}(p) \\
\end{array}%
\right)A_{k,\nu,0, \rho, \varepsilon, r}\ B_{k, \nu,0, \rho,
\varepsilon, r}^*
\end{equation}
as claimed.
\end{proof}
\begin{proof}[Proof of the Theorem] The theorem is already proven
if $m+min\{m_1, \ldots , m_p\}>0$. Also if $m+min\{m_1, \ldots ,
m_p\}=0$, we have a formula for the coefficients. We only have to
notice that the formula in the case in which $m$ and $m_j$ are 0
has to be understood in the previous lemma.
\end{proof}

\begin{thm}Let $b_1,\ldots , b_{\mu}$ be a set of column vectors
such that $b_j \in \mathbb{C}^p$, $b_j\neq 0$, $\rho:\Gamma
\longrightarrow GL(p,\mathbb{C})$ a $p$-dimensional complex
representation , $\varepsilon$ a Multiplier System , and $r>
2\alpha$ (\ref{eq:alpha}), $r,\mu \in \mathbb{Z^+}$ then if
 \begin{equation}
 \begin{split}G(\tau )=\left(%
\begin{array}{c}
  \sum_{\nu =1}^\mu b_\nu(1) e^{2 \pi i (m_1-\nu)\tau} \\
  \vdots \\
  \sum_{\nu =1}^\mu b_\nu(p) e^{2 \pi i (m_p-\nu)\tau} \\
\end{array}%
\right) +\sum_{m=0}^\infty \left(%
\begin{array}{ccc}
  e^{2 \pi i (m + m_1)\tau} & \cdots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & e^{2 \pi i (m + m_p)\tau} \\
\end{array}%
\right). \\ \sum_{\nu =1}^\mu\left(%
\begin{array}{ccc}
  b_\nu(1) & \cdots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \cdots & b_\nu(p) \\
\end{array}%
\right)C_{k,\nu,m, \rho, \varepsilon, r}
\end{split}
\end{equation},
where
\begin{equation}\label{Eq:sum AknumBknum}
C_{k,\nu,m, \rho, \varepsilon, r} = 2 \pi \sum_{k=1}^{\infty}\frac
{1}{k}A_{k,\nu,m, \rho, \varepsilon}B_{k,\nu,m, \rho, \varepsilon,
r}
\end{equation}
and $A_{k,\nu,m, \rho, \varepsilon}$ and  $B_{k,\nu,m, \rho,
\varepsilon, r}$  are defined as in (\ref{Eq:defam})
(\ref{Eq:defBjknum}) respectively, we have
\begin{enumerate}
\item $G(\tau)$ is regular in the complex upper half-plane $
\mathcal{H}$ \item $G(\tau)$ satisfies
\begin{equation}
G(\tau)- \varepsilon^{-1}(M)(-i(c \tau +d))^r \rho^{-1} (M)
G(M\tau)= p_M(\tau)
\end{equation},
for all
\begin{equation}
M=\left(%
\begin{array}{cc}
  a & b \\
  c & d \\
\end{array}%
\right)\in \Gamma
\end{equation}

\end{enumerate}
\end{thm}
\begin{lem}Define $C_m(l)$ by
\begin{equation}
C_m(l)=2 \pi \sum_{\tiny{\begin{array}{c}
  k=1 \\
  k\equiv 0
\pmod {l}\\
\end{array}}}^{\infty}\frac {1}{k}A_{k,\nu,m, \rho,
\varepsilon}\left(%
\begin{array}{c}
  \left( \frac{\nu -
m_1}{m+m_1}\right)^{\frac{r+1}{2}}I_{r+1}\left(\frac{4 \pi}{k}(\nu
- m_1)^{\frac{1}{2}}(m+m_1)^{\frac{1}{2}}\right) \\
  \vdots \\
  \left( \frac{\nu -
m_p}{m+m_p}\right)^{\frac{r+1}{2}}I_{r+1}\left(\frac{4 \pi}{k}(\nu
- m_p)^{\frac{1}{2}}(m+m_p)^{\frac{1}{2}}\right) \\
\end{array}%
\right)
\end{equation}
then
\begin{equation}
|C_m(l)|\sim A_{l,\nu,m, \rho, \varepsilon}\left(%
\begin{array}{c}
  \frac{(\nu-m_1)^{\frac{r}{2}+\frac{1}{4}}e^{\frac{4 \pi \sqrt{(\nu-m_1)(m+m_1)}}{l}}}{\sqrt{2l}(m+m_1)^{\frac{r}{2}+\frac{3}{4}}} \\
  \vdots \\
  \frac{(\nu-m_p)^{\frac{r}{2}+\frac{1}{4}}e^{\frac{4 \pi \sqrt{(\nu-m_p)(m+m_1)}}{l}}}{\sqrt{2l}(m+m_p)^{\frac{r}{2}+\frac{3}{4}}}  \\
\end{array}%
\right)
\end{equation}
\end{lem}
\begin{proof}
From the power series definition for $I_{\rho}(z)$ in
(\ref{Eq:powerseriesof Iro}), we can see that for every
nonnegative integer $r$, we have that

\begin{equation}
I_{r+1}(z) \leq z^r \sinh z
\end{equation}
Also we will use the fact that
\begin{equation}
\sinh z \leq \frac{z \sinh B}{B}, \quad for \quad 0\leq z \leq B
\end{equation}
now
\begin{eqnarray*}
\lefteqn{|C_m(l)-first \quad term|}\\
 &&= \left| 2 \pi
\sum_{\tiny{\begin{array}{c}
  k=2l \\
  k\equiv 0
\pmod {l}\\
\end{array}}}^{\infty}\frac {1}{k}A_{k,\nu,m, \rho,
\varepsilon}\left(%
\begin{array}{c}
  \left( \frac{\nu -
m_1}{m+m_1}\right)^{\frac{r+1}{2}}I_{r+1}\left(\frac{4
\pi}{k}\sqrt{(\nu
- m_1)(m+m_1)}\right) \\
  \vdots \\
  \left( \frac{\nu -
m_p}{m+m_p}\right)^{\frac{r+1}{2}}I_{r+1}\left(\frac{4
\pi}{k}\sqrt{(\nu
- m_1)(m+m_1)}\right) \\
\end{array}%
\right)\right|\\
&&\leq C_1  \sum_{\tiny{\begin{array}{c}
  k=2l \\
  k\equiv 0
\pmod {l}\\
\end{array}}}^{\infty}\frac {1}{k}\left|A_{k,\nu,m, \rho,
\varepsilon} \right| \left|\left(%
\begin{array}{c}
  \left( \frac{\nu -
m_1}{m+m_1}\right)^{\frac{r+1}{2}}\left(\frac{4 \pi}{k}\sqrt{(\nu
- m_1)(m+m_1)}\right)^{r+1}\frac{\sinh\left(\frac{4
\pi}{2l}\sqrt{(\nu - m_1)(m+m_1)}\right)}{\left(\frac{4
\pi}{2l}\sqrt{(\nu
- m_1)(m+m_1)}\right)} \\
  \vdots \\
  \left( \frac{\nu -
m_p}{m+m_p}\right)^{\frac{r+1}{2}}\left(\frac{4 \pi}{k}\sqrt{(\nu
- m_p)(m+m_p)}\right)^{r+1}\frac{\sinh\left(\frac{4
\pi}{2l}\sqrt{(\nu - m_p)(m+m_p)}\right)}{\left(\frac{4
\pi}{2l}\sqrt{(\nu
- m_p)(m+m_p)}\right)} \\
\end{array}%
\right)\ \right|\\
&& \leq C_2 (m+min(m_1,\ldots, m_p))^{- \frac{1}{2}}
\end{eqnarray*}
\end{proof}
\begin{proof}

\begin{eqnarray}
\end{eqnarray}
\end{proof}
\begin{equation}
\end{equation}
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\end{document}