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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}
\newtheorem{thm}{Theorem}
\begin{thm}...
\end{thm}

asdf

\newtheorem{lem}[thm]{Lemma}
\begin{lem}...
\end{lem}
\begin{proof}...
\end{proof}
\begin{lem}....
\end{lem}
\begin{proof}...
 \end{proof}
\begin{proof}[Proof of the Main Theorem]....
\end{proof}




\begin{thm}Let $F(\tau)$ be a vector-valued modular form, then
\begin{equation}
a_m=2 \pi .... see 7.2
\end{equation}
\end{thm}


\begin{lem}
\begin{equation}
a_m= \sum_{k=1}^{\infty}k^r \sum_{\tiny{\begin{array}{c}

  \nu \geq min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}^{-1}\left(%
\begin{array}{ccc}
  a_\nu(1) & \ldots & 0 \\
  \vdots & \ddots & \vdots \\
  0 & \ldots & a_\nu(p) \\
\end{array}%
\right)A_{k,\nu,m}\left( L_{k, \mu,m}+H_{k, \mu,m} \right)
\end{equation}
where
\begin{equation}
A_{k,\nu,m}=\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}
\begin{equation}
L_{k, \mu,m}=\frac{1}{i}
\int_{-\infty}^{(0+)}f(\omega,k,\nu)d\omega, \quad H_{k,
\mu,m}=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 $Q_m$, under the condition

\begin{equation}
m+max\{m_1,m_2\}>0
\end{equation}
todavia no se muy bien porque, parece ser que sera necesario para
la convergencia de cierta integral

Now if we make the substitution

\begin{equation}
\omega=N^{-2}-i\varphi
\end{equation}
in (\ref{Eq:defQm})We have,
\begin{equation}\begin{split}
Q_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}}\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}\\
&&\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}{cc}
  e^{2\pi m_1\omega}e^{-\frac{2\pi m_1}{k^2\omega}} & 0 \\
  0 & e^{2\pi m_2\omega}e^{-\frac{2\pi m_2}{k^2\omega}} \\
\end{array}%
\right)\\
\left(%
\begin{array}{c}
  \sum_{\tiny{\begin{array}{c}
  \nu \geq min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}^{-1}a_{-\nu} (1)e^\frac{2\pi i h' \nu}{k}e^\frac{2\pi  \nu}{k^2\omega} \\
  \sum_{\tiny{\begin{array}{c}
  \nu \geq min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}^{-1}a_{-\nu} (2)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{\begin{array}{c}

  \nu \geq min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}^{-1}e^\frac{2\pi i h'
\nu}{k}\left(%
\begin{array}{cc}
  a_{-\nu} (1) & 0 \\
  0 & a_{-\nu} (2) \\
\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 \\
  \frac{1}{i}\int_{N^{-2}-i\theta_{h,k}^{''}}^{N^{-2}+i\theta_{h,k}^{'}} \omega^{r}e^\frac{2\pi  (\nu-m_2)}{k^2\omega}e^{2\pi  (m+m_2) \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 \\
  I_{k,m,\nu}^2 \\
\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*}
\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-J_2^j-J_3^j-J_4^j-J_5^j-J_6^j
\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 $J_2^j$ 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| \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| <2^{\frac{r}{2}}k^{-r-1}N^{-r-1}
\end{equation}
In the integral $J_3^j$, 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| \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|  \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 +J_6^j=\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 +J_6^j=-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}) and using the fact that

\begin{eqnarray*}
\lefteqn{Q_m}\\
&&\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{\begin{array}{c}
  \nu \geq min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}^{-1}e^\frac{2\pi i h'
\nu}{k}\left(%
\begin{array}{cc}
  a_{-\nu} (1) & 0 \\
  0 & a_{-\nu} (2) \\
\end{array}%
\right)(L_{k, \mu,m}+H_{k, \mu,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}}k^{r}\sum_{\tiny{\begin{array}{c}
  \nu \geq min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}^{-1}e^\frac{2\pi i h'
\nu}{k}\left(%
\begin{array}{cc}
  a_{-\nu} (1) & 0 \\
  0 & a_{-\nu} (2) \\
\end{array}%
\right)(O(N^{-r-1})k^{-r-1})
\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{\begin{array}{c}

  \nu \geq min \mu_j \\
  0 \leq j \leq p \\
\end{array}}}^{-1}e^\frac{2\pi i h'
\nu}{k}\left(%
\begin{array}{cc}
  a_{-\nu} (1) & 0 \\
  0 & a_{-\nu} (2) \\
\end{array}%
\right)(L_{k, \mu,m}+H_{k, \mu,m})+ O(N^{-r+2\alpha})
\end{eqnarray*}

\end{proof}
\begin{proof}...
\end{proof}

\begin{equation}
\end{equation}





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\end{document}