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% was@math.harvard.edu, verrill@math.ku.dk                       %
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\begin{document}

\par\noindent
{\bf \Large \sc Cuspidal modular symbols are transportable}
\vspace{3ex}
\par\noindent
{\large William A.\ Stein}\\
{\small Harvard University, USA}\\
{\small \sf was@math.harvard.edu\\
            http://modular.fas.harvard.edu}
\vspace{2ex}
\par\noindent
{\large Helena A.\ Verrill}\\
{\small Universit\"at Hannover}\\
{\small \sf verrill@math.uni-hannover.de}
\vspace{2ex}

\begin{abstract}
Modular symbols of weight~$2$ for a congruence subgroup
$\Gamma$ satisfy the identity 
$\{\alpha, \gamma(\alpha)\} = \{\beta,\gamma(\beta)\}$
for all $\alpha, \beta$ in the extended 
upper half plane and $\gamma\in \Gamma$.  
The analogue of this identity is false for
modular symbols of weight greater than~$2$. 
In this paper we define transportable modular symbols,
which are symbols for which an analogue of 
the above identity holds, and
prove that every cuspidal symbol can be written
as a transportable symbol.
As a corollary, we obtain an algorithm for computing
periods of cuspforms.
\end{abstract}

\section*{Introduction}
It is well known that 
modular symbols of weight~$2$ for a congruence subgroup~$\Gamma$ 
satisfy the identity 
$\{\alpha, \gamma(\alpha)\} = \{\beta,\gamma(\beta)\}$
for all $\alpha, \beta$ in the extended upper half plane and
$\gamma\in \Gamma$.  
The analogue of this identity is, in general, false for
modular symbols of weight greater than~$2$. 
To investigate further, we define transportable modular symbols,
which are symbols that can be expressed in such a way 
that the above identity holds.  We then 
prove that every cuspidal symbol is transportable.
As a corollary we obtain an algorithm for computing
periods of cuspforms.

In Section~\ref{sec:modsym} we review the definition of 
modular symbols.  In Section~\ref{sec:transportable}
we define transportable modular symbols and prove 
our main theorem.
Section~\ref{sec:computingperiods}
contains an application of our transportability result to
the computation of periods of modular forms.  Finally,
Section~\ref{sec:examples} contains two examples in which
we verify the assertion of Theorem~\ref{thm:onlyoo} and
apply the period computation algorithm.

{\bf Acknowledgement.}
The authors would like to thank L.~Merel
and J.~Cremona for helpful advice, 
I.~Kiming for improving the proof of Theorem~\ref{thm:onlyoo},
and the referee for several helpful remarks.

\section{Modular symbols}\label{sec:modsym}
In Section~\ref{sec:modsymdef} we recall the definition
of modular symbols given in \cite{merel:1585}, then
in Section~\ref{sec:extmodsymdef} we introduce a slight
generalization of the definition.
Let~$N$ and~$k$ be positive integers with $k\geq 2$,
and let $\eps:\Z/N\Z\ra \C$ be a Dirichlet character modulo~$N$.

\subsection{Definition}\label{sec:modsymdef}
Let~$\sM$ be the abelian group generated by all symbols
$\{\alpha, \beta\}$ with $\alpha, \beta\in\PP^1(\Q)$,
modulo the relations 
$\{\alpha,\beta\}+\{\beta,\gamma\}+\{\gamma,\alpha\} = 0,$
and modulo any torsion. 
Let $V_{k-2}$ denote
the group of homogeneous polynomials in $\Z[X,Y]$ of degree $k-2$. 
Then each element $\gamma=\abcd{a}{b}{c}{d}\in\SL(2,\Z)$ acts on the left 
on $V_{k-2}$ by
$$\gamma(P(X,Y)) = P(dX-bY,-cX+aY),$$
and on $\sM_{k} = V_{k-2}\tensor \sM$ by
$$\gamma(P\tensor\{\alpha,\beta\})
    = \gamma(P)\tensor\{\gamma(\alpha),\gamma(\beta)\}.$$
Fix a Dirichlet character $\eps:\Z/N\Z\ra\C$,
and denote by $\Z[\eps]$ the ring generated by the image
of~$\eps$.
We also view~$\eps$ as a homomorphism $\Gamma_0(N)\ra \C^*$ by setting
$\eps\abcd{a}{b}{c}{d} = \eps(d)$.

The space $\sM_k(N,\eps)$ of \defn{modular symbols} of level~$N$ 
and character~$\eps$ is
the quotient of the $\Z[\eps]$-module 
$\sM_k\tensor\Z[\eps]$ by the $\Z[\eps]$-submodule generated
by $\gamma(x) - \eps(\gamma) x$
for all $x\in \sM_k$, all $\gamma\in\Gamma_0(N)$,
and by any torsion.
Denote by $P\{\alpha,\beta\}$ the image of
$P\tensor\{\alpha,\beta\}$ in $\sM_k(N,\eps)$.
The $\Q[\eps]$-vector space
 $$\sM_k(N,\eps;\Q)=\sM_k(N,\eps)\otimes_\Z \Q$$
contains $\sM_k(N,\eps)$.

Let $\sB$ be the free abelian group
generated by all symbols $\{\alpha\}$, for $\alpha\in\PP^1(\Q)$. 
Define a left action of $\SL(2,\Z)$ on 
$\sB_k=V_{k-2}\tensor \sB$ by 
$$\gamma(P\otimes\{\alpha\})=\gamma(P)\otimes\{\gamma\alpha\}.$$
The space $\sB_k(N,\eps)$ of \defn{boundary symbols} is the quotient
of $\sB_k\tensor\Z[\eps]$ by the $\Z[\eps]$-submodule generated
by $\gamma(x)-\eps(\gamma)x$ 
for all $x\in \sB_k$, all $\gamma\in\Gamma_0(N)$, and
by any torsion.
The subspace $\sS_k(N,\eps)$ of \defn{cuspidal symbols} 
is the kernel of the map
$\delta:\sM_k(N,\eps) \ra \sB_k(N,\eps)$
given by
  $\delta(P\{\alpha,\beta\}) = 
          P\{\beta\} - P\{\alpha\}.$

When $\eps=1$ is the trivial character, we will also 
write $\sM_k(\Gamma_0(N))$ for $\sM_k(N,1)$, and similarly for
$\sS_k$ and $\sB_k$.


\subsection{Extended modular symbols}\label{sec:extmodsymdef}
It is useful to extend the notion of modular symbols 
to allows symbols of the form
$P\{z,w\}$ where~$z$ and~$w$ are 
arbitrary elements of $\h^*=\h\union\PP^1(\Q)$.  
\begin{definition}[Extended modular symbols]
The group $\esM_k$ of \defn{extended modular symbols} 
is the free abelian group with basis the set of
all symbols $P\{z,w\}$ with $z,w \in \h^*$,
subject to the relations $P\{u,v\}+P\{v,w\}+P\{w,u\}=0$.
\end{definition}
Note that $\esM_k$ is of uncountable rank over $\Z$.
It is equipped with
an action of $\gzero$; we 
let $\esM_k(N,\eps)$ be the 
largest torsion-free quotient of $\esM_k$ by 
the relations $\gamma x = \eps(\gamma)x$ for
$\gamma\in\gzero$.

\section{Transportable modular symbols}\label{sec:transportable}
In Section~\ref{sec:deftrans} we define transportable modular
symbols, and we prove an elementary proposition that motivates
the definition.  Section~\ref{sec:character}, which is the heart
of this paper, contains a proof that every cuspidal
modular symbol is transportable.

\subsection{Definition}\label{sec:deftrans}

\begin{definition}[Transportable]
\label{transdef}
A modular symbol is {\em transportable} if 
it can be written in the form
$$\sum_{i=1}^m P_i\{\infty,\gamma_i(\infty)\},$$
for $\gamma_i\in \Gamma_0(N)$
and $P_i\in V_{k-2}$ with
$$\sum_{i=1}^m P_i\{\infty,\gamma_i(\infty)\} =
\sum_{i=1}^m P_i\{\alpha,\gamma_i(\alpha)\}$$
for all $\alpha\in{\Hu}^*$, where
the equality takes place in $\esM_k$.
\end{definition}


When $k=2$,
the identity $\{\infty,\gamma(\infty)\}=\{\alpha,\gamma(\alpha)\}$
holds for any $\alpha\in\h^*$, so in
weight $2$ there is a plentiful supply of transportable modular
symbols.  This identity sometimes fails when $k>2$, as indicated below.
\begin{proposition}\label{prop:modsym-errorterm}
For any $\gamma\in \Gamma_0(N)$, $P\in V_{k-2}$ and $\alp\in\h^*$,
\begin{eqnarray}
P\{\oo, \gamma(\oo)\}
  &=& P\{\alp,\gamma(\alp)\} + (P -
\eps(\gamma)\gamma^{-1}P)\{\oo,\alp\}\nonumber\\
  &=& \eps(\gamma)(\gamma^{-1}P)\{\alp, \oo\} - P\{\gamma(\alp),\oo\}.
\label{errorterm}
\end{eqnarray}
In particular,
\begin{equation}
\label{trivialerror}
P\{\infty,\gamma(\infty)\}=P\{\alpha,\gamma(\alpha)\}\Leftrightarrow 
P=\eps(\gamma)\gamma^{-1}P.
\end{equation}
\end{proposition}
\begin{proof}
If $x\in\sM_k(N,\eps)$ is a modular symbol 
and $\gamma\in\Gamma_0(N)$ then $\gamma{}x=\eps(\gamma)x$, where,
as usual,~$\eps$ is viewed as a homomorphism
$\Gamma_0(N)\ra \C^*$ via $\eps(\abcd{a}{b}{c}{d}) = \eps(d)$.
In particular, $\eps(\gamma)\gamma^{-1}x=x$, so 
\begin{eqnarray*}
P\{\oo, \gamma(\oo)\}
  &=& P\{\oo,\alp\} + P\{\alp,\gamma(\alp)\} +
P\{\gamma(\alp),\gamma(\oo)\}\\
  &=& P\{\oo,\alp\} + P\{\alp,\gamma(\alp)\} +
\eps(\gamma)\gamma^{-1}(P\{\gamma(\alp),\gamma(\oo)\})\\
  &=& P\{\oo,\alp\} + P\{\alp,\gamma(\alp)\} +
\eps(\gamma)(\gamma^{-1}P)\{\alp, \oo\}\\
  &=& P\{\alp,\gamma(\alp)\} + P\{\oo,\alp\}  -
\eps(\gamma)(\gamma^{-1}P)\{\oo, \alp\}\\
  &=& P\{\alp,\gamma(\alp)\} + (P - \eps(\gamma)\gamma^{-1}P)\{\oo,\alp\}.
\end{eqnarray*}
The remaining statements of the proposition now follow easily.
\end{proof}

\begin{example}\label{ex:constructsyms}
In some cases it is easy to give a formula for symbols
that are obviously transportable.
Suppose~$k\geq 2$ is an even integer.
If~$P$ is a polynomial such that $\gamma (P)=P$ for some
$\gamma\in\Gamma_0(N)$, then 
$P\{\oo,\gamma^{-1}(\infty)\}$ is transportable.
Given $\gamma\in\Gamma_0(N)$, an example of such a~$P$ is 
$$P(X,Y) = (cX^2 + (d-a)XY - bY^2)^{\frac{k-2}{2}}.$$
We found this polynomial by viewing
$V_{k-2}$ as the $(k-2)$th symmetric product of the $2$-dimensional
space on which $\gzero$ acts naturally.  
If~$\gamma$, which has determinant~$1$, 
has eigenvalues~$\alpha$ and $\alpha^{-1}$, 
then the eigenvalues of the 
${k-2}$ fold symmetric product of $\gamma$ are given
by $\alpha^{k-2-2j}$ for $0\le j\le k-2$.
Although we have not been able to find a counterexample,
the authors see no reason to believe that transportable
symbols of the form given in this example
always span $\sS_k(N;\Q)$.
\end{example}

However, by definition, any transportable symbol can be written
in the form $\sum_{i=1}^n P_i \{\oo,g_i\oo\}$ for some finite sequence
$\{g_1,\dots,g_n\}\subset \Gamma_0(N)$.
To find such transportable symbols we just have
to find the kernel of the map
$\bigoplus_{i=1}^n(1-g_i)$ from $\bigoplus_{i=1}^nV_{k-2}$
to $V_{k-2}$.  This is a simple matter of linear algebra.
%If~$k$ is even and $\alpha\not=1$, there is
%a unique eigenvector with eigenvalue~$1$, which is given by $P$ as above.
%If $\eps(\gamma)\neq 1$, then  the existence of a nonzero
%polynomial~$P$ with $\gamma(P)=\eps(\gamma) P$ implies that 
%$\gamma$ has order dividing~$N$,  which is generally not the case.
%When~$\eps$ is nontrivial one can choose~$\gamma$ so that $\eps(\gamma)=1$.


\subsection{Characterization of transportable modular symbols}
\label{sec:character}
\begin{lemma}
\label{lem:trans-condition}
A modular symbol in $\sM_k(N,\eps;\Q)$
is transportable if and only if it can be
written in the form $\sum_{i=1}^m P_i\{\infty,\gamma_i(\infty)\}$
with $$\sum P_i = \sum \eps(\gamma_i)\gamma_i^{-1} P_i.$$
\end{lemma}
\begin{proof}
This follows from Proposition~\ref{prop:modsym-errorterm}.
\end{proof}
Figure~\ref{transfig} illustrates
Lemma~\ref{lem:trans-condition}
with a trivial-character example.
\begin{figure}
\begin{center}
\begin{picture}(305,150)(0,0)
%\put(-20,10){\parbox[t][5mm][l]{2in}{
%$$I_1=\int_\infty^{\gamma \infty}P(z,1)f(z)dz
%+\int_{\gamma \infty}^{\beta \infty}Q(z,1)f(z)dz$$}
%}
%\put(-20,200){\parbox[t][5mm][l]{2in}{
%$$I_2=\int_\alpha^{\gamma \alpha}P(z,1)f(z)dz
%+\int_{\gamma \alpha}^{\beta \alpha}Q(z,1)f(z)dz$$}}
\put(-10,10){\line(1,0){200}}
\put(10,10){\circle*{3}}
\put(10,0){$\infty$}
\qbezier(10,10)(50,40)(75,10)
%thickening
%\qbezier(10,11)(50,41)(75,11)
%endthickening
\put(49,25){\vector(1,0){1}}
\put(45,28){$P$}
\put(75,10){\circle*{3}}
\put(75,0){$\gamma \infty$}
\qbezier(75,10)(130,50)(140,10)
\put(120,30){\vector(1,0){1}}
\put(115,34){$Q$}
\put(140,10){\circle*{3}}
\put(140,0){$\beta \infty$}
\put(20,90){\circle*{3}}
\put(17,96){$\alpha$}
\qbezier(20,90)(50,120)(55,60)
\put(40,100){\vector(1,0){1}}
\put(40,104){$P$}
\put(55,60){\circle*{3}}
\put(60,54){$\gamma \alpha$}
\qbezier(55,60)(70,180)(100,140)
\put(87,150){\vector(1,0){1}}
\put(85,154){$Q$}
\put(100,140){\circle*{3}}
\put(100,146){$\beta \alpha$}
\comment{
%first vertical wiggle
\qbezier(10,10)(-10,30)(10,50)
\qbezier(10,50)(40,70)(20,90)
\put(5,45){\vector(2,3){1}}
%2nd vertical wiggle
\qbezier(75,10)(90,30)(75,40)
\qbezier(75,40)(60,50)(55,60)
\put(69,45){\vector(1,-1){1}}
\put(80,35){\vector(-1,2){1}}
%3rd vertical wiggle
\qbezier(140,10)(160,30)(140,50)
\qbezier(140,50)(90,80)(110,90)
\qbezier(110,90)(130,100)(100,120)
\qbezier(100,120)(90,130)(100,140)
\put(110,90){\vector(-1,-1){1}}
}
%vertical arrows
\put(100,50){\vector(-1,4){10}}
\put(50,40){\vector(-1,4){10}}
\put(140,155){\parbox[t][4in][l]{2in}{
The modular symbol 
$$P\{\infty ,\gamma \infty \}+Q\{\gamma \infty ,\beta \infty \}$$
$$\hspace{2em}=P\{\infty ,\gamma \infty \}+Q\{\infty ,\beta \infty \}-Q\{\infty ,\gamma \infty \}$$
can be ``transported'' to 
$$P\{\alpha,\gamma \alpha\}+Q\{\gamma \alpha,\beta \alpha\},$$
provided that\\
$$P\,+\,Q\,-\,Q=\gamma ^{-1}P\,+\,\beta ^{-1}Q\,-\,\gamma ^{-1}Q.$$\\
%In this case, $I_1=I_2$.
}}
\end{picture}
\end{center}
\caption{``Transporting'' a transportable modular symbol.}
\label{transfig}
\end{figure}

\begin{theorem}
\label{thm:onlyoo}
A modular symbol is transportable if and only if it
is cuspidal.
\comment{previous statement, without transportable symbols:
Any element of $\sS_k(N,\eps)$ can be written in the form
$$\sum_{i=1}^n P_{i}\{\infty,\gamma_i(\infty)\}$$
with $P_i\in V_{k-2}$ and $\gamma_i\in\gzero.$}
\end{theorem}
\begin{proof}
By Lemma~\ref{lem:trans-condition} 
every transportable modular symbols is cuspidal,
so we must prove that every cuspidal symbol is transportable.
	
Let $I=I_{N,\eps}$ be the ideal in the 
group ring of~$\gzero$ generated by all
elements of the form $\eps(\gamma) -\gamma$
for $\gamma\in\gzero$. 
Suppose $v\in\sS_k(N,\eps)$.  Use the relation 
$\{\alp,\beta\}=\{\oo,\beta\}-\{\oo,\alp\}\in\sM$
to see that any $v$ is the image 
of an element $\tilde{v}\in \sM_k$ of the form
  $$\tilde{v} = \sum_{\beta\in\Q}P_\beta\tensor \{\oo,\beta\}\in \sM_k$$
with only finitely many $P_\beta$ nonzero.  For later convenience
we set $P_\infty=0$, and take sums over all $\beta\in P^1(\Q)$.
The boundary map $\delta$ lifts in a natural way
to $\sM_k = V_{k-2}\tensor\sM$, as illustrated.
$$\xymatrix{
&I(V_{k-2}\otimes\sM)\ar[r]\ar[d]&
  I(V_{k-2}\otimes\sB)\ar[d] \\
&V_{k-2}\otimes\sM\ar[r]^{\tilde{\delta}}\ar[d] & V_{k-2}\otimes\sB\ar[d] \\
*++{\sS_k(N,\eps)}\ar@{^{(}->}[r] 
&\sM_k(N,\eps)\ar[r]^{\delta}
&\sB_k(N,\eps)\\
}\qquad\qquad\mbox{}$$
Bearing in mind torsion, our assumption that $\delta(v)=0$ implies that
for some nonzero $M\in{\Z}$ we have
$M\tilde{\delta}(\tilde{v})\in I(V_{k-2}\otimes\sB)$.
So there are $Q_{\gamma,\beta}\in V_{k-2}$,
for $\gamma\in\gzero$ and $\beta\in\PP^1(\Q)$, only
finitely~$\beta$ many nonzero, such that
$$M\tilde{\delta}(\tilde{v})
  = \sum_{\gamma,\beta}(\eps(\gamma)-\gamma)
                      (Q_{\gamma,\beta}\tensor\{\beta\}).$$
We now use a summation trick.
\begin{eqnarray}
M\tilde{\delta}(\tilde{v})
&=& 
M\sum_{\beta}\Bigl(P_\beta\tensor \{\beta\}-P_\beta\tensor \{\oo\}\Bigr)\nonumber\\ 
&=& \sum_{\gamma, \beta} 
  \Bigl(\eps(\gamma) Q_{\gamma,\beta}\tensor \{\beta\}
-
(\gamma Q_{\gamma,\beta})\tensor \{\gamma\beta\}\Bigr)\nonumber\\
&=&
\sum_{\gamma, \beta} \eps(\gamma) Q_{\gamma,\beta}\tensor \{\beta\}
-
(\gamma{}Q_{\gamma,\gamma^{-1}\beta})\tensor \{\beta\}\nonumber \\
&=&
\sum_{ \gamma, \beta}\Bigl( \eps(\gamma)Q_{\gamma,\beta}
    -\gamma{}Q_{\gamma,\gamma^{-1}\beta}\Bigr)\tensor \{\beta\}.\nonumber
\end{eqnarray}
This shows that 
\begin{equation}
\label{equateinfbet}
M\sum_{\beta}\Bigl(P_\beta\tensor \{\beta\}-P_\beta\tensor \{\oo\}\Bigr)
 = \sum_{ \gamma, \beta}\Bigl( \eps(\gamma)Q_{\gamma,\beta}
    -\gamma{}Q_{\gamma,\gamma^{-1}\beta}\Bigr)\tensor \{\beta\}.
\end{equation}
Equating terms we deduce that for $\beta\not=\infty$,
\begin{equation}
\label{equatebeta}
MP_\beta=\sum_{\gamma}
\Bigl(\eps(\gamma)Q_{\gamma,\beta}-\gamma{}Q_{\gamma,\gamma^{-1}\beta}\Bigr).
\end{equation}
Using this expression for $P_\beta$ and
that $\eps(\gamma)\gamma^{-1}$ acts trivially
on $\sM_k(N,\eps)$, we find that
\begin{eqnarray}
Mv = M\sum_{\beta}
P_\beta
\{\oo,\beta\}
&=&
\sum_{\gamma,\beta}
\Bigl(\eps(\gamma)Q_{\gamma,\beta}  
     -\gamma{}Q_{\gamma{},\gamma^{-1}\beta}\Bigr)
\{\oo,\beta\}\nonumber \\
&=&
\sum_{\gamma,\beta}
\eps(\gamma)Q_{\gamma,\beta}  
     -\eps(\gamma)\gamma^{-1}
      \Bigl((\gamma{}Q_{\gamma{},\gamma^{-1}\beta})
\{\oo,\beta\}\Bigr)\nonumber \\
&=&
\sum_{\gamma,\beta}
\eps(\gamma)Q_{\gamma,\beta}\{\oo,\beta\}
-\eps(\gamma)Q_{\gamma,\gamma^{-1}\beta}\{\gamma^{-1}\oo,\gamma^{-1}\beta\}\nonumber \\
&=&
\sum_{\gamma,\beta}
\eps(\gamma)Q_{\gamma,\beta}\{\oo,\beta\}
-\eps(\gamma)Q_{\gamma,\beta}\{\gamma^{-1}\oo,\beta\}\nonumber \\
&=&
\sum_{\gamma,\beta}
\eps(\gamma)Q_{\gamma,\beta}
\{\oo,\gamma^{-1}\oo\}.\label{vexpression}
\end{eqnarray}
Equating coefficients of $\{\oo\}$ in Equation~\ref{equateinfbet}, we have 
$$-M\sum_{\beta}P_\beta = \sum_{\gamma}
\Bigr(\eps(\gamma)Q_{\gamma,\oo}
-\gamma Q_{\gamma,\gamma^{-1}\oo}\Bigr),$$
which, combining with Equation~\ref{equatebeta}, and
recalling that $P_\infty = 0$, means that
$$-\sum_{\gamma,\beta\not=\oo}
\Bigl(\eps(\gamma)Q_{\gamma ,\beta}-\gamma Q_{\gamma ,\gamma ^{-1}\beta}\Bigr)
= \sum_{\gamma}
\Bigr(\eps(\gamma)Q_{\gamma ,\oo}
-\gamma Q_{\gamma ,\gamma ^{-1}\oo}\Bigr),$$
and hence
$$\sum_{\gamma,\beta}
\Bigl(\eps(\gamma) Q_{\gamma ,\beta}-\gamma Q_{\gamma ,\beta}\Bigr)=0.$$
Using the expression 
$v=-\frac{1}{M}
\sum_{\beta,\gamma}
\eps(\gamma) Q_{\gamma ,\beta}
\{\oo,\gamma ^{-1}\oo\}$
obtained from 
Equation~\ref{vexpression}, we see this is the condition for 
$v$ 
to be transportable.
\end{proof}
 

\begin{corollary}\label{cor:span}
Fix $\alpha\in\h^*$.  
Every element of $\sS_k(N,\eps)$ is a sum of modular symbols 
of the form $P\{\alpha,\gamma(\alpha)\}$.
\end{corollary}
\begin{proof}
Let $x\in \sS_k(N,\eps)$.  Proposition~\ref{prop:modsym-errorterm}
implies that~$x$ is transportable, so there exists $P_i$ and
$\gamma_i$ such that 
$$x = \sum P_i\{\infty,\gamma_i(\infty)\} = 
 \sum P_i\{\beta,\gamma_i(\beta)\}$$ 
for any $\beta \in \h^*$.  Taking $\beta=\alpha$ proves the corollary.
\end{proof}
\begin{remark}\label{rem:span}\mbox{}\vspace{-3ex}\\
\begin{enumerate}
\item When $k=2$, the corollary follows from 
\cite[\S1]{manin:parabolic}, which asserts that map
$\Gamma_0(N)\ra \sS_2(\Gamma_0(N))=H_1(X_0(N),\Z)$ sending~$\gamma$ to
$\{\alpha,\gamma(\alpha)\}$ is a surjective group homomorphism.
%When $k>2$, we have not
%found a similar group-theoretic statement involving the semidirect product 
%of $\Gamma_0(N)$ and $V_{k-2}$.  In fact, the natural map from 
%the semidirect product is not even a homomorphism. 
\item In Proposition~\ref{prop:span_alpha} below, we will prove more generally that
every element of $\sM_k(N,\eps)$ is a sum of modular symbols
of the form $P\{\alpha,\gamma(\alpha)\}$, as long as we allow~$\alpha$ 
to vary over $\PP^1(\Q)$.
\end{enumerate}
\end{remark}




\subsection{What space do the symbols $P\{\infty,\gamma(\infty)\}$ span?} 
\label{sec:whenspan}
Suppose $N$ and $k$ are positive integers, with~$k$ even.
\begin{definition}
For any $\alpha\in\PP^1(\Q)$, let~$\sW_\alpha$ 
denote the subspace of
$\sM_k(\Gamma_0(N);\Q)$ spanned by symbols of the form
$P\{\alpha,\gamma(\alpha)\}$, for $P\in V_{k-2}$ and
$\gamma\in\Gamma_0(N)$.
\end{definition}
Corollary~\ref{cor:span} draws our attention to $\sW_\infty$.
Since $\sW_\infty$ contains $\sS_k(\Gamma_0(N))$, it is natural 
to ask how much bigger it is.
As mentioned in Remark~\ref{rem:span}, 
when $k=2$, Manin proved that for any $\alpha\in\PP^1(\Q)$, 
we have $\sW_\alpha=\sW_\infty=\sS_2(\Gamma_0(N);\Q)$.
We now compute $\sW_\alpha$ for any weight $k>2$:

\begin{proposition}\label{prop:span_alpha}
Suppose $k>2$.  Then the space $\sW_{\alpha}$ is equal to
the inverse image under the boundary map~$\delta$ of
the one-dimensional subspace 
$V_{k-2}\{\alpha\}\subset \sB_k(\Gamma_0(N);\Q).$
Hence $\displaystyle \dim \sW_\alpha = \dim \sS_k(\Gamma_0(N);\Q)+1$
and $\sM_k(\Gamma_0(N);\Q) = \sum_{\alpha\in \PP^1(\Q)} \sW_\alpha.$
\end{proposition}
\begin{proof}
In \cite[\S1.4]{merel:1585}, Merel shows that $V_{k-2}\{\alpha\}$ has
dimension~$1$ (see the proof of \cite[Prop.~4, \S1.4]{merel:1585}),
and that $P(X,Y)\{\frac{u}{v}\}$ is nonzero if $P(u,v)\neq 0$.

Corollary~\ref{cor:span} implies that~$\sW_\alpha$ contains the kernel
$\sS_k(\Gamma_0(N))$ of the boundary map~$\delta$.  
It thus suffices
to show that $\delta(\sW_\alpha)=V_{k-2}\{\alpha\}$.
For $P\in V_{k-2}$ and $\gamma\in \Gamma_0(N)$, we have
$$\delta(P\{\alpha,\gamma(\alpha)\})
=P\{\gamma(\alpha)\} - P\{\alpha\} 
= (\gamma^{-1}P - P)\{\alpha\}\in V_{k-2}\{\alpha\},$$
so $\delta(\sW_{\alpha}) \subset V_{k-2}\{\alpha\}$.
For $\gamma=\abcd{1}{0}{N}{1}\in\Gamma_0(N)$, we have
\begin{align*}
\delta(X^{k-3}Y\{\alpha,\gamma(\alpha)\}) 
   &= (\gamma^{-1}(X^{k-3}Y) - X^{k-3}Y)\{\alpha\}\\
   &= (X^{k-3}(NX+Y)-X^{k-3}Y)\{\alpha\}\\
   &= NX^{k-2}\{\alpha\}.
\end{align*}
If $\alpha\neq 0$, then, as mentioned above,
$X^{k-2}\{\alpha\}\neq 0$.
(If $\alpha=0$, use $XY^{k-3}$ and $\gamma=\abcd{1}{N}{0}{1}$ instead.)
Because there is a nonzero element in $\delta(\sW_\alpha)$ 
and $V_{k-2}\{\alpha\}$ has dimension~$1$,
it follows that $\delta(\sW_\alpha)=V_{k-2}\{\alpha\}$.
The final claim of the proposition is true because
$\sB_k(\Gamma_0(N);\Q) = \sum_{\alpha\in\PP^1(\Q)} V_{k-2}\{\alpha\}$.

\end{proof}

\begin{corollary}\label{cor:whenspan}
Fix $\alpha\in\PP^1(\Q)$.
Then $\sW_\alpha=\sM_k(\Gamma_0(N);\Q)$ if and only if $N=1$.
\end{corollary}
\begin{proof}
When $N=1$, $\gamma$ can be any element of $\SL_2(\Z)$,
so the assertion is clear.
Next suppose that $\sW_\alpha=\sM_k(\Gamma_0(N);\Q)$.  
If $k=2$ then by \cite[\S1]{manin:parabolic} 
$\sW_\alpha=\sS_k(\Gamma_0(N);\Q)$, so $N=1$ since
there is always a weight~$2$ Eisenstein series when $N>1$.
Next suppose that $k>2$.
By \cite[Prop.~5, \S1.4]{merel:1585},~$\delta$ is surjective and by 
\cite[Prop.~5, \S1.4]{merel:1585} the dimension of the image of~$\delta$
equals $\#\Gamma_0(N)\backslash \PP^1(\Q)$.
Combining Proposition~\ref{prop:span_alpha} with our assumption that
$\sW_\alpha=\sM_k(\Gamma_0(N);\Q)$ implies that
$\#\Gamma_0(N)\backslash \PP^1(\Q) = 1$, so
$N=1$, as claimed.
\end{proof}


\section{Application to computing periods of newforms}
\label{sec:computingperiods}%
The authors were led to introduce transportable modular symbols
in order to study the error term  
$(P - \eps(\gamma)\gamma^{-1}P)\{\oo,\alp\}$ 
of Equation~\ref{errorterm} of Proposition~\ref{prop:modsym-errorterm}
in the context of computing periods of newforms. 
There are many ways to compute periods of newforms, but
we hope that the method given below will be of value in 
some contexts.  

Section~\ref{sec:algperiod} contains an algorithm for computing
periods that relies on Theorem~\ref{thm:onlyoo}.  We present
a potentially more efficient method in Section~\ref{sec:wntrick}.

\subsection{An algorithm for computing periods}\label{sec:algperiod}
Let $f=\sum a_n q^n\in S_k(N,\eps)$ be a cuspform 
and let $x \in \sM_k(N,\eps)$ be a modular symbol.  Then 
$\langle f, x\rangle$ (see~\cite[\S1.5]{merel:1585})
is a linear combination of integrals of the form
\begin{equation}
\label{intsum}
\langle f, X^mY^{k-2-m}\{\alpha,\infty\}\rangle = 
2\pi i \int_{\alpha}^{i\infty} f(z)z^m dz,
\end{equation}
where $\alpha\in \h^*$ and the integer~$m$ 
satisfies $0\le m\le k-2$.
If $\alpha\in\h$ then the imaginary part of~$\alpha$ is positive, so 
$$2\pi i \int_{\alpha}^{i\infty} f(z)z^m dz = \sum_{n\geq 1} a_n c_n,$$
where
$$c_n=2\pi{}i \int_{\alpha}^{i\infty} z^m e^{2\pi{}inz}dz.$$
The reversal of summation and integration is justified 
because the sum converges absolutely. 
We compute the $c_n$ using the following formula, which we
obtain using repeated integration by parts.
\begin{lemma}
\label{lem:intexp}
$$ 
 \int_{\alpha}^{i\infty} e^{2\pi i n z} z^m dz
   = e^{2\pi i n \alpha} 
      \sum_{s=0}^m \left\{
          \frac{(-1)^s \alpha^{m-s}}
              {(2\pi i n)^{s+1}}
          \cdot \prod_{j=(m+1)-s}^m j\right\}.$$
\end{lemma}
If~$\alpha$ has large imaginary part, the $c_n$ will rapidly converge
to~$0$ as $n\ra\infty$.  
However, the reversal of summation and integration above need
not be valid when~$\alpha$ is a real number, so for computational
purposes we are led to express periods in terms of integrals with end
points that are in~$\h$.  When $k=2$ this is easy because
of the identity $\{\infty,\gamma(\infty)\}=\{\alpha,\gamma(\alpha)\}$,
which is valid for any $\alpha\in\h^*$. 
However, this identity can fail
when $k>2$; the failure is made precise 
in Proposition~\ref{prop:modsym-errorterm}.

Since we can take the real part of~$\alpha$ to be
greater than~$0$, each of the terms on the right hand side of 
Equation~\ref{errorterm} can be computed using the sum 
given by Lemma~\ref{lem:intexp}. 

We showed in Section~\ref{sec:transportable}
that every cuspidal modular symbol can be expressed as a sum of
symbols of the form $P\{\infty,\gamma(\infty)\}$.  Periods of modular
symbols of this form can then be computed using the following algorithm.

\begin{algorithm}\label{alg:compute_period}
{\em Given a triple $\gamma\in\Gamma_0(N)$,  $P\in V_{k-2}$ and 
$g\in S_k(N,\eps)$ this algorithm computes the period integral 
$\langle g, \,P\{\oo, \gamma(\oo)\}\rangle.$}
Express $\gamma$ as $\abcd{\hfill a}{b}{cN}{d}\in\gzero$ and set
$\alp = \frac{-d+i}{cN}$ in Proposition~\ref{prop:modsym-errorterm}.
Replacing $\gamma$ by $-\gamma$ if necessary,
we find that the imaginary parts of $\alp$ and 
$\gamma(\alp)=\frac{a+i}{cN}$
are both equal to the positive number $1/cN$. 
Equation~\ref{intsum} and Lemma~\ref{lem:intexp} can now be
used to compute the period integrals provided by
Proposition~\ref{prop:modsym-errorterm}.
\end{algorithm}

\subsection{The $W_N$-trick}\label{sec:wntrick}%
In this section, in order to obtain a potentially more
efficient way of computing periods than Algorithm~\ref{alg:compute_period}, 
we generalize the method of 
Cremona~\cite{cremona:periods} to even integer weight $k\geq 2$. 
In Algorithm~\ref{alg:compute_period}, 
with $\gamma=\abcd{\hfill{}a}{b}{cN}{d}$, 
the endpoints of the corresponding integrals  
have imaginary part $\frac{1}{cN}$.
However, using the following trick one can increase the imaginary part
of all endpoints involved to
at least $\frac{1}{d\sqrt{N}}$,
which is sometimes a significant improvement.

Recall that the Atkin-Lehner involution $W=W_N$ is induced by the matrix
$\abcd{0}{-1}{N}{\hfill0}$; it acts on modular forms by 
sending a cuspform $f\in S_k(N,\eps)$ to the form
$$f|_W(z) = N^{-k/2}z^{-k} f(-1/(Nz)) \in S_k(N,\eps^{-1}).$$
If~$f$ is an eigenvector for~$W$, then necessarily $\eps=\eps^{-1}$.
For the rest of this section we assume that $\eps^2=1$. 
Then~$W$ acts on $\sM_k(N,\eps)$ by
 $$W\Bigl(P(X,Y)\{\alp,\beta\}\Bigr)
    =\frac{P(Y,-NX)}{N^{k/2-1}}
        \left\{-\frac{1}{N\alp},-\frac{1}{N\beta}\right\},$$
and this action is compatible with the integration pairing.

\begin{proposition}\label{wntrick}
Let $g \in S_k(N,\eps)$ be a cuspform that is
an eigenform for the Atkin-Lehner involution~$W$ 
having eigenvalue~$w$.
Then for any transportable modular symbol 
$\sum_{j=1}^m P_j\{\infty,\gamma_j(\infty)\}$ with
$\gamma_j\in\Gamma_0(N)$ and $P_j\in V_{k-2}$, we have 
for any $\alp\in\h$ the following formula:
\begin{eqnarray*}
\langle g, \sum_{j=1}^m P_j\{\oo,\gamma_j(\oo)\}\rangle &=&
\left\langle g,\,\, \sum_{j=1}^m w \frac{P_j(Y,-NX)}{N^{k/2-1}}
\left\{W(\alp),\oo\right\}\right. \\
&& +
\sum_{j=1}^m\left( P_j - w 
       \frac{P_j(Y,-NX)}{N^{k/2-1}}\right)
      \left\{\frac{i}{\sqrt{N}},\oo\right\} \\
&&\left. - \sum_{j=1}^m P_j\left\{\gamma_j(\alp),\oo\right\} \right\rangle.
\end{eqnarray*}
Here $W(\alp) = -1/(N\alp)$. 

If $\gamma_j=\abcd{a_j}{b_j}{c}{d}$, 
where~$c$ and~$d$ are fixed integers that do not
depend on~$j$, then
\begin{eqnarray*}
\langle g, \sum_{j=1}^m P_j\{\oo,\gamma_j(\oo)\}\rangle &=&
\left\langle g, \sum_{j=1}^m w \frac{P_j(Y,-NX)}{N^{k/2-1}}\left\{\frac{c}{d}+\frac{i}{d\sqrt{N}},\oo\right\}\right. \\
&& +
\sum_{j=1}^m\left( P_j - w \frac{P_j(Y,-NX)}{N^{k/2-1}}\right)\left\{\frac{i}{\sqrt{N}},\oo\right\} \\
&&\left. - \sum_{j=1}^m P_j\left\{\frac{b_j}{d} + \frac{i}{d\sqrt{N}},\oo\right\} \right\rangle.
\end{eqnarray*}
\end{proposition}
\begin{proof}
By Proposition~\ref{prop:modsym-errorterm} our condition of transportability
implies that we have
$\sum_{j=1}^m P_j\{\oo,\gamma_j(\oo)\}
=\sum_{j=1}^m P_j\{\alp,\gamma_j(\alp)\}$.  
The steps of the following computation are described below.\vspace{1ex}\\
$\langle g, P_j\{\alp,\gamma_j(\alp)\}\rangle$\vspace{-1ex}
\begin{eqnarray*}
  &=&\langle g, P_j\{\alp,\frac{i}{\sqrt{N}}\} + P_j\{\frac{i}{\sqrt{N}},W(\alp)\}+P_j\{W(\alp),\gamma_j(\alp)\}
           \rangle \\
  &=&\langle g, w \frac{W(P_j)}{N^{k/2-1}}
        \{W(\alp),\frac{i}{\sqrt{N}}\} + P_j\{\frac{i}{\sqrt{N}},W(\alp)\}+P_j\{W(\alp),\gamma_j(\alp)\}
           \rangle \\
  &=&\langle g, \left(w \frac{W(P_j)}{N^{k/2-1}}-P_j\right)
        \{W(\alp),\frac{i}{\sqrt{N}}\} +P_j\{W(\alp),\oo\} - P_j\{\gamma_j(\alp),\oo\}\rangle\\
  &=&\langle g, w \frac{W(P_j)}{N^{k/2-1}}\{W(\alp),\oo\}
        +\left(P_j - w \frac{W(P_j)}{N^{k/2-1}}\right)\{\frac{i}{\sqrt{N}},\oo\} 
        -P_j\{\gamma_j(\alp),\oo\} \rangle.\\
\end{eqnarray*}
In the first step, we break the path from~$\alpha$ to $\gamma_j(\alpha)$
into three paths.
In the second step, we apply the $W$-involution to the first
term, and use that the action of~$W$ is compatible with
the pairing $\langle \,,\, \rangle$. The third step involves
combining the first two terms and breaking up the third.
In the final step, we replace $\{ W(\alp), i/\sqrt{N}\}$
by $\{W(\alp),\infty\}+\{\infty,i/\sqrt{N}\}$ and regroup.
Taking the sum of both sides of the expression over $j$ from $1$ to $m$
gives the first result of the proposition.

Now, following Cremona \cite[\S2.10.8]{cremona:algs}, in order
to simultaneously maximize the imaginary parts of~$\gamma_j(\alp)$ and 
$W(\alp)$, we take~$\alp =
\gamma_1^{-1}\left(\frac{b_1}{d_{\phantom{1}}}+\frac{i}{d\sqrt{N}}\right)$.
In this case we have $W(\alp) = \frac{c}{d}+\frac{i}{d\sqrt{N}}$,
and 
$\gamma_j(\alp) = 
\frac{b_j}{d_{\phantom{}}} + \frac{i}{d\sqrt{N}}$.
The second formula then follows.
\end{proof}

\begin{remark}\label{rem:wntrick}
Let $\gamma=\abcd{a}{b}{c}{d}\in \Gamma_0(N)$.  
Since the imaginary parts of the terms
$i/\sqrt{N}$, $\gamma_j(\alp)$ and $W(\alp)$ in the 
second part of the proposition
are all relatively large, the sums appearing in 
Equation~\ref{intsum} converge relatively quickly if~$d$ is small.
However, we emphasize that {\em it is extremely important 
to choose~$\gamma_j$ 
in Proposition~\ref{wntrick} with~$d$ small; otherwise,
the series will converge very slowly.}
\end{remark}

\comment{
\subsection{The $*$-involution}\label{sec:starinvolution}
The matrix $j=\abcd{-1}{0}{\hfill0}{1}$ defines 
an involution~$*$ of $\sM_k(N,\eps)$ given by 
$x\mapsto x^*=j(x)$.  Explicitly,
$(P(X,Y)\{\alp,\beta\})^* = P(X,-Y)\{-\alp,-\beta\}.$
The space of modular symbols is constructed as a quotient, so
it is not obvious that the $*$-involution is well defined.
\begin{proposition}
The $*$-involution is well defined.
\end{proposition}
\begin{proof}
We give a proof, since this is not proved in \cite{merel:1585}.
Recall that $\sM_k(N,\eps)$ is the largest torsion-free quotient of the
quotient of the
free $\Z[\eps]$-module generated by symbols 
$x=P\{\alp,\beta\}$ by the submodule generated by
relations $\gamma x - \eps(\gamma)x$ for
all $\gamma\in \Gamma_0(N)$ and all $x\in\sM_k(N,\eps)$.
In order to check that the operator~$*$ is well defined, it
suffices to check, for any $x\in\sM_k$, that 
$*(\gamma x - \eps(\gamma)x)$ is of
the form $\gamma' y - \eps(\gamma') y$, for some~$y$ in $\sM_k$
and $\gamma'\in\Gamma_0(N)$. 
Note that if $\gamma=\abcd{a}{b}{c}{d}\in \Gamma_0(N)$, then 
$j\gamma j^{-1} = \abcd{\hfill a}{-b}{-c}{\hfill d}$ is also in $\Gamma_0(N)$
and $\eps(j\gamma j^{-1}) = \eps(\gamma)$.  We have
\begin{align*}
    *(\gamma x - \eps(\gamma)x) &= j(\gamma x - \eps(\gamma) x) 
        = j \gamma x - j \eps(\gamma) x \\
        &= j \gamma j^{-1} j x - \eps(\gamma) j x
        = (j\gamma j^{-1}) (j x) - \eps(j \gamma j^{-1}) (jx).
\end{align*}
\end{proof}

Let~$f$ be a modular form and denote by $f^*$ the holomorphic
function $\overline{f(-\overline{z})}$, where the bar
denotes complex conjugation.
The Fourier coefficients
of $f^*$ are the complex conjugates of those of~$f$; though $f^*$
is again a holomorphic modular form, its character
is $\overline{\eps}$ instead of~$\eps$. 
The pairing $\langle \,\, , \,\rangle$
is the restriction of a pairing on the full spaces without
character, and we have the following proposition.
\begin{proposition}\label{prop:starpairing}
With the above notation, we have
\begin{equation*}
\langle f^*,  x^* \rangle = \overline{\langle f, x\rangle}.
\end{equation*}
\end{proposition}
}


\section{Examples}\label{sec:examples}
The example of Section~\ref{sec:ram} illustrates some of the results
of this paper for the weight-$12$ modular form $\Delta$, and
Section~\ref{sec:11} concerns a nonrational form of level $11$ and
weight~$4$.  The computations below were done using the first author's
implementation of the algorithms of \cite{stein:phd} in \magma{} 
\cite{magma}.
\subsection{The weight-$12$ form $\Delta$}\label{sec:ram}
Let $f=\Delta=q\cdot\prod(1-q^n)^{24}$ be the unique normalized
eignform in $S_{12}(1)$.  The space $\sM_{12}(1;\Q)$ of modular symbols
has dimension~$3$ and is spanned by $a_1=X^{10}\{0,\infty\}$, $a_2=X^8
Y^2\{0,\infty\}$, and $a_3=X^9 Y\{0,\infty\}$, and the cuspidal
subspace $\sS_{12}(1;\Q)$ has dimension~$2$ and is spanned by $a_2$ and
$a_3$.

As explained in Example~\ref{ex:constructsyms},
there is a transportable modular symbol associated
to each nonidentity element $\gamma\in \SL_2(\Z)$.
The transportable symbol 
$(2X^2+2XY - Y^2)^5\{\infty,\frac{1}{2}\}  
= -300X^9Y\{0,\infty\}$
is attached to $\abcd{1}{1}{2}{3}$, and
$-4665600X^8Y^2\{0, \infty\} -87300X^9Y\{0, \infty\}$
is attached to $\abcd{-8}{\hfill{}5}{19}{-12}$. Together these
two transportable symbols span $\sS_{12}(1;\Q)$.

The period map $\Phi_f$ sends $X^i Y^{10-i}\{0,\infty\}$ to
$2\pi i \int_0^{\infty} z^i f(z) \dz$.  These integrals are, up to
scalars, special values of $L(f,s)$ at critical integers, so they
could be computed using any of the standard methods.  
In any case, we obtain an approximation for the period map:
$\Phi_f(a_1) \sim 0.0374412812$,
$\Phi_f(a_2) \sim -0.0159703242$,
$\Phi_f(a_3) \sim -0.0232962319i$.
The period lattice $\Lambda$ of~$f$ is spanned by
$\Phi_f(\frac{1}{14}a_2)$ and $\Phi_f(\frac{1}{48}a_3)$.   
(The fractions appear because $\sS_{12}(1;\Z)$ has
basis $\frac{1}{14}a_2$ and $\frac{1}{48}a_3$.)
Since $\C/\Lambda$ is a one-dimensional torus, it makes sense to 
consider the corresponding elliptic curve over $\C$.  This is
the elliptic curve $y^2 = x^3 + c_4 x + c_6$, where
$c_4 \sim 28091951348793344.58$  and $c_6\sim 0$.
The $j$-invariant of this curve is 
approximately $2592849.394270$. 
Is~$j$ a transcendental number?

\subsection{Level $11$, weight $4$}\label{sec:11}
The unique normalized eigenform in $S_4(\Gamma_0(11))$ is
$$f = q + \alpha q^2 + (-4\alpha + 3)q^3 
      + (2\alpha - 6)q^4 + (8\alpha - 7)q^5 + \cdots,$$
where $\alpha^2-2\alpha-2=0$.
The space $\sM_4(\Gamma_0(11);\Q)$ has basis
$a_1=X^2\{0, \infty\}$,
$a_2=(64X^2 + 16XY + Y^2)\{-\frac{1}{8}, 0\}$,
$a_3=(49X^2 + 14XY + Y^2)\{-\frac{1}{7}, 0\}$,
$a_4=(25X^2 + 10XY + Y^2)\{-\frac{1}{5}, 0\}$,
$a_5=(100X^2 + 20XY + Y^2)\{-\frac{1}{10}, 0\}$,
$a_6=Y^2\{\infty, 0\}$.
The subspace $\sS_4(\Gamma_0(11);\Q)$ has basis
$b_1=a_2-a_6$, $b_2=a_3-a_6$, $b_3=a_4-a_6$, $b_4=a_5-a_6$. 

As explained in Example~\ref{ex:constructsyms},
there is a transportable modular symbol associated
to each nonidentity element $\gamma\in \Gamma_0(11)$.
For example the transportable symbol
$(11X^2-11XY+Y^2)\{\infty,\frac{10}{11}\}=11(a_5-a_6)$
is associated to $\gamma=\abcd{10}{-1}{11}{-1}$.
The symbol $-\frac{5}{4} b_1 + \frac{5}{4} b_2 
  -\frac{1}{4} b_3 + \frac{1}{4} b_4$ is
the transportable symbol 
associated to $\abcd{\hfill{}5}{-1}{11}{-2}$.
The symbol 
$-\frac{9}{8} b_1 - \frac{19}{8} b_2 + \frac{19}{8} b_3 + \frac{99}{8}b_4$
is associated to $\abcd{4}{1}{11}{\hfill{}3}$,
and 
$-\frac{27}{8}b_1 + \frac{11}{8} b_2 + \frac{9}{8} b_3 + \frac{49}{8} b_4$
is associated to $\abcd{\hfill{}3}{-2}{11}{-7}$.
Together these four transportable symbols 
span $\sS_4(\Gamma_0(11);\Q)$.

In order to illustrate Section~\ref{sec:whenspan},
we remark that symbols of the form $P\{\infty,\gamma(\infty)\}$
do not span all of $\sM_4(\Gamma_0(11);\Q)$, but they do span
a space bigger than $\sS_4(\Gamma_0(11);\Q)$.
Corollary~\ref{cor:span} implies that their span
contains $\sS_4(\Gamma_0(11);\Q)$;
however, the symbol $Y^2\{\infty, \frac{1}{11}\}$
does not lie in $\sS_4(\Gamma_0(11);\Q)$.

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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\bibitem{cremona:periods}
\bysame, \emph{Computing periods of cusp forms and modular elliptic curves},
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\bibitem{magma}
W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
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\bibitem{manin:parabolic}
J.\thinspace{}I. Manin, \emph{Parabolic points and zeta functions of modular
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\bibitem{merel:1585}
L.~Merel, \emph{Universal \protect{F}ourier expansions of modular forms}, On
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\bibitem{stein:phd}
W.\thinspace{}A. Stein, \emph{Explicit approaches to modular abelian
  varieties}, Ph.D. thesis, University of California, Berkeley (2000).

\end{thebibliography}
\end{document}