\chapter*{Preface}
\begin{quote}
The object of numerical computation is theoretical advance.\\
\hfill --{\em A.\thinspace{}O.\thinspace{}L. Atkin}
\end{quote}
The definition of the spaces of modular
forms as functions on the upper half plane satisfying
a certain equation is very abstract. The definition
of the Hecke operators even more so.
Nevertheless, one wishes to carry out explicit
computational investigations on these objects.
We are fortunate that we now have methods available which
allow us to transform the vector space of cusp forms of given weight
and level into a concrete object, which can be
explicitly computed. We have the work of Atkin-Lehner,
Birch, Swinnerton-Dyer, Manin, Merel, and many others to thank for this
(see \cite{antwerpiv, cremona:algs, merel:1585}).
The Eichler-Selberg trace formula, as extended
in \cite{hijikata:trace}, can be used to compute
characteristic polynomials of
Hecke operators and hence gain some information about
spaces of modular forms. It is also possible to compute
Hecke operators and $q$-expansions using Brandt matrices and
quaternion algebras as in \cite{kohel:hecke, pizer:alg};
another
closely related method relies on the module
of enhanced supersingular elliptic curves as exploited by
Mestre-Oesterl\'{e}~\cite{mestre:graphs} and extended
by Edixhoven in his Ph.D. thesis.
I take the approach of Cremona's book
\cite{cremona:algs} and view modular symbols as central.
No single computational method provides complete information
about the package of invariants attached to a modular form.
The various methods must be organized into a symphony;
for example, the Mestre-Oesterl\'e method combined with
modular symbols can be used to compute component groups.
In this thesis, I present methods for explicitly
computing with modular forms, Hecke operators, and the abelian
varieties attached to them. In Chapter~\ref{chap:bsd},
I give an example of how to coordinate these methods to
gain data about real world problems.
Chapters~\ref{chap:modsym} and~\ref{chap:computing}
owe much to the publications of Merel and Cremona.