  This is a book about computational aspects of modular forms and the Galois
representations attached to them. The main result is the following: Galois
representations over finite fields attached to modular forms of level one can,
in almost all cases, be computed in polynomial time in the weight and the size
of the finite field. As a consequence, coefficients of modular forms can be
computed fast via congruences, as in Schoof's algorithm for the number of
points of elliptic curves over finite fields. The most important feature of the
proof of the main result is that exact computations involving systems of
polynomial equations in many variables are avoided by approximations and height
bounds, i.e., bounds for the accuracy that is necessary to derive exact values
from the approximations.
