We present an improved method of computing the periods of a newform for {\mathversion{normal}$\Gamma$}$_0(N)$, which converges faster than the method used in [Cremona 1992] (and originally in [Tingley 1975]). We also present some shortcuts that speed up the process of computing all modular elliptic curves of a given conductor $N$. As an application of these methods, we report on the extension of the systematic computation of modular elliptic curves to all conductors up to 5077.

We show the existence of points in the Mandelbrot cardioid that have the one-equator property, a property useful for the study of quaternionic dynamics. The question whether the Julia set is homeomorphic to a codimension-one sphere becomes a good deal more subtle in quaternionic dynamics.

We discuss the Hausdorff convergence of hyperbolic components in parameter space as a one-parameter family of transcendental functions is dynamically approximated by polynomials. This convergence is strongly suggested by computer experiments and is proved in a weaker form, which is illustrated with exponential, sine and cosine families. Furthermore, we consider the convergence of subhyperbolic components. Our result also applies to the antiholomorphic exponentials, which allows us to investigate the limit shape of the unicorns.

We construct closed, embedded, totally geodesic surfaces in two hyperbolic manifolds obtained as knot complements.

A generalization of Kneser's neighboring method allows us to classify two interesting genera at the same time. The new method is used to determine the genus of Hermitian unimodular lattices of rank 8 over the Hurwitz order $\MM $ and the genus of those $\MM $-lattices corresponding to unimodular $\Z$-lattices.

We list here Hecke eigenvalues of several automorphic forms for congruence subgroups of $\SL(3,{}$\funnyZ$)$. To compute such tables, we describe an algorithm that combines techniques developed by Ash, Grayson and Green with the Lenstra--Lenstra--Lovász algorithm. With our implementation of this new algorithm we were able to handle much larger levels than those treated by Ash, Grayson and Green and by Top and van Geemen in previous work. Comparing our tables with results from computations of Galois representations, we find some new numerical evidence for the conjectured relation between modular forms and Galois representations.