  We introduce a new technique for the study of the distribution of modular
symbols, which we apply to congruence subgroups of Bianchi groups. We prove
that if $K$ is a quadratic imaginary number field of class number one and
$\mathcal{O}_K$ its ring of integers, then for certain congruence subgroups of
$\mathrm{PSL}_2(\mathcal{O}_K)$, the periods of a cusp form of weight two obey
asymptotically a normal distribution. These results are specialisations from
the more general setting of quotient surfaces of cofinite Kleinian groups,
where our methods apply. We avoid the method of moments. Our new insight is to
use the behaviour of the smallest eigenvalue of the Laplacian for spaces
twisted by modular symbols. Our approach also recovers the first and the second
moment of the distribution.
