  Using finite-size-scaling methods, we study the quantum chain version of the
spin-$1$-Blume-Capel model coupled to an imaginary field. The aim is to realize
higher order non-unitary conformal field theories in a simple Ising-type spin
model. We find that the first ground-state level crossing in the
high-temperature phase leads to a second-order phase transition of the Yang-Lee
universality class (central charge $c=-22/5$). The Yang-Lee transition region
ends at a line of a new type of tricriticality, where the {\em three} lowest
energy levels become degenerate. The analysis of the spectrum at two points on
this line gives good evidence that this line belongs to the universality class
of the ${\cal M}_{2,7}$-conformal theory with $c=-68/7$.
