  Certain integrable models are described by pairs (X,Y) of ADET Dynkin
diagrams. At high energy these models are expected to have a conformally
invariant limit. The S-matrix of the model determines algebraic equations,
whose solutions are mapped to the central charge and scaling dimensions of the
corresponding conformal field theory. We study the equations of the (D_m,A_n)
model and find all solutions explicitly using the representation theory of Lie
algebras and related Yangians. These mathematically rigorous results are in
agreement with the expectations arising from physics. We also investigate the
overlap between certain q-hypergeometric series and modular functions. We study
a particular class of 2-fold q-hypergeometric series, denoted f_{A,B,C}. Here A
is a positive definite, symmetric, 2x2 matrix, B is a vector of length 2, and C
is a scalar, all three with rational entries. It turns out that for certain
choices of the matrix A, the function f_{A,B,C} can be made modular. We
calculate the corresponding values of B and C. It is expected that functions
f_{A,B,C} arising in this way are characters of some rational conformal field
theory. We show that this is true in at least one case.
