  We consider a pivotal monoidal functor whose domain is a modular tensor
category (MTC). We show that the trace of such a functor naturally extends to a
representation of the corresponding tube category. As irreducible
representations of the tube category are indexed by pairs of simple objects in
the underlying MTC, the simple multiplicities of this representation form a
candidate modular invariant matrix. In general, this matrix will not be modular
invariant, however it will always commute with the T-matrix. Furthermore, under
certain additional conditions on the original functor, it is shown that the
corresponding representation of the tube category is a haploid, symmetric,
commutative Frobenius algebra. Such algebras are known to be connected to
modular invariants, in particular a result of Kong and Runkel implies that the
matrix of simple multiplicities commutes with the S-matrix if and only if the
dimension of the algebra is equal to the dimension of the underlying MTC.
Finally, we apply these techniques to certain pivotal monoidal functors arising
from module categories over the Temperley-Lieb category and the associated MTC.
This provides a novel explanation of the A-D-E pattern appearing in the
classification of $ A_1^{(1)} $ modular invariants.
