For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:

1. (1) ${\mathcal{C}}$ always contains a simple projective object;

2. (2) if ${\mathcal{C}}$ is in addition ribbon, the internal characters of projective modules span a submodule for the projective $\text{SL}(2,\mathbb{Z})$-action;

3. (3) the action of the Grothendieck ring of ${\mathcal{C}}$ on the span of internal characters of projective objects can be diagonalised;

4. (4) the linearised Grothendieck ring of ${\mathcal{C}}$ is semisimple if and only if ${\mathcal{C}}$ is semisimple.

Results (1)–(3) remain true in positive characteristic under an extra assumption. Result (1) implies that the tensor ideal of projective objects in ${\mathcal{C}}$ carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of $S$-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular $S$-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.