Two character formulas for $\widehat{s{l}_2}$ spaces of coinvariants

We consider $\widehat{s{l}_2}$ spaces of coinvariants with respect to two kinds of ideals of the enveloping algebra $U(s{l}_2\otimes \text{C}[t])$. The first one is generated by $s{l}_2\otimes t^N$, and the second one is generated by $e\otimes P(t), f\otimes R(t)$where$P(t), R(t)$ are fixed generic polynomials. (We also treat a generalization of the latter.) Using a method developed in our previous paper, we give new fermionic formulas for their Hilbert polynomials in terms of the level-restricted Kostka polynomials and $q$-multinomial symbols. As a byproduct, we obtain a fermionic formula for the fusion product of $s{l}_3$-modules with rectangular highest weights, generalizing a known result for symmetric (or anti-symmetric) tensors.