  We prove new cases of the inverse Galois problem by considering the residual
Galois representations arising from a fixed newform. Specific choices of weight
$3$ newforms will show that there are Galois extensions of $\mathbb{Q}$ with
Galois group $PSL_2(\mathbb{F}_p)$ for all primes $p$ and
$PSL_2(\mathbb{F}_{p^3})$ for all odd primes $p \equiv \pm 2, \pm 3, \pm 4, \pm
6 \pmod{13}$.
