Modular forms and some cases of the inverse Galois problem

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 $PS{L}_2({\mathbb{F}}_p)$ for all primes $p$ and $PS{L}_2({\mathbb{F}}_{p^3})$ for all odd primes $p \equiv \pm 2, \pm 3, \pm 4, \pm 6 \pmod{13}$.