For the following weights $(j,k)$, the space of non-lifted forms $S_{j,k}$ is two-dimensional and the characteristic polynomial, for $p=2$ and $p=3$, of Frobenius acting on $S_{j,k}$ is reducible over the integers: \[ (18,7), (10,13), (0,24), (0,26). \] The last two examples were found by Skoruppa. The splitting in the case $(10,13)$ is due to the fact that this space contains $\text{Sym}^3$ of the elliptic eigenform of level 1 and weight 12.

For the weight $(j,k)=(4,15)$ the space $S_{j,k}$ is one-dimensional, and the spinor $L$-factor acting on $S_{j,k}$ is reducible over the integers for $p=2$ but not for $p\in\{3,5,7,11,13\}$.

For the weight $(j,k)=(2,20)$ the space $S_{j,k}$ is three-dimensional, and the spinor $L$-factor acting on $S_{j,k}$ is reducible over the integers for $p=2$.