For the following weights and representations of $S_6$, $(j,k,s[P])$, the $s[P]$ subspace of non-lifted forms in $S_{j,k}$ is two-dimensional and the characteristic polynomial, for $p=3$, of Frobenius acting on the $s[P]$ subspace of $S_{j,k}$ is reducible over the integers: $\begin{array}{l} (4, 6, s[3, 2, 1])\\ (10, 4, s[3, 2, 1])\\ (10, 4, s[3, 1, 1, 1])\\ (10, 4, s[2, 2, 1, 1])\\ (8, 5, s[3, 1, 1, 1])\\ (6, 6, s[4, 1, 1])\\ (2, 8, s[3, 2, 1])\\ (2, 8, s[3, 1, 1, 1])\\ (14, 3, s[2, 2, 1, 1])\\ (10, 5, s[4, 1, 1])\\ (8, 6, s[4, 1, 1])\\ (6, 7, s[4, 2])\\ (4, 8, s[4, 1, 1])\\ (2, 9, s[4, 1, 1])\\ (2, 9, s[3, 2, 1])\\ (0, 10, s[3, 1, 1, 1])\\ (16, 3, s[3, 2, 1])\\ (16, 3, s[3, 1, 1, 1])\\ (8, 7, s[5, 1])\\ (6, 8, s[3, 3])\\ (4, 9, s[5, 1])\\ (4, 9, s[4, 2])\\ (18, 3, s[3, 2, 1])\\ (14, 5, s[5, 1])\\ (12, 6, s[5, 1])\\ (4, 10, s[5, 1])\\ (2, 11, s[5, 1])\\ (2, 11, s[3, 1, 1, 1])\\ (0, 12, s[3, 2, 1])\\ (22, 3, s[4, 1, 1])\\ (20, 4, s[5, 1])\\ (2, 13, s[5, 1])\\ (2, 13, s[4, 2])\\ (0, 14, s[4, 2])\\ (0, 14, s[2, 2, 2])\\ (16, 7, s[1, 1, 1, 1, 1, 1])\\ (2, 14, s[3, 3])\\ (0, 15, s[3, 2, 1])\\ (26, 3, s[3, 3])\\ (32, 3, s[5, 1])\\ (0, 19, s[5, 1])\\ (0, 20, s[5, 1]). \end{array}$ The examples with $s[P]=s[6]$ can be found on the page for level 1. For $(j,k,s[P])=(4, 7, s[4, 1, 1])$ and $(j,k,s[P])=(4, 7, s[3, 3])$, the $s[P]$ subspace of $S_{j,k}$ is spanned by $\text{Sym}^3$ of the elliptic eigenform of level 4 and weight 6.

For the following weights and representation $(j,k,s[P])$, the $s[P]$ subspace of $S_{j,k}$ is one-dimensional, and the spinor $L$-factor acting on $S_{j,k}$ is reducible over the integers for $p=3$ but not for $p\in\{5,7,11,13\}$: $(2, 6, s[3, 1, 1, 1]), (4, 6, s[4, 2]), (4, 6, s[2, 2, 2]), (8, 4, s[3, 2, 1]).$