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Let ${\cal{A}}^{lev}_{11}$ be the moduli space of $(1,11)$-polarized abelian surfaces with a canonical level structure. Let $\chi$ be a primitive character of order 5 with conductor 11. In this paper we construct five endoscopic lifts $\Pi_i$, $0\le i\le 4$ from two elliptic modular forms $f\otimes\chi^i$ of weight 2 and $g\otimes\chi^i$ of weight 4 with complex multiplication by ${\Bbb{Q}} (\sqrt{-11})$ such that ${\Pi_i}_\infty$ gives a non-holomorphic differential form on ${\cal{A}}^{lev}_{11}$ for each $i$, $0\le i\le 4$. Then their spinor $L$-functions are of form $L(s-1,f\otimes\chi^i)L(s,g\otimes\chi^i)$ such that $L(s,g\otimes\chi^i)$ does not appear in the $L$-function of ${\cal{A}}^{lev}_{11}$ for any $i$, $0\le i\le 4$. The existence of such lifts is motivated by the computation of the $L$-function of Klein's cubic threefold which is a birational smooth model of ${\cal{A}}^{lev}_{11}$.

Endoscopic lifts to the Siegel modular threefold related to Klein’s cubic threefold