Endoscopic lifts to the siegel modular threefold related to Klein's cubic threefold

Let Alev A₁₁^lev be the moduli space of (1, 11)-polarized abelian surfaces with a canonical level structure. Let χ be a primitive character of order 5 with conductor 11. In this paper we construct five endoscopic lifts Π_i, 0≤i≤4 from two elliptic modular forms f ⊗χi of weight 2 and g⊗Xⁱ of weight 4 with complex multiplication by Q(√ -11) such that Πi∞ gives a non-holomorphic differential form on Alev A₁₁^lev for each i, 0 ≤ i ≤ 4. Then their spinor L-functions are of form L(s-1,f ⊗χi)L(s,g⊗χi) such that L(s,g⊗χi) does not appear in the L-function of Alev A₁₁^lev for any i, 0 ≤ i ≤ 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 Alev A₁₁^lev.