This talk serves as a sequel to the preceding presentation by Jeng-Daw Yu. While the previous talk established the general theory of finite monodromic mixed Hodge structures and the structural decomposition of such motives, this talk will focus on their explicit applications. Our main object of study is the exponential motive associated with the symmetric powers of the Airy sheaf, denoted by $\mathrm{Sym}^k \mathrm{Ai}$, which exhibits finite monodromy.

By relating the structure of $\mathrm{Sym}^k \mathrm{Ai}$ to confluent hypergeometric sheaves, we will demonstrate how to reconstruct it by piecing together these constituent parts with appropriate twists. Relying on this structural description, we will pass to the arithmetic setting to study the corresponding Galois representations. Consequently, this approach provides a clear picture of the arithmetic properties of the symmetric powers of the Airy sheaf, including the explicit relationship between their Euler factors and Airy moments, as well as their connections to modular forms.