Abstract: We shall give a distributional version of Motohashi's formula by presenting a compact variant. Then we give an application of the formula to the Weyl-type hybrid subconvexity for Hecke characters of cube-free level over totally real number fields, which includes:

(1) description of a large class of local admissible weight functions at archimedean places on the cubic moment side;

(2) a local archimedean transformation formula from the cubic moment side to the fourth moment side;

(3) a bound of the local archimedean dual weight functions on the fourth moment side.

Details of (3) will be given. Other details will be given according to the audience's interests if time permits. Relations and differences of our methods with other methods in the literature will be emphasized.

Joint result with Olga Balkanova and Dmitry Frolenkov.

The link for the talk is https://zoom.us/j/94752830725, the password is the order of $\mathrm{SL}_2(\mathbb{F}_{97})$.