Local factors in the vector-valued Waldspurger formula

Compute the local factors in the explicit Waldspurger formula at the supercuspidal places 2 and 3 of the automorphic representation associated with $f_{36}$, as well as at the archimedean place, in the vector-valued normalization used for $S_{3/2}(\bar\rho_4)$, and verify the translation between lattice coefficients and classical Fourier coefficients.

Background

The section-layer Petersson scalar is numerically identified with a Chowla–Selberg expression through a chain involving Waldspurger’s theorem, Damerell’s theorem, Shimura period relations, and the local-to-global comparison of Fourier coefficients.

The global identities and numerical checks are in place, but the local normalization at the ramified primes 2 and 3 and at infinity has not been established in the vector-valued setting. Resolving these local factors would supply the missing step needed to make the claimed section-layer transcendental identity unconditional.