The Harris-Venkatesh conjecture for derived Hecke operators III: local constants

The first two papers in this series prove the Harris-Venkatesh conjecture and its refinement with the Stark conjecture for imaginary dihedral modular forms of weight $1$. This paper explicitly describes the constants appearing in the Harris-Venkatesh (plus Stark) conjecture for dihedral modular forms by evaluating $\mathrm{GL}(2) \times \mathrm{GL}(2)$ Rankin--Selberg periods and zeta integrals on newforms and optimal forms. One consequence is a formula for the ratio between Petersson norms and adjoint $L$-values. Our calculations also extend to exotic modular forms whose level is odd or whose Deligne-Serre representation is $2$-ordinary.