  We prove a compatibility theorem between the Stark conjecture and the
Harris-Venkatesh conjecture for imaginary dihedral modular forms of weight $1$.
The key technical input is a general two-variable $\mathrm{PGL}_2$ Siegel-Weil
formula that precisely gives the Laurent series coefficients of Eisenstein
series as a dual theta lift of Eisenstein series. This two-variable Siegel-Weil
formula is applied to Rankin-Selberg periods of imaginary dihedral optimal
forms and a refinement of Stark's formula relating $L$-values with elliptic
units, yielding compatibility between derived Hecke operators and adjoint
$L$-values for Deligne-Serre representations. We formulate a general unified
conjecture encompassing the Stark and Harris-Venkatesh conjectures and
conceptually reinterpret the action of derived Hecke operators as modular Stark
unit data at almost all primes.
