Title: The Harris-Venkatesh conjecture for derived Hecke operators II: a unified Stark conjecture

Abstract: We formulate a unified conjecture for weight-$1$ modular forms that combines the Stark conjecture and the Harris-Venkatesh conjecture, suggesting that the leading coefficient of Artin $L$-functions and the action of derived Hecke operators respectively give real and mod $p$ data of the same algebraic units. First, we prove a refinement of Stark's theorem for imaginary quadratic number fields and a new Siegel-Weil formula for $\mathrm{PGL}(2)$. We then prove our new "Harris-Venkatesh plus Stark" conjecture for imaginary dihedral modular forms.