  The Harris-Venkatesh conjecture posits a relationship between the action of
derived Hecke operators on weight-one modular forms and Stark units. We prove
the full Harris-Venkatesh conjecture for imaginary dihedral weight-one modular
forms. This reproves results of Darmon-Harris-Rotger-Venkatesh, extends their
work to the adelic setting, and removes all assumptions on primality and
ramification from the imaginary dihedral case of the Harris-Venkatesh
conjecture. This is accomplished by introducing two new key ingredients: the
Harris--Venkatesh period on modular curves and the two-variable optimal form.
