  We present two families of lattice theta functions accompanying the family of
lattice theta functions studied by Montgomery in [H.~Montgomery. Minimal theta
functions. \textit{Glasgow Mathematical Journal}, 30(1):75--85, 1988]. The
studied theta functions are generalizations of the Jacobi theta-2 and theta-4
functions. Contrary to Montgomery's result, we show that, among lattices, the
hexagonal lattice is the unique maximizer of both families of theta functions.
As an immediate consequence, we obtain a new universal optimality result for
the hexagonal lattice among two-dimensional alternating charged lattices and
lattices shifted by the center of their unit cell.
