  It has long been known that galaxy shapes align coherently with the
large-scale density field. Characterizing this effect is essential to
interpreting measurements of weak gravitational lensing, the deflection of
light from distant galaxies by matter overdensities along the line of sight, as
it also produces coherent galaxy alignments that we wish to interpret in terms
of a cosmological model. Existing direct measurements of intrinsic alignments
using galaxy samples with high-quality shape and redshift measurements
typically use well-understood but sub-optimal projected estimators, which do
not make good use of the information in the data when comparing those
estimators to theoretical models. We demonstrate a more optimal estimator,
based on a multipole expansion of the correlation functions or power spectra,
for direct measurements of galaxy intrinsic alignments. We show that even using
the lowest order multipole alone increases the significance of inferred model
parameters using simulated and real data, without any additional modeling
complexity. We apply this estimator to measurements of parameters of the
non-linear alignment model using data from the Sloan Digital Sky survey,
demonstrating consistent results with a factor of $\sim$2 greater precision in
parameter fits to intrinsic alignments models. This result is functionally
equivalent to quadrupling the survey area, but without the attendant costs --
thereby demonstrating the value in using this new estimator in current and
future intrinsic alignments measurements using spectroscopic galaxy samples.
