  Tungsten is an important element for magnetically confined fusion plasmas but
has the potential to cool, or even quench the plasma due to it being an
efficient radiator. Total and level-resolved dielectronic recombination (DR)
rate coefficients, for all ionization stages, are essential to model tungsten.
We describe a set calculations performed using the distorted wave code
AUTOSTRUCTURE for the tungsten ions W$^{37+}$ to W$^{28+}$. We demonstrate the
importance of relativistic configuration mixing in such calculations. In
particular, we show that the partial DR rate coefficients calculated in level
and configuration resolution can differ by as little as 5%, and up to as much
as 75%. Using the new data, we calculate a revised steady-state ionization
fraction for tungsten. We find that, relative to the ionization fraction
calculated using the recombination rate coefficients of Putterich et al.
(Plasma Phys. Control. Fusion, 50, 085016), the peak temperatures of W$^{37+}$
to W$^{28+}$ ionization states are shifted to lower temperatures spanning
0.9-1.6keV. This temperature range is important for understanding the
performance of large tokamaks, such as ITER, because the temperatures in the
pedestal, edge, scrape-off-layer and divertor region fall in this range.
