  We discuss the effect of interatomic interactions on the condensation
temperature $T_c$ of a laboratory atomic Bose-Einstein condensate under the
influence of an external trapping magnetic field. We predict that accounting
for hyperfine interactions mediated Zeeman term in the mean-field approximation
produces, in the case of the $403 \, G$ Feshbach resonance in the $|F,m_F> =
|1,1>$ hyperfine state of a $^{39}K$ condensate, with $F$ the total spin of the
atom, an experimentally observed (and not yet explained) shift in the
condensation temperature $\Delta T_{c}/T_{c}^{0}=b^{*}_0+b^{*}_1
(a/\lambda_{T}) + b^{*}_2 (a/\lambda_{T})^2$ with $b^{*}_0 \simeq 0.0002$,
$b^{*}_1 \simeq -3.4$ and $b^{*}_2 \simeq 47$, where $a$ is the s-wave
scattering length, and $\lambda_T$ is the thermal wavelength at $T_{c}^{0}$.
Generic expressions for the coefficients $b^*_0$, $b^*_1$ and $b^*_2$ are also
obtained, which can be used to predict the temperature shift for other Feshbach
resonances of $^{39}K$ or other atomic condensates.
