We have already discussed why $e^{(\pi\sqrt{163})}$ is an almost integer.

http://mathoverflow.net/questions/4775/why-are-powers-of-exppisqrt163-almost-integers

Basically $j(\frac{1+\sqrt{-163}}{2} ) \simeq 744 - e^{\pi\sqrt{163}}$, where $j(\frac{1+\sqrt{-163}}{2} )$ is a rational integer.

But $j(\sqrt {\frac{-232}{2}})$ and $j(\sqrt {\frac{-232}{4}})$ are not integers.  They are algebraic integers of degree $2$, but they are also almost integers themselves.  The same phenomenon happens with Class $2$ numbers $88$ and $148$.

Is there another modular function that explains why these numbers are almost integers?