We have already discussed why Exp(Pi*Sqrt(163)) is an almost integer.

Why are powers of $\exp(\pi\sqrt{163})$ almost integers?

Basically j((1+√(-163))/2) ~ 744 - Exp[Pi*Sqrt[163]], where j((1+√(-163))/2) is a rational integer.

But j(√-232/2) and j(√-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?