Your going to need a scan tool to diagnose it since the compressor is controlled by the dme.

If its not happy it wont come on. To turn on initially it requires the pressure on the pressure sensor and a good aux fan circuit. I say good because the fan can be fried but it has to be plugged in. If the dme doesn't see the fan it wont turn on the compressor. The fan can be bad as in doesn't spin but it has to see the fan.

That part is a bit hard to call since a non spinning fan can be caused from it not being plugged in which will cause the compressor to not be turned on.

The fan can be fried in a way that makes it not show up to the dme thus no compressor engagement.

But the fan can be fried but in a way that shows up to the dme as plugged in but not functioning which in that case it will turn on the compressor.

So a scan tool is pretty much required.

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Aftermarket E53 Radio Install