Rattle and roar while coasting in gear

[2003 M3]

My 46K mile, 6 speed coupe has developed an alarming issue which occurs when coasting in 1st gear (that is, removing one's foot from the accelerator while still in gear). A slight shuddering as well as an audible humming/groaning sound occurs immediately and lasts as long as I coast. This effect is also present in 2nd gear, although less obviously, but is not noticeable in any other gear. I can feel the vibration through the seat, although I can't quite tell what part of the car it is coming from. It happens with the A/C on or off.

Any ideas?

Thanks.

[Grimlok]

I typed that, but then also recall that some people here (or that used to be here...) have had their rear differential bolts come loose causing the diff to shift. I would think that there might be a slight alignment issue with the diff that could cause noise and vibration as well. Is it possible that the noise and vibration might be present at all speeds, but that it is less noticeable at higher speeds because of more ambient noise and ambient vibration associated with that speed?