If you unplug the interconnect completely, does it still hum? If it doesn't, the problem is most likely a lose ground shield in your interconnect.
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I would recommend that you have someone skilled in soldering re-solder the connectors, in the amp, to be sure you are not being subjected to a cold solder joint. The symptoms you describe are typical of this elusive beast. Even looking at the joints cannot tell you if they are not well connected or soldered. Best way is to remove all existing, re-strip and clean connections, get to new wire if you can, and re-solder. Be sure to very tightly crimp the wire to the connector, solder is there to seal and insure great connection, not to be the medium.
As Mr L mentioned; the problem could be a cold solder joint on a ground connection, or a loose ground terminal on one of the RCA jacks. That the problem didn't occur at the shop, without your interconnects, does make them suspect as well. Cheaper RCA connectors can easily lose their grip at the ground. You don't mention much about yours, other than that you can vary the problem by manipulating them. Does the hum affect both channels? If not; interchange the cables, channel for channel, and see if the hum follows.
Re soldering all the connections from the RCA's inputs is the
first choice, grounds especially, then if not solved, the second choice is the filter capacitors in the power supply feeding the preamp-driver stage.Those could have a bad solder joint too.If its not a bad solder joint, then the electrolytic (caps) for the preamp stage might be bad.Its probably not in the output stage because it would most likely hum all the time,whether the volume is low or loud.Preamp stage most likely.
DON'T FORGET!!!TUBE AMP VOLTAGES COULD GET OVER 600 VOLTS+++
If you don't know how to check and discharge the HIGH VOLTAGES
keep out. The caps could stay charged like a high voltage battery for many hours. Turning the amp off while the music is playing (you'll here the sound fad out)sometimes helps.If
you don't know how to discharge the high voltages be warned,and don't sue me,or haunt me if something goes wrong.
You say you are only using cd as source. Assuming the amp has more than one line input, try hooking your cdp to an aux input. If the problem does not present itself the issue is most likely the cd input connectors. If problem persists, there is most likely a problem with the amp as you have stated ICs have been swapped.