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When an Xbox 360 shows an E66 error on screen, it is signaling a general hardware failure. A single red blinking light will be present in section 4 of the ring of light on the front of the console, as shown in the picture. An on-screen message, which depicts the E66 error, will instruct a user to contact Microsoft Customer Support.
An E66 error is known from reports online to indicate a problem with the console's DVD-ROM drive. In particular, this problem can present itself when a DVD-ROM fails to pair with the system due to a mismatch in certain details. For example, a drive being a difference make and model than the console expects (incorrect spoofing during flashing may cause this, as would replacing it would a replacement drive.)
This could also be the case for the DVD-ROM's firmware - if the firmware version is different to that expected by the system it may result in an E66 error in cases. With an E66 error, the drive usually doesn't have a problem ejecting or reading media (failure to eject or read a disc could indicate bad firmware.)
If this problem was caused by an attempt to flash firmware, then this problem cannot be fixed with warranty. However, restoring the original firmware could fix this problem.
An E66 error is usually associated with an error code of 1002. To get an error code from an Xbox 360 console that is displaying an E66 error (or 3ROL) you simply press and hold the "sync up" button (used to connect wireless controllers) and then press the Eject button for the DVD drive. This will bring up the first digit of the error code (usually it is one light blinking red, which represents 1, please note that four lights blinking red represents a 0, not 4). With the sync up button still held, press the Eject button again and you get the second digit. Do this twice more to get the full 4 digits of the error code.
For more information on the error code you receive, or on Xbox 360 hardware problems in general, read this Xbox-Scene forum thread.