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Here's a couple of pages from the manual supplement that appear to cover turning off the display. I have not checked this to see if it matches the car ('13 Eco), but stuff I've looked up in the past has been accurate.

 

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I'll check it on mine on the way home today and report back.

 

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I can confirm this works on a 2013. Press the config button, scroll to screen settings, select screen off.

It works the same with the radio on or off.

Touch any radio button and it comes back on.

Sent from my SAMSUNG-SGH-I727 using AutoGuide.Com Free App

 

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Dammit! Maybe an update is available that would bring your system to the latest version? You might want to PM the Chevy rep on the forum and inquire.

Looking back at your picture the display is different from mine. Mine has the bowtie, but says MyLink.

It's interesting that the '13 doesn't have the day/night/auto setting. Mine seems to be auto. Have you tried that setting? Mine's not crazy bright at night.