The idiot who blackmailed rapper J. Cole on Twitter yesterday -- by threatening to kill
 his little sister if J. Cole didn't retweet him -- is now being investigated by Texas police, TMZ reports.

A rep for the New Braunfels Police Department says they have opened an investigation into the 19-year-old responsible for the tweet ... after the picture (showing the fan aiming a gun at his sister's face) was brought to their attention.

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The rep says the teenager in question "is cooperating and it seems this was just a bad lapse in judgment." Police say the gun in the picture is a BB gun, not an actual handgun.

The rep adds, "For now this is still an open investigation, and for the time being no charges are being filed."

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