Since you had to provide the bank routing and account number for the direct deposit of the state tax refund, only you would know the name of the institution.
Look at the State Filing Instructions page included with your tax return for the bank name.
To access your current or prior year online tax returns sign onto the TurboTax website with the userID you used to create the account - https://myturbotax.intuit.com/
Scroll down to the bottom of the screen and on the section Your tax returns & documents click on Show. Click on the Year and Click on Download/print return (PDF)
If you used the desktop CD/Download editions installed on your computer, the only copy of your tax data file and any PDF's will be on the computer where the return was created. TurboTax does not store online any returns completed using the desktop editions.
First of all.......has your state sent it yet?
Check for a state refund https://ttlc.intuit.com/community/refund-status/help/how-do-i-track-my-state-refund/00/25571
Make sure you are using the refund amount from your actual state tax return and not the combined amount Turbo Tax showed you at the end. That was for federal and state combined.
Contact your state