Yes, dividends when taxed by the state are taxed in the state of residence.
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A tax exempt mutual fund holds bonds from many states, if your state is one of them , you get the percentage from your mutual fund company. If they did not provide a breakdown, you check the box “I earned tax exempt dividends in more than one state” ("Multiple States" in the online program) on the first screen after entering the 1099-INT or 1099-DIV. Then select "More than one state" at the bottom of the state scroll down list.
Your state only exempts the interest from that state.
If your mutual fund company provided you a breakdown*, you are only interested in your home state*. Multiply the % for your state by your total tax exempt dividends to get a $ amount (you can't enter the % in TurboTax [TT]). When asked which state, check the box "I earned tax exempt dividends in more than one state". In the drop down menu, select your state and enter the $ amount you calculated. In the 2nd box, select "More than one state*" (at the bottom of the scroll down list) and enter the remaining dollar amount.
*Most mutual funds will provide a breakdown. But you usually have to ask for it, or find it on their web site.
**Your state will tax all the dividends except the dividends from municipal bonds from your state and US Territories. In Illinois, only certain of its municipal are state tax-exempt.