Get food in the cafe in front of the mosque. Not expensive and VERY authentic and delicious!
Also, souvenirs in that area are cheaper that in down town Sultanahmet, so it would be a good chance for you.
The last, thing, if you are a photograper, go away from the mosque in the direction where the souvenirs and cafes are, pass through the arch and the first building on the left is a student dormitory or some kind of student campus. You can go there and ask the students to go on the balcony and take an awesome picture of the mosque!! If you dare of course:)
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