Hatchling turtles are pretty sensitive and hard to take care of, they have a lower survival rate than adults...but here are some tips.
If he is just a hatchling then a ten gallon tank filled almost all the way up should be enough space. (they should have 10gallons of water per inch of shell)
Make sure you get an acurate thermometer (temp gun or digital thermometer from walmart) to measure temps. The basking spot should be between 88-90. Being that its a small tank make sure you are not overheating the whole tank and that your turtle can escape the heat if he wants.
You mentioned he had a basking light but did not mention a UVB bulb, these are VERY important, especially in baby turtles that are growing rapidly. Make sure to buy one of these (most comonly available at any pet store in the form of a flurescent tube, they usually cost about 20-30$. Zoo-med 5.0 or 8.0 are good bulbs) Put this over the basking spot so on top of the screen lid, UVB rays will not pass through glass at all and only through water a little bit.
At this age he should be eating mostly protein (small crickets, turlte pellets, krill, blood worms, beefheart, chopped up earth worms etc...) with small amounts of veggies just to make sure he aquires a taste for them as when he gets older his diet will be mostly greens. You can try collard, mustard, and dandelion greens. Along with endive, escarole and many others.
Make sure to leave him alone as much as possible, you do not want to stress the little guy out by handling him a lot, tappin on the tank or even staring at him exesseively.
Set the lights on a timer so your turtle gets a set amount of light and dark each day.
And finally set up a good water changing schedual, i would change 1/3 of the water weekly. More or less frequently depending on the size of your filter. Buy a good test kit or locate a good tropical fish store that uses good quality test kits (Sera are my personal favorite. The germans rock at fish keeping skills!) and keep a good eye on your levels of ammonia, nitrate and phostphates.
Im sure i missed something, but someone else will pick up on it!!
Hope this helps!
~Mim
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0.1 Bearded dragon
0.1 mexican kingsnake
1.0.2 Leopard Gecko's
0.0.1 Rose Hair Tarantula
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4.1 Cats