The mantis in your photo is a Chinese Mantid, an introduced species commonly found in the eastern part of the United States.
I've raised many a clutch when I was a kid, but only outside; I never brought them inside to store, and I'd advise against doing so for 3 main reasons:
When the egg cases (called an ootheca) does hatch, you will have anywhere from about 150 to as many as 400 tiny baby mantids from each ootheca. These 1st instar (meaning that they have not yet shed their skin) babies will be able to fit through screen mesh or other very small openings, and the chances are excellent that they will swarm out of your fish tank, and spread throughout the room.
Temperate region mantids, like the Chinese Mantid, require a phase called a diapause, during which colder weather slows development of the babies within the individual eggs. This helps to regulate their growth to time the hatching with spring, when food is naturally available (aphids, fruit flies, etc.). It is possible to simulate the temperature drops to trigger an artificial diapause, but I have no experience in trying this with Chinese Mantids. One source I found suggests dropping the temperature to 12-15 degrees Celsius for at least 8 weeks.
Mantis oothecae are a bit finicky with regards to humidity. Keeping them inside runs the risk of them drying out, and killing the eggs. You may be able to regulate the humidity by periodically spritzing them with a water mister, but be careful not to overdo it.
Note also that you may not get any babies even if you do everything perfectly. Female mantids will lay oothecae even if they have not mated, in which case the eggs will be unfertilized, and will not hatch.
My suggestion would be to put the fish tank with the oothecae outside, in an area outside of direct sunlight. Check it daily once the weather starts to turn nice, and then be prepared to deal with hundreds of teeny, tiny baby mantids running everywhere.