It is not advisable to remove the patina from a coin when it preserves its appearance; otherwise it would just diminish its value.
A good way to clean an oxidized coin is to use electrolysis. There are kits sold on the net.
But know that most of the coins that will have been eaten by the oxidation will be totally smooth, only the gold coins will have been preserved.
If you think you have come across a rare item of high value, it is advisable to go with a specialist.
Avoid corrosive products; white spirit is not necessarily a good idea. Just like the “mirror“. The ajax can help if you rub gently, but you can also polish the piece if you do not measure your strength well enough.
Clean the shekel thoroughly with water and then put it in an airtight plastic.
Avoid over-manipulating them, the skin can deposit acids on the metals that will slowly gnaw them.
Ultrasonic cleaning is awesome if you want to limit the number of products on your coins. It also makes it possible to brighten any type of metallic object. An ultrasonic cleaner costs about 35-40 dollars.