Many times when we have a contactor in an application and it is not pulling in (turning on), we assume the whole contactor or just the coil is bad. While this may be the case, it is important to ensure the contactor is receiving the necessary voltage to actuate the coil. Many times the contactor "not working" is a result of it not receiving voltage to the coil which could be a result of a faulty switch elsewhere.
NOTE: If you are not a trained electrician DO NOT ATTEMPT to do this on your own. You might be good at plugging in a lamp or maybe even wiring a light switch, but plain and simple, this stuff can and probably will electrocute you,causing anywhere from a light tickle to your spouse angry at you for not taking out a larger life insurance policy
A simple way to test the coil is to make sure the A1 connection is receiving voltage using a volt meter or multimeter. If it is, then you can test your A1 to A2 coil connection and if the A2 connection is not receiving voltage (and A1 is), then you have a bad coil.