making friends and making mistakes in friendships and learning how to differentiate between good and bad friends and how to negotiate with friends are all crucial tools in learning to become a well rounded confident adult. whilst at school you mum is not in charge your teachers are, so if they allow you to make friends (and i believe they will strongly encourage you to) then there is no reason why you can't make friends to hang out with during breaks, lunch etc, just, for now do not invite them back to yours after school.
if you get invited back to theirs then you need to decide if you ask your parents permission or politely decline the invitation.
I am curious as to why your mum does not want you to make any friends, i would have thought that if she was any sort of decent mother she would want to see you flourish into a social butterfly.
it could however be she is scared you may get in with a bad crowd, if this is the case then you need to be extra open with her about who you are socialising with, talk to her about how great you new friends are, and about any concerns you have around the let your mum help you sort out how to handle the problems that come with school friendships, that way your mum will still feel she has some control even if she hasn't got much and she will feel more able to trust your decisions around who you choose as friends and who you don't. her trust may take some time to win round but it is worth sticking with the above until eventually your mum should see you are being responsible and mature about choosing your friends and lay off you a bit.