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How many levels will n=4 hydrogen atom split into due to the spin orbit coupling?

i know J = L + S

i dont know if i have understood the concepts properly - i know if n = 1,2,3....

l=0,1,2 ... n-1

m= -l,-l+1 ..0.. l-1,l

and aslo thas s = +-1/2

so for instance n=1 level,will have l=0 so hence m=0 and hence j = l + s (which is 0 +or-1/2) so is it right in me saying j=-1/2 or j=1/2 but j cannot be -ve so i state from eigenfunctions at level n=1 which is j=1/2

now for n=2 , we have 1 state in l=0 level as before one state ,but now for l=1 we have 3 m states(m=-1,0,+1) and for each of these in j = 1 (+or - 1/2) we have j=1/2 or j=3/2 so we have 7 states all together

COULD ANYONE HELP ME AND TELL ME IF IM ON THE RIGHT LINES AND IF I HAVE GONE WRONG ANYWHERE?

THANKS LOGAN