Irreducible characters of symmetric group

Characters of the irreducible representation of the symmetric group indexed by the partition evaluated at elements of cycle type indexed by the partition can be evaluated as scalar products of Schur functions with power sums

(1)

due to Frobenius characteristic map. For partitions of 5 the character table obtained this way is as follows 

Sym = SymmetricFunctions(QQ)
s = Sym.s()
p = Sym.p()

M = Matrix([[s[mu.conjugate()].scalar(p[nu.conjugate()])
             for nu in Partitions(5)]
            for mu in Partitions(5)])

M
(2)

It can be compared with the character table calculated via internal Sage methods 

SymmetricGroup(5).character_table() == M

True