  The Hamming weight enumerator function of the formally self-dual even, binary
extended quadratic residue code of prime p = 8m + 1 is given by Gleason's
theorem for singly-even code. Using this theorem, the Hamming weight
distribution of the extended quadratic residue is completely determined once
the number of codewords of Hamming weight j A_j, for 0 <= j <= 2m, are known.
The smallest prime for which the Hamming weight distribution of the
corresponding extended quadratic residue code is unknown is 137. It is shown in
this paper that, for p=137 A_2m = A_34 may be obtained with out the need of
exhaustive codeword enumeration. After the remainder of A_j required by
Gleason's theorem are computed and independently verified using their
congruences, the Hamming weight distributions of the binary augmented and
extended quadratic residue codes of prime 137 are derived.
