From MathSciNet:

MR1037906 (90m:11065) Reviewed
McKay, John(3-CONC); Strauss, Hubertus(3-CONC)
The q-series of monstrous moonshine and the decomposition of the head characters.
Comm. Algebra 18 (1990), no. 1, 253–278.
11F22 (20C15 20D08 33A99)

The authors tabulate the first fifty coefficients of the q-expansions of the Hauptmoduln which arise in the Monster-modular connection, and provide the decompositions of the corresponding Monster characters into irreducibles. This work to some extent duplicates earlier computations of S. D. Smith [in Finite groups—coming of age (Montreal, PQ, 1982), 303–313, Contemp. Math., 45, Amer. Math. Soc., Providence, RI, 1985; MR0822245 (87h:20037)] but the calculations are based on different methods which are also briefly discussed. The authors observe a surprising periodicity of the signs of the q-coefficients, a phenomenon which as yet remains unexplained.