THETA PRODUCTS AND ETA QUOTIENTS OF LEVEL 2 4 AND WEIGHT 2

We find bases for the spaces M-2 (Gamma(0)(24), (d/center dot)) (d = 1, 8, 1 2, 2 4) of modular forms. We determine the Fourier coefficients of all 3 5 theta products phi[alpha(1), alpha(2), alpha(3), alpha(4)] (z) in these spaces. We then deduce formulas for the number of representations of a positive integer n by diagonal quaternary quadratic forms with coefficients 1, 2, 3 or 6 in a uniform manner, of which 1 4 are Ramanujan's universal quaternary quadratic forms. We also find all the eta quotients in the Eisenstein spaces E-2 (Gamma(0)(24), (d/center dot)) (d = 1, 8, 1 2, 2 4) and give their Fourier coefficients.