The number of representations of integers by generalized Bell ternary quadratic forms

摘要

For a positive definite ternary integral quadratic form f, let r(n, f) be the number of representations of an integer n by f. A ternary quadratic form f is said to be a generalized Bell ternary quadratic form if f is isometric to x(2) + 2(alpha)y(2) + 2(beta)z(2) for some nonnegative integers alpha, beta. In this paper, we give a closed formula for r(n, f) for a generalized Bell ternary quadratic form f(x, y, z) = x(2) + 2(alpha)y(2) + 2(beta)z(2) with 0 = alpha = beta = 6 and class number greater than 1 by using the Minkowski-Siegel formula and bases for spaces of cusp forms of weight 3/2 and level 2(t) with t = 6, 7, 8 consisting of eta-quotients.