Counting factor-neutral portfolios

Four stocks have market betas 0.5, 1.0, 1.5 and 2.0. You want long-short portfolios $w$ (dollar weights) that are dollar neutral, $\sum w_i = 0$, and beta neutral, $\sum \beta_i w_i = 0$.

(a) What is the dimension of the set of such portfolios? Give a basis.

(b) A risk model covers 500 stocks. Each stock has a market beta and belongs to exactly one of 11 sectors. You require dollar neutrality, beta neutrality and zero net exposure to each sector, which is 13 linear constraints. How many dimensions of portfolios are left?