Moving average models are expressed as depended of lagged white noise ($a_t$) values.
$$ X_t = \mu + a_t - \theta_1 a_{t-1} - \theta_2 a_{t-2} - ... - \theta_q a_{t-q} $$The operator form (zero-mean) is
$$ X_t = \left( 1- \theta_1 B - ... - \theta_{q} B^q \right) a_t $$Which provides the characteristic
$$ 1- \theta_1 z - ... - \theta_{q} z^q = 0 $$The autocorrelations for MA(1) are as follows
$$ \rho_1 = \frac{-\theta_1}{1 + \theta_1^2} $$$$ \rho_k = 0 |_{k > 1} $$Note: The max absolute value of $\rho_1 = 0.5$ for MA(1).
The autocorrelations for MA(2) are as follows
$$ \rho_1 = \frac{-\theta_1 + \theta_1 \theta_2}{1 + \theta_1^2 + \theta_2^2} $$$$ \rho_1 = \frac{-\theta_2}{1 + \theta_1^2 + \theta_2^2} $$$$ \rho_k = 0 |_{k > 2} $$Due to model multiplicity in MA models, MA models are restricted to have all roots outside the unit circle. This is to make models identifible i.e. a singlar mapping between parameters and models.