var/covar of Linear Lease Square estimates

Goal: Find, $Var(\hat B_0)$ and $Covar(\hat B_1, \hat B_0)$

$($ Note: $y_i = B_0 + x_i B_1 +e_i,$ with $e_i$ ~ $N(0,\sigma ^2)$ So, everything is fixed, but $e_i$ and $y_i.$ Now I found unbiased estimates $\hat B_0$ and $\hat B_1.$ Also, I found the variance of $\hat B_1.)$

$\hat B_0 = \bar y - {\bar x} \hat B_1$ and,

$\hat B_1 = \frac{\sum x_i y_i -{\bar y} \sum x_i}{\sum x_i ^2 - {\bar x} \sum x_i}$

Work:

$E[\hat B_0 ^2] = E[(\bar y - {\bar x}\hat B_1)^2] = E[\bar y ^2] - 2{\bar x}E[{\bar y}\hat B_1] + \bar x E[\hat B_1^2]$

Each part is easy enough, I just don't know how to start $E[{\bar y}\hat B_1]....?$

Lastly, how to approach $E[\hat B_0\hat B_1]....?$