Linear combinations of iid trivariate normal vectors

Suppose $X_1,X_2,X_3$ and $X_4$ are independent and identically distributed tri-variate random vectors having normal distribution. Then the distribution of $a_1X_1+a_2X_2+a_3X_3+a_4X_4$, where $a_1,a_2,a_3,a_4$ are real numbers, is: