OMCPLP 2016 Problem 3

Suppose a real number $a$ is a root of a polynomial with integer coefficients $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ . Let $G=|a_n|+|a_{n-1}|+...+|a_1|+|a_0|$ . We say that $G$ is a gingado of $a$ . For example, as $2$ is root of $P(x)=x^2-x-2$ , $G=|1|+|-1|+|-2|=4$ , we say that $4$ is a gingado of $2$ . What is the fourth largest real number $a$ such that $3$ is a gingado of $a$ ?