Lying Algebras – Imaginary roots in \$ widetilde {E} _8 \$

Consider the root system of a Kac-Moody algebra. Designate with $$alpha_i$$ the simple root associated with the node $$i$$
from for $$i in {1, ldots, n-1 }$$ and from $$beta$$ the simple root associated with $$n$$,

The dynkin diagram for $$widetilde {E} _8$$ is
begin {align} circ – circ – & circ – circ – circ – circ – circ – circ \ & | \ & bullet end
from where $$bullet$$ corresponds to the simple root $$beta$$, The degree of a root is the coefficient of the root at $$beta$$,

Is there evidence of imaginary type roots? $$widetilde {E} _8$$? I only find an imaginary root $$gamma = 3 beta + 2 alpha_1 + 4 alpha_2 + 6 alpha_3 + 5 alpha_4 + 4 alpha_5 + 3 alpha_6 + 2 alpha_7 + alpha_8$$, The origin $$gamma$$ has grad $$3$$,

Many thanks.