cubic diophantine equations $x^3+11y^3=1$

Question

Many books solve this equation looking roots of $x^3-11$ in $\mathbb{Q}_{19}$. But there is a point not so clear. These roots correspond to $$a_1=-2+8*19 \pmod{19^2} \\ a_2=-3+5*19 \pmod{19^2}\\ a_3=5+6*19 \pmod{19^2}.$$ Now we know $\epsilon=1+4\theta-2\theta^2$ is fundamental unit ($\theta=11^{1/3}$) and then every book says these correspond to 3 embeddings of $\epsilon$ and without calculation these are $$v_1=4 \pmod{19^2} \\ v_2=9+2*19 \pmod{19^2} \\ v_3=9+16*19 \pmod{19^2}$$ I made calculation by me but i'm not sure i used correct embeddings (for me they correspond to $v_i=1+4a_i-2a_i^2$ as element of $\mathbb{Q}_{19}$. But the results are wrong. What are these embeddings? Thanks.