@WindomEarle Great thanks also for your warm-hearted help, dear Windom Earle.

@YemonChoi Great thanks for your kindly comments.

Dear Prof. Loeffler, I have, yes, searched many papers, however it seems that there is no any account on the associated parameters $\alpha_1,\alpha_2,\alpha_3,\alpha_4. $ And, to be frankly, I am not familiar with the group representations. It's maybe not an easy exercise to work out the Langlands parameter to any given $2\times 2 $ diagonal matrix, for which I really need some help from the top experts like you here.

Dear Prof. Loeffler, thanks for explanation, definitely I was concerned about the Langlands parameters just from the point of view of the functional equation of the $L$-function of $L(s, \text{sym}^3f)$. Particularly, I need the exact forms of the Gamma factors of the $L$-function from the symmetric cube lifts of the $GL_2$-cusp forms. So, could you please give some more specific information on the associated parameters $\alpha_1,\alpha_2,\alpha_3,\alpha_4$? This is really what I am concerned about. Much obliged!

@LSpice Many thanks!

Thanks for impressive answer! Please permit me to greatly appreciate your so kindly help in the acknowledgements. Anyway, much obliged for so warm-heated and so professional reply! Great great thanks again and again, dear prof. Sawin!

@WillSawin Dear prof. Sawin, could you take some of your time to give a detailed description of the Newton polyhedron nondegeneracy argument? an answer? I really need this as part of the article which seems very hard for me for the time being. Many thanks.

@PeterHumphries Thanks for your comment!

@WillSawin Dear prof. Sawin, much obliged. This is what I would like to expect. Yes, my puzzle if whether or not the matrix like $$\gamma =\begin{pmatrix}0&1\\1&1\end{pmatrix}$$ can be applied to Remark 6.2 of Michel's paper to get the general bound of $q^{3/2}$. Thanks for telling me this works!!

@WillSawin Sorry, have edited. Many thanks.