I have a question concerning the Langlands parameters of the symmetric cube lifts. Lets $f$ be a $GL_2$-cusp form, and $\text{sym}^2f$ the symmetric-square lift of it. Assume that $\alpha_1,\alpha_2,\alpha_3$ refer to the associated Langlands parameters. It is known that \begin{split}&\alpha_1=k-1,\alpha_2=0,\alpha_3=-(k-1) \text{, when }f \text{ is a holomorphic form of weight } k; \\ &\alpha_1=2t_j,\alpha_2=0,\alpha_3=-2t_j \text{, when }f \text{ is a Maass form of Laplace eigenvalue } 1/4+t^2_j.\end{split}

My question is how about the symmetric cube lifts $\text{sym}^3f$ ? If one now assumes that $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ are the associated Langlands parameters of $\text{sym}^3f$, what are the exact shapes of these parameters when $f$ being a holomorphic form (resp. a Maass form)?

If some expert leans something on this question, please give some comments or guide a reference. Great great thanks in advance! Your help is highly appreciated!

I have a question concerning the Langlands parameters of the symmetric cube lifts. Lets $f$ be a $GL_2$-cusp form, and $\operatorname{sym}^2f$ the symmetric-square lift of it. Assume that $\alpha_1,\alpha_2,\alpha_3$ refer to the associated Langlands parameters. It is known that \begin{split}&\alpha_1=k-1,\alpha_2=0,\alpha_3=-(k-1) \text{, when }f \text{ is a holomorphic form of weight } k; \\ &\alpha_1=2t_j,\alpha_2=0,\alpha_3=-2t_j \text{, when }f \text{ is a Maass form of Laplace eigenvalue } 1/4+t^2_j.\end{split}

My question is how about the symmetric cube lifts $\operatorname{sym}^3f$ ? If one now assumes that $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ are the associated Langlands parameters of $\operatorname{sym}^3f$, what are the exact shapes of these parameters when $f$ being a holomorphic form (resp. a Maass form)?

If some expert knows something on this question, please give some comments or guide a reference.

I have a question concerning the Langlands parameters of the symmetric cube lifts. Lets $f$ be a $GL_2$-cusp form, and $\text{sym}^2f$ the symmetric-square lift of it. Assume that $\alpha_1,\alpha_2,\alpha_3$ refer to the associated Langlands parameters. It known that \begin{split}&\alpha_1=k-1,\alpha_2=0,\alpha_3=-(k-1) \text{, when }f \text{ is a holomorphic form of weight } k; \\ &\alpha_1=2t_j,\alpha_2=0,\alpha_3=-2t_j \text{, when }f \text{ is a Maass form of Laplace eigenvalue } 1/4+t^2_j.\end{split}

My question is how about the symmetric cube lifts $\text{sym}^3f$ ? If one now assumes that $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ are the associated Langlands parameters of $\text{sym}^3f$, what are the exact shapes of theses parameters when $f$ being a holomorphic form (resp. a Maass form)?

If some expert leans something on this question, please give some comments or guide a reference. Great great thanks in advance! Your help is highly appreciated!

I have a question concerning the Langlands parameters of the symmetric cube lifts. Lets $f$ be a $GL_2$-cusp form, and $\text{sym}^2f$ the symmetric-square lift of it. Assume that $\alpha_1,\alpha_2,\alpha_3$ refer to the associated Langlands parameters. It is known that \begin{split}&\alpha_1=k-1,\alpha_2=0,\alpha_3=-(k-1) \text{, when }f \text{ is a holomorphic form of weight } k; \\ &\alpha_1=2t_j,\alpha_2=0,\alpha_3=-2t_j \text{, when }f \text{ is a Maass form of Laplace eigenvalue } 1/4+t^2_j.\end{split}

My question is how about the symmetric cube lifts $\text{sym}^3f$ ? If one now assumes that $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ are the associated Langlands parameters of $\text{sym}^3f$, what are the exact shapes of these parameters when $f$ being a holomorphic form (resp. a Maass form)?

If some expert leans something on this question, please give some comments or guide a reference. Great great thanks in advance! Your help is highly appreciated!