I am interested in the relationship between the Hecke eigenvalue at $p$ and at $p^k$ for $k \geq 2$ in the unramified and ramified situation for modular/Maass forms.


More precisely, I know from a representation theoretic perspective that the Hecke eigenvalue at $p$ determines uniquely the factor of the automorphic representation at $p$, and consequently should determine the Hecke eigenvalue at $p^k$. Are there formulas, which express this quantitatively, i.e., does there exist a polynomial $P\_k$ such that
$$ \lambda( p^k) = P\_k( \lambda(p) )?$$



Hecke eigenvalue at p and at p^k
For each $k$ there is such a polynomial and can be determined recursively from the usual relation $\lambda(m)\lambda(n)=\sum\_{d\mid (m,n)}\chi(d)\lambda(mn/d^2)$, where $\chi$ is the nebentypus (for weight zero Maass newforms). Am I missing something?


Actually, for unramified primes $P\_k(x)=U\_k(x/2)$, where $U\_k$ is the $k$-th Chebyshev polynomial of the second kind, while for ramified primes $P\_k(x)=x^k$. See Section 2.3 in [Young's paper](https://arxiv.org/abs/0708.4042).


