Analogues of Hecke relations for Maass forms

Question

If a (suitably normalised) holomorphic cusp newform has q-expansion $$f(z) = \sum_n \lambda_f(n) e(nz),$$

then we know the Hecke relations for $(mn,q)=1$, $$(\star) \qquad \lambda_f(m)\lambda_f(n) = \sum_{d | (m, n)} \lambda_f\left( \frac{mn}{d^2} \right).$$

In the case of a Maass cusp form, we can write the expansion $$u(z) = \sum_{n \geqslant 1} \rho_u(n) W_s(nz),$$

where $s$ is related in an explicit way to the associated eigenvalue and $W_s$ is a Whittaker function. Similarly, the continuous part of the spectrum made of Eisenstein series at a cusp $a$ admits expansion of the form $$E_a(s,z) = \phi_a y^s + \phi_a(s) y^{1-s} + \sum_{n \geqslant 1} \phi_a(n,s)W_s(nz)$$

I am interested in the following question :

Are there analogous "Hecke relations" for the coefficients of Maass forms, $\rho_u(n)$ and $\phi_a(n,s)$?

I suppose so, but I do not have any good reference for these matters.

Answer 1

Two remarks supplementing the excellent comments below your post.

1. The relation $(\star)$ holds for every $m$ and $n$ if your restrict the sum to $(d,q)=1$. This is because $f$ is a newform (and we assume that $f$ has trivial nebentypus). The same is true for Maass cuspidal newforms.

2. You can get a Hecke eigenbasis of Eisenstein series if you induce the basis vectors from Dirichlet characters modulo $q$. See Section 2.1.1 of this article for a quick introduction. I also recommend you to study Chapter 7 of Miyake's book, which discusses these induced Eisenstein series in detail and in classical language.