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 :

Is 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.

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.