Suppose $f$ is a Maass cusp form on $\Gamma_0(D)$. The associated symmetric square $L$-function $L(\mathrm{sym}^2 f, s)$ has a pole at $s = 1$ if and only if $f = \overline{f}$ (if $f$ is self-dual).

One way for $f$ to be self-dual is if $f$ comes from a Hecke character $\eta$ (sometimes called a grossencharacter) on a real quadratic field and $L(s, f) = L(s, \eta)$. This is a construction of Maass forms that was described by Maass. I know it better from section 1.9 of Bump's *Automorphic Forms and Representations*. I don't use this terminology often, but I think this is equivalent to $f$ being a CM Maass form, or to $f$ corresponding to a dihedral Galois representation.

Is this the

onlyway to get a self-dual Maass form?

I believe this is true, but I don't know if this is folklore or conjectural or known.