No Exceptional Eigenvalues of Weight 1/2 Maass Forms on $\Gamma_0(4)$?

Question

Some colleagues and I were wondering if there is a citation out there which shows there are no exceptional eigenvalues, $\lambda$, of classical weight 1/2 Maass forms on $\Gamma_0(4)$, which is to say $\lambda=1/4+t^2>1/4$, so $t \in \mathbb{R}_{>0}$.

A resource where somebody has computed the lowest eigenvalue for this case, like is done in LMFDB for the weight zero case, would be optimal. We're not looking to go through the guts of analogous methods to compute this for ourselves, if we can help it.

Answer 1

A weight $1/2$ Hecke-Maass form of eigenvalue $(1-s^2)/4$ on $\Gamma_0(4)$ has a Shimura lift to a weight $0$ even Hecke-Maass form of eigenvalue $1/4-s^2$ on $\Gamma_0(1)$, see e.g. the proof of Theorem 1.5 in Baruch-Mao: A generalized Kohnen-Zagier formula for Maass forms (manuscript here). So the database for weight $0$ forms furnishes the required information for weight $1/2$ forms.

Answer 2

Edit: Oh, I see only now that you are interested in half weight Maass forms (damn cellphone browsers;). My answer applies to weight zero Maass forms only. Probably the Shimura lift GH addresses generalizes, thus, translating half weight Maass forms to integer weight one of some different (lower?) level.

For $\Gamma_1(n)$ and $n\leq 18$, the Selberg eigenvalue conjecture for weight zero/even Maass forms is due to Huxley (1985). All eigenvalues are $> 1/4$ here.

Check for example page 12 in Blomer, Brumley - The role of the Ramanujan conjecture in analytic number theory, Bulletin AMS 50 (2013), 267-320

Booker and Strömbergsson verified the Selberg eigenvalue conjecture for weight zero Maass forms for $\Gamma_1(n)$ and $n \leq 857$ squarefree.

For weight one/odd Maass forms, the generalization of the Selberg eigenvalue conjecture holds trivially, because the infinite component of the corresponding automorphic representation is a ramified principal series. These are all tempered.

There exists an even Maass form of eigenvalue $1/4$ for $\Gamma(23)$, I was told, because the class group of $\mathbb{Q}( \sqrt{-23})$ is $\mathbb{Z}/3$.

Using the Shimura lift (as GH) mentions, this yields similar results for half integer weight forms.