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

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.

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.

• Yes, I think that will do the trick. Thanks. May 2 '13 at 12:56
• @GH: Hope this isn't a breach of etiquette, but we lack other apparent means of communication. We need this fact for a paper we're near-done with and we'd like to make reference to your help and the help of others in a short "thank you" paragraph. Would you be ok with our thanking you and, if so, would you prefer we refer to you as GH or would you rather we use your real name? May 4 '13 at 1:28
• @Thomas: Thanks for the note. I sent you an email. May 4 '13 at 8:49

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.

• @Marc: I don't think that is known, as this is the famous Selberg conjecture. A good reference about what is known numerically is Booker-Strömbergsson: Numerical computations with the trace formula and the Selberg eigenvalue conjecture May 2 '13 at 9:30