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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 holds trivially, because the infinite component of the corresponding automorphic representation is an unramified 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.

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.

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 $n\leq 18$, this 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 $\Gamma_1(n)$ and $n \leq 857$ squarefree.

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

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 holds trivially, because the infinite component of the corresponding automorphic representation is an unramified 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.

Edit: Oh, I see only now that you are interested in half weight Maass forms. My answer applies to weight zero Maass forms only.

For $\Gamma_1(n)$ and $n\leq 18$, this 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 Strombergsson verified the Selberg eigenvalue conjecture for $\Gamma_1(n)$ and $n \leq 857$ squarefree.

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

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 $n\leq 18$, this 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 $\Gamma_1(n)$ and $n \leq 857$ squarefree.

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