Maass forms of weight $0$ with trivial character
A Maass form on a subgroup \(\Gamma\) of \(\GL_{2}(\R)\) is a smooth, squareintegrable, automorphic eigenfunction of the LaplaceBeltrami operator $\Delta$. In other words, $$f\in C^\infty(\mathcal{H}),\quad f\in L^2(\Gamma\backslash{\mathcal H}),\quad f(\gamma z)=f(z)\ \forall\gamma\in\Gamma,\quad (\Delta+\lambda)f(z)=0 \textrm{ for some } \lambda \in \C.$$
Maass forms of weight $k$ with trivial character
A Maass form $f$ of weight $k$ and multiplier system $v$ on a group $\Gamma$ is a smooth function $f:\mathcal{H} \rightarrow\C$ with the following properties:

it transforms according to a unitary weight \( k\) slashaction: $ f[\gamma,k]( z) = v(\gamma) f(z)$ for all $\gamma \in \Gamma$ where \( f[\gamma,k]( z) = \exp(ik\mathrm{Arg}(c z+d)) f(\gamma z)\) for $\gamma=\begin{pmatrix}a & b \\ c & d \end{pmatrix}$

it is an eigenfunction of the corresponding weight $k$ Laplacian \( \Delta_k = y^2\left( \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)  iky\frac{\partial}{\partial x} \)
If the level is equal to 1, the only possible multiplier systems are given by the Dedekind eta function. A compatible multiplier system for weight $k\in \R$ is given by
\[ v(A) = v_{\eta}^{2k}:=\eta(Az)^{2k}/\eta(z)^{2k} \]
or one of its 6 conjugates. One can show that $v(A)$ is welldefined and independent of the point $z$ in the upper halfplane.
Remark:
One can also consider Maass forms transforming with the usual "holomorphic" weight $k$ slashaction. In this case the corresponding weight $k$ Laplacian will look slightly different: \( \Delta_k = y^2\left( \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)  iky\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right). \) This convention is usually used in the context of Harmonic weak Maass forms.
 Review status: reviewed
 Last edited by Nathan Ryan on 20190501 11:07:32
 lfunction.underlying_object
 mf.maass.mwf.dimension
 mf.maass.mwf.eigenvalue
 mf.maass.mwf.fourierexpansion
 mf.maass.mwf.level
 mf.maass.mwf.precision
 mf.maass.mwf.spectralparameter
 mf.maass.mwf.symmetry
 mf.maass.mwf.weight
 rcs.cande.lfunction
 rcs.rigor.lfunction.maass
 rcs.source.lfunction.maass
 lmfdb/lfunctions/templates/MaassformGL2.html (line 5)
 lmfdb/modular_forms/maass_forms/maass_waveforms/views/templates/mwf_browse_all_eigenvalues.html (line 156)
 lmfdb/modular_forms/maass_forms/maass_waveforms/views/templates/mwf_navigate.html (line 26)
 20190501 11:07:32 by Nathan Ryan (Reviewed)
 20190429 23:35:41 by Nathan Ryan
 20181219 06:34:33 by Alex J. Best