For each Maass newform $f$, we use domain coloring to show the plot of $f$. The form $f$ is approximated using the first $50$ coefficients across a $200 \times 200$ mesh.
At each point $z$ in the unit disk $\mathcal{D}$, we assign a color based on the value of $f(\mu(z))$, where $\mu$ is the Möbius transform $\mu: \mathcal{D} \longrightarrow \mathcal{H}$ given by $\mu(z) = \frac{1 - iz}{z - i}$. This transform preserves the apparent orientation of the imaginary axis in $\mathcal{H}$: the point $(0, -1)$ in $\mathcal{D}$ corresponds to $0$ in $\mathcal{H}$, the center of $\mathcal{D}$ corresponds to $i$ in $\mathcal{H}$, and the point $(0, 1)$ in $\mathcal{D}$ corresponds to $i\infty$ in $\mathcal{H}$.
Writing $f(\mu(z)) = re^{i \theta}$, the hue is determined by the angle $\theta$. In addition, small changes in lightness/darkness represent logarithmically spaced contours: consecutive contours correspond to magnitudes $r$ that differ by a factor of $2$. The relationship is implicit, but these Maass forms all vanish near the top of the disk.
If the Maass form $f$ has squarefree level, then $f$ takes only real values. In this case, the portrait will have only two colors, corresponding to positive and negative values.
Maass form portraits are made in essentially the same way as classical modular form portraits, except they use a set of colors that looks better when the Maass form is real-valued. See [MR:4427977] for more on the methods of visualization.
Code that makes this type of portrait is available at github.com/davidlowryduda/maass_portraits.