For what I have heard, Maass forms of (Laplacian) eigenvalue $1/4$ on modular surfaces are somewhat special. But I don't know where to look for explicit examples. (In fact, one form came here on MO Does anyone want a pretty Maass form?, but I am not sure I can figure out what it is from the description.) So, can anyone help me with references?
One place you can look is the paper by Booker and Strombergsson which appeared in Crelle. Their aim is to verify the Selberg eigenvalue conjecture in a number of cases, and when there are Maass forms of eigenvalue $1/4$ these must be accounted for by finding corresponding Galois representations. See Section 5 of the paper for some explicit examples; of course there is a lot of other work on this, but the paper will give you more references.