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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?

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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.

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Thank you. Indeed, this is good reference. It would be nice though if someone published coefficients of some form. –  Alex Gavrilov Apr 22 at 15:04
    
I mean, I would like coefficients and can, if necessary, compute them. But who isn't lazy, and busy? –  Alex Gavrilov Apr 22 at 15:39

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