Non-liftable weight one modular forms over  $\mathbb{F}_p$

This section was written by Gabor Wiese.

This problem is closely connected to the "big images" challenge, and could/should be treated in collaboration. Modular forms of weight 1 over $\mathbb{F}_p$ behave completely differently from forms of higher weights. One feature is that they are very often NOT reductions of holomorphic modular forms. In the course it will be explained how to compute modular forms of weight one. By looking at the image of a weight one form, one can often prove that it is such a non-liftable form. So far, there are many examples over $\mathbb{F}_2$ , but only one example for an odd prime, namely for $p=199$ .

Problem 2.2.1   Find examples in small odd characteristics!