This question has its origins in this entertaining discussion on MO.

There are many programs (CAS) and libraries that are able to handle algebraic expressions. These are both a verification tool for (sometimes nightmarish) computations and a way to explore objects for which a good intuition is not available.

Among them, I have in mind Mathematica, Maple, Magma, and also the Python-interface Sage (including lots of packages like NumPy, SciPy, Maxima, GAP, etc.). Sage has the appeal to be free and open source, however I wonder if it is at the same level of the others.

To narrow the question to a more specific field, for every language has its advantages somewhere, I am interested in automorphic forms and number theory. Thus the basic uses will be to manipulate automorphic forms for different groups and congruence subgroups, to locate zeros of the associated $L$-functions, compute Fourier coefficients, etc.

What are the pros and cons of these programs to work with automorphic forms?

Every direction of answer is welcome, in particular taking into account

• ease of use
• available literature (not on the program itself, but related to automorphic forms)
• regular updating of packages and functions
• community size for support and discussion