There's been a lot of hyperbole due to Mathematica 7.0's recent release. A colleague of mine got a personal email from Stephen Wolfram himself, asking him to try out Mathematica 7.0, and instead my colleague forwarded the message to me and remarked that it was too late, since he had switched to Sage.

I looked over the Mathematica 7.0 release notes... and noticed that they added support for computing with Dirichlet characters. I implemented the code in Magma and Sage, and wrote a chapter in my modular forms book about computing with Dirichlet characters. So I followed the "what's new" to this Mathematica page about their new functionality for Dirichlet characters. It's

*sad*. They give no way of specifying a character, except to give the "

*i*th character", which is meaningless and random (and they say so) -- that's like giving a matrix over a finite field at random. All they give is a function to evaluate characters at numbers -- they don't give functions for arithmetic with them, or computing invariant such as the conductor, which is where all the real fun comes in. Boggle. Sage is light years ahead of Mathematica here.

The Mathematica release notes also brag about finally having something for finite groups, but again it is very minimal compared to what Sage provides (via GAP). Basically all they have is a bunch of tables of groups, but no real algorithms or functionality. The whole approach seems all backwards -- first one should implement algorithms for computing with groups, then use them to make tables in order to really robustify the algorithms, then compare those tables to existing tables, etc. I wonder whether the group theory data in Mathematica was computed using Gap or Magma?