A time ago I was intrigued by the following remark: "... one ends up with a (non-sparse) system of equations in about 10000 real variables. One important practical point is that solving such systems is now well within the capabilities of a standard desctop PC;...". The above is a quote from Notices of AMS 55 no. 9 (October 2008), page 1093 (left). Occasionally, I have to solve a system or two which are not the size of an undergraduate excercise in linear algebra. In my experience, Maple become pretty useless for the purpose as soon as the number of variables approaches 1000.

*QUESTION*: What kind of software I need to solve such a big system
on a "standard desctop PC"?

P.S. It's a pity that in the quoted article ("Uncovering a New L-function" by A.R.Booker) there is no single word about this.