Computing the dimensions of algebraic modular form spaces for F4

Our main program is F4.gp, the other two GAP programs are used for the enumeration of conjugacy classes of some permutation groups.

Reading F4.gp in PARI/GP, then you can use the function RepDim to calculate the dimensions of invariant subspaces:

Input: A vector [a, b, c, d] with non-negative entries, which means the dominant weight w=a*w_1+b*w_2+c*w_3+d*w_4 of F4 as in Bourbaki's notations.
Output: [d1, d2], where d1 is the dimension of the irreducible representation with highest weight w under the finite group G_e(Z), and d2 is that for G_e'(Z). Their sum vecsum(RepDim([a,b,c,d])) is the dimension d(w) of algebraic modular form spaces for F4 with weight w.

Here we also provide a larger table containing the list of [w, d1(w), d2(w), d(w)], where w=[a, b, c, d] is a dominant weight of F4 such that the d(w) is nonzero and 2*a+3*b+2*c+d<=40.

The conjectural formula for F4(w)

Admitting a series of conjectures of Arthur and Langlands, we have the program CountPara.gp to calculate the number F4(w) in the paper (need to load F4.gp at first):

Input: A vector [a,b,c,d] with non-negative entries and satisfying 2*a+3*b+2*c+d<=22, which means the dominant weight w=a*w_1+b*w_2+c*w_3+d*w_4 of F4 as in Bourbaki's notations.
Output: The number F4(w) of level one automorphic representations with weight w whose Sato-Tate groups are F4.

In this program we use some tables in Discrete series multiplicities for classical groups over Z and level 1 algebraic cusp forms so if you want to calculate for larger weights you need to ask them for larger tables.