Number of lattice points in a hyperball

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Problem 596

Let T(r) be the number of integer quadruplets x, y, z, t such that x2 + y2 + z2 + t2r2. In other words, T(r) is the number of lattice points in the four-dimensional hyperball of radius r.

You are given that T(2) = 89, T(5) = 3121, T(100) = 493490641 and T(104) = 49348022079085897.

Find T(108) mod 1000000007.