An enormous factorial

For any prime p the number N(p,q) is defined by N(p,q) = ∑_{n=0 to q} T_n*pⁿ with T_n generated by the following random number generator:

S₀ = 290797 S_n+1 = S_n² mod 50515093 T_n = S_n mod p

Let Nfac(p,q) be the factorial of N(p,q). Let NF(p,q) be the number of factors p in Nfac(p,q).

You are given that NF(3,10000) mod 3²⁰=624955285.

Find NF(61,10⁷) mod 61¹⁰

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