On the Modular Equation of j(z)1/3

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In our previous papers [6], [7], we have studied the modular equation <I>((X, Y) of j(z) where j (z) is the most basic modular function with respect to SL2 (Z). Now we study about the modular equation <I>~3) (X, Y) of j (z) 1/3 which is a modular function for r(3). Especially, we have otained the explicit form of <I>~3\X, Y) for all primes f :::; 131 and found that certain ongruences of their coefficients (like those noted in [6]) hold for f E P = {2, 5, 7,13,19, 31}. This is remarkable since if we include 3 in P then these primes in P coincide with those primes that arise in connection to the Monster simple group.

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