 
Summary: On the 2n
divisibility of the Fourier coefficients of
Jq functions and the Atkin conjecture for p = 2
Shigeki Akiyama
College of General Education, Niigata University
Niigata, 95021, JAPAN
§1. Introduction
Let f be the holomorphic modular form of weight 2k, which is a normalized
common eigenform with respect to Hecke operators. Then it is well known that the
Fourier coefficient (n) of f satisfies the equation
(np)  (n)(p) + p2k1
(n/p) = 0, (1)
for any prime p and any positive integer n. Here (n/p) is defined to be zero when
n/p is not an integer. In [2] and [3], Atkin made a similar conjecture for a modular
function:
Conjecture (Atkin).
Let j(z) be the modular invariant:
j(z) =
n1
c(n)xn
