This paper continues the work done in Olofsson [Commun Math Phys 286(3):1051-1072, 2009] about the supremum norm of eigenfunctions of desymmetrized quantized cat maps. N will denote the inverse of Planck's constant and we will see that the arithmetic properties of N play an important role. We prove the sharp estimate parallel to psi parallel to(infinity) - O(N-1/4) for all normalized eigenfunctions and all N outside of a small exceptional set. We are also able to calculate the value of the supremum norms for most of the so called newforms. For a given N = p(n), with n > 2, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about 2/root 1 +/- 1/p and the supremum norm of the newforms in the second half have at most three different values, all of the order N-1/6. The only dependence of A is that the normalization factor is different if A has eigenvectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms.
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