be a newform of weight for the full modular group, and let

be a cusp form of weight and level 4 which is associated to by the Shimura map. We normalize by requiring that and we normalize by requiring that

Then for squarefree with we have, by the formula of Kohnen and Zagier,

where

In particular, the Riemann Hypothesis for implies that

for an appropriate choice of . If the bound

holds, then a similar bound for will hold (but with replaced by ). The question here is to decide which (if either) of these bounds represents the true state of affairs.

If has a square factor, then can be determined from values of where .
It is known, by Deligne's theorem, that

where is the number of divisors of . Of course, so that we can write

where is real. The conjecture of Sato and Tate about the distribution of the asserts that

for . A consequence of the Sato-Tate conjecture is that there are infinitely many (actually a positive proportion of ) for which

for any given . The maximum order of is given by

The Sato-Tate conjecture implies the same assertion with replaced by .

Thus, in particular, we find that

and we expect this bound to be sharp in the sense that the cannot be replaced by anything smaller.

Thus, if the ``smaller'' bound is true for then there is a great variation between the distribution of the Fourier coefficients according to whether is square or squarefree.

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