Fourier Coefficients of a Class of Eta Quotients of Weight 16 with Level 12 ()
The divisor function is defined for a positive integer i by
(1)
The Dedekind eta function is defined by
(2)
where
(3)
And an eta quotient of level n is defined by
(4)
It is interesting and important to determine explicit formulas of the Fourier coefficients of eta quotients since they are the building blocks of modular forms of level n and weight k. The book of Köhler [3] (Chapter 3, p. 39) describes such expansions by means of Hecke Theta series and develops algorithms for the determination of suitable eta quotients. One can find more information in [4] -[8] . I have determined the Fourier coefficients of the theta series associated to some quadratic forms, see [9] - [14] .
Recently, Williams, see [1] discovered explicit formulas for the coefficients of Fourier series expansions of a
class of 126 eta quotients in terms of and. One example is as follows:
gives the expansion found by Williams.
Then Yao, Xia and Jin [2] expressed the even Fourier coefficients of 104 eta quotients in terms of
and. One example is as follows:
where the even coefficients are obtained. Motivated by these two results, we find that we can express the even Fourier coefficients of 360 eta quotients in terms of and,
see Table 2. One example is as follows:
We see that the odd Fourier coefficients of 875 eta quotients are zero and even coefficients can be expressed by simple formula. Let
Now we can state our main Theorem:
Theorem 1 Let be non-negative integers satisfying
(5)
Define the integers by
(6)
(7)
(8)
(9)
(10)
(11)
They are functions of q by (3). Now define integers
by
(12)
(13)
(14)
(15)
(16)
Define the rational numbers
and as in Table 1. Here
and
Table 1. Coefficients of eisenstein series and some eta quotients.
where for
In particular,
for
Proof. It follows from (6)-(11) that
(17)
(18)
Now we will use p-k parametrization of Alaca, Alaca and Williams, see [15] :
(19)
where the theta function is defined by
Setting x = p in (12), and multiplying both sides by k16 we obtain
Alaca, Alaca and Williams [16] have established the following representations in terms of p and k:
(20)
(21)
(22)
(23)
(24)
(25)
Therefore, since
we immediately obtain:
It is easy to check the following expressions by (20)-(25)
Obviously, are functions of q, see (3), (19). We see that
by [17] . Now
where
So
Therefore, for
since it is easy to see that
hence,
and, for
Remark 2 We have found 360 eta quotients, see Table 2, such that, for
and 875 eta quotients, such that for
Remark 3 If f is an eta quotient, then is also an eta quotient, so the coefficients of
are exactly the even coefficients of f. In particular, it means that we have obtained all coefficients of some sum of 360 eta quotients.
Remark 4 is 27 dimensional, is 33 dimensional, see [18] (Chapter 3, p. 87 and Chapter 5, p. 197), and generated by
where is the unique newform in; is the unique newform in;
are the unique newforms in, is the unique newform in, are
the unique newforms in and are the unique newforms in. By
taking t as a root of, we see as linear combinations in Table 3.