Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12
Baris Kendirli
Dept. of Math,. Aydın University, Istanbul, Turkey
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To cite this article:
Barış Kendirli. Fourier Coefficients of a Class of Eta Quotients of Weight 18 with Level 12.Pure and Applied Mathematics Journal.Vol.4, No. 4, 2015, pp. 178-188. doi: 10.11648/j.pamj.20150404.17
Abstract: Williams [16] and later Yao, Xia and Jin[15] discovered explicit formulas for the coefficients of the Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quotients in terms of and and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of 104 eta quotients in terms of and Here, we will express the even Fourier coefficients of 324 eta quotients in terms of and
Keywords: Dedekind Eta Function, Eta Quotients, Fourier Series
1. Introduction
The divisor function is defined for a positive integer by
(1)
The Dedekind eta function is defined by
(2)
Where
(3)
and an eta quotient of level is defined by
(4)
It is interesting and important to determine explicit formulas of the Fourier coefficients of eta quotients, because they are the building blocks of modular forms of level n and weight k. The book of Köhler [13] (Chapter 3, pg.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 [3], [6], [14]. I have determined the Fourier coefficients of the theta series associated to some quadratic forms, see [7], [8], [9][10], [11] and [12].
It is known that Williams, see [16] 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 [15] expressed the even Fourier coefficients of 104 eta quotients in terms of and One example is as follows:
where the even coefficients are obtained. After that we find that we can express the even Fourier coefficients of 324 eta quotients in terms of and see Table 3. One example is as follows:
We also see that the odd Fourier coefficients of 650 eta quotients are zero and even coefficients can be expressed by simple formula.
2. Main Body
Now 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 by (3). Now define integers
by
(12)
(13)
(14)
(15)
(16)
Define the rational numbers
and as in www.fatih.edu.tr/~bkendirli/weight18/Table 1. Here and
where for n
In particular,
for n
Proof. It follows from (6-11) that
(17)
(18)
Now we will use p-k parametrization of Alaca, Alaca and Williams, see [1]:
(19)
where the theta function is defined by
etting in (12), and multiplying both sides by we obtain
Alaca, Alaca and Williams [2] have established the following representations in terms of p and k:
(20)
(21)
(22)
(23)
(24)
(25)
Therefore, since
we have
We can also similarly determine and in terms of and as in www.fatih.edu.tr/~bkendirli/weight18/Table 3. Obviously, are functions of , see (3),( 19). We see that by [4]. Now
where
So
Therefore, for n=1,2,…
since it is easy to see that
hence,
and, for n=1,2,…
3. Conclusion
1. We have found 324 eta quotients, see www.fatih.edu.tr/~bkendirli/weight18/Table 4, such that, for n=1,2,
and 650 eta quotients, such that for n=1,2,
2. is 31 dimensional, is 37 dimensional,see [5] (Chapter 3, pg.87 and Chapter 5, pg.197), and generated by
where is the unique newform in ; is the unique newform in ; and are the unique newforms in , are the unique newforms in , and are the unique newforms in and are the unique newforms in . By taking as a root of , and as the root of we see as linear combinations of this basis in www.fatih.edu.tr/~bkendirli/weight18/Table 3.
References