﻿ Fourier Coefficients of a Class of Eta Quotients of Weight 16 with Level 12

Applied Mathematics
Vol.06 No.08(2015), Article ID:58484,67 pages
10.4236/am.2015.68133

Fourier Coefficients of a Class of Eta Quotients of Weight 16 with Level 12

Barış Kendirli

Istanbul Kultur University, Istanbul, Turkey

Email: baris.kendirli@gmail.com   Received 30 May 2015; accepted 28 July 2015; published 31 July 2015

ABSTRACT

Recently, Williams  and then Yao, Xia and Jin  discovered explicit formulas for the coef- ficients 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, by using the method of proof of Williams, we will express the even Fourier coefficients of 360 eta quotients i.e., the Fourier coefficients of the sum, f(q) + f(−q), of 360 eta quotients in terms of and .

Keywords:

Dedekind Eta Function, Eta Quotients, Fourier Series 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  (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  - . I have determined the Fourier coefficients of the theta series associated to some quadratic forms, see  -  .

Recently, Williams, see  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  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  :

(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  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  . 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.

Table 2. The eta quotients whose even coefficients can be explicitly calculated.

Remark 4 is 27 dimensional, is 33 dimensional, see  (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

Table 3. Expression of fi in terms of newforms.

taking t as a root of, we see as linear combinations in Table 3.

Cite this paper

BarışKendirli, (2015) Fourier Coefficients of a Class of Eta Quotients of Weight 16 with Level 12. Applied Mathematics,06,1426-1493. doi: 10.4236/am.2015.68133

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