_{1}

^{*}

Here, we determine formulae, for the numbers of representations of a positive integer by certain sextenary quadratic forms whose coefficients are 1, 2, 3 and 6.

The divisor function

The Dedekind eta function and the theta function are defined by

where

and an eta quotient of level N is defined by

It is important and interesting to determine explicit formulas of the representation number of positive definite quadratic forms.

Here we give the following Lemma, see ( [

Lemma 1. An eta quotient of level N is a meromorphic modular form of weight

a)

b)

c)

For

Clearly

Now, let’s consider sextenary quadratic forms of the form

where

We write

Formulae for

First, by the following Theorem, we characterize the facts that

are in

Theorem 1. Let

where,

Moreover, it is in

Proof. It follows from the Lemma 1, holomorphicity criterion in ( [

(0 2 2 12), | (0 2 4 10), | (0 2 6 8), | (0 2 8 6) |
---|---|---|---|

(0 2 10 4), | (0 2 12 2), | (0 2 14 0), | (0 4 2 10) |

(0 4 4 8), | (0 4 6 6), | (0 4 8 4), | (0 4 10 2) |

(0 4 12 0), | (0 6 2 8), | (0 6 4 6), | (0 6 6 4) |

(0 6 8 2), | (0 6 10 0), | (0 8 2 6), | (0 8 4 4) |

(0 8 6 2), | (0 8 8 0), | (0 10 2 4), | (0 10 4 2) |

(0 10 6 0), | (0 12 2 2), | (0 12 4 0), | (0 14 2 0) |

(1 1 1 13), | (1 1 3 11), | (1 1 5 9), | (1 1 7 7) |

(1 1 9 5), | (1 1 11 3), | (1 1 13 1), | (1 3 1 11) |

(1 3 3 9), | (1 3 5 7), | (1 3 7 5), | (1 3 9 3) |

(1 3 11 1), | (1 5 1 9), | (1 5 3 7), | (1 5 5 5) |

(1 5 7 3), | (1 5 9 1), | (1 7 1 7), | (1 7 3 5) |

(1 7 5 3), | (1 7 7 1), | (1 9 1 5), | (1 9 3 3) |

(1 9 5 1), | (1 11 1 3), | (1 11 3 1), | (1 13 1 1) |

(2 0 0 14), | (2 0 2 12), | (2 0 4 10), | (2 0 6 8) |

(2 0 8 6), | (2 0 10 4), | (2 0 12 2), | (2 0 14 0) |

(2 2 0 12), | (2 2 2 10), | (2 2 4 8), | (2 2 6 6) |

(2 2 8 4), | (2 2 10 2), | (2 2 12 0), | (2 4 0 10) |

(2 4 2 8), | (2 4 4 6), | (2 4 6 4), | (2 4 8 2) |

(2 4 10 0), | (2 6 0 8), | (2 6 2 6), | (2 6 4 4) |

(2 6 6 2), | (2 6 8 0), | (2 8 0 6), | (2 8 2 4) |

(2 8 4 2), | (2 8 6 0), | (2 10 0 4), | (2 10 2 2) |

(2 10 4 0), | (2 12 0 2), | (2 12 2 0), | (2 14 0 0) |

(3 1 1 11), | (3 1 3 9), | (3 1 5 7), | (3 1 7 5) |

(3 1 9 3), | (3 1 11 1), | (3 3 1 9), | (3 3 3 7) |

(3 3 5 5), | (3 3 7 3), | (3 3 9 1), | (3 5 1 7) |

(3 5 3 5), | (3 5 5 3), | (3 5 7 1), | (3 7 1 5) |

(3 7 3 3), | (3 7 5 1), | (3 9 1 3), | (3 9 3 1) |

(3 11 1 1), | (4 0 0 12), | (4 0 2 10), | (4 0 4 8) |

(4 0 6 6), | (4 0 8 4), | (4 0 10 2), | (4 0 12 0) |

(4 2 0 10), | (4 2 2 8), | (4 2 4 6), | (4 2 6 4) |

(4 2 8 2), | (4 2 10 0), | (4 4 0 8), | (4 4 2 6) |

(4 4 4 4), | (4 4 6 2), | (4 4 8 0), | (4 6 0 6) |

(4 6 2 4), | (4 6 4 2), | (4 6 6 0), | (4 8 0 4) |

(4 8 2 2), | (4 8 4 0), | (4 10 0 2), | (4 10 2 0) |

(4 12 0 0), | (5 1 1 9), | (5 1 3 7), | (5 1 5 5) |

(5 1 7 3), | (5 1 9 1), | (5 3 1 7), | (5 3 3 5) |

(5 3 5 3), | (5 3 7 1), | (5 5 1 5), | (5 5 3 3) |

(5 5 5 1), | (5 7 1 3), | (5 7 3 1), | (5 9 1 1) |

(6 0 0 10), | (6 0 2 8), | (6 0 4 6), | (6 0 6 4) |
---|---|---|---|

(6 0 8 2), | (6 0 10 0), | (6 2 0 8), | (6 2 2 6) |

(6 2 4 4), | (6 2 6 2), | (6 2 8 0), | (6 4 0 6) |

(6 4 2 4), | (6 4 4 2), | (6 4 6 0), | (6 6 0 4) |

(6 6 2 2), | (6 6 4 0), | (6 8 0 2), | (6 8 2 0) |

(6 10 0 0), | (7 1 1 7), | (7 1 3 5), | (7 1 5 3) |

(7 1 7 1), | (7 3 1 5), | (7 3 3 3), | (7 3 5 1) |

(7 5 1 3), | (7 5 3 1), | (7 7 1 1), | (8 0 0 8) |

(8 0 2 6), | (8 0 4 4), | (8 0 6 2), | (8 0 8 0) |

(8 2 0 6), | (8 2 2 4), | (8 2 4 2), | (8 2 6 0) |

(8 4 0 4), | (8 4 2 2), | (8 4 4 0), | (8 6 0 2) |

(8 6 2 0), | (8 8 0 0), | (9 1 1 5), | (9 1 1 3) |

(9 1 5 1), | (9 3 1 3), | (9 3 3 1), | (9 5 1 1) |

(10 0 0 6), | (10 0 2 4), | (10 0 4 2), | (10 0 6 0) |

(10 2 0 4), | (10 2 2 2), | (10 2 4 0), | (10 4 0 2) |

(10 4 2 0), | (10 6 0 0), | (11 1 1 3), | (11 1 3 1) |

(11 3 1 1), | (12 0 0 4), | (12 0 2 2), | (12 0 4 0) |

(12 2 0 2), | (12 2 2 0), | (12 4 0 0), | (13 1 1 1) |

(14 0 0 2), | (14 0 2 0), | (14 2 0 0), | (16 0 0 0) |

that

The condition

Now let,

Theorem 2. The set

is a basis of

the two unique newforms in

and the three unique newforms in

Proof.

where

As a consequence of this Theorem, we have obtained the following Corollary.We have used Magma for the calculations.

The following representation numbers formulae are valid.

Barış Kendirli, (2016) Representations by Certain Sextenary Quadratic Forms Whose Coefficients Are 1, 2, 3 and 6. Advances in Pure Mathematics,06,212-296. doi: 10.4236/apm.2016.64018