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\begin{document}
\title{\large{\gt ͭ���ξ�λؿ��·׻����르�ꥺ��}}
\author{��ǰ ����}
\date{1997.10.21}
\maketitle

\noindent{\large \S {\bf 1. }{\gt �Ϥ����}}
%(section 1)

\bigskip
$p$ ���ǿ�, $q=p^k$ $(k\geq 2)$, ${\bf F}_q$ �򸵤θĿ� $q$ ��ͭ���ΤȤ���. �ܹƤǰ���ͭ���� ${\bf F}_q$ ��λؿ��¤Ȥ�, ���η����¤Τ��ȤǤ���:
$$S(f)=\sum_{x\in{\bf F}_q}e\left(\tr f(x)\right),$$
������ $f(X)\in{\bf F}_q[X]$, $e(y)=e^{\frac{2\pi iy}p}$, �ޤ� $\alpha\in{\bf F}_q$ ���Ф���
$$\tr\alpha=\alpha+\alpha^p+\cdots +\alpha^{p^{k-1}}\in{\bf F}_p.$$
������, ���ˤ�ä� $x$ ��ư���ϰϤ� ${\bf F}_q^\times$ �Τ��Ȥ⤢��. 
����򼡤Τ褦�ʷ����ѷ����뤳�Ȥ���ɸ�Ȥ���:
$$\sum_{r=0}^{p-1}t(r)e(r),$$
������,
$$t(r):=\sharp\left\{x\in{\bf F}_q \ |\  \tr f(x)=r\right\}
\quad(0\leq r\leq p-1).$$
���Τ褦�ʷ׻���, ����ͭ���ξ��¿�༰�η׻���ޤळ�Ȥˤʤ뤬, �׻������Τ���Ƥ���, $\tr$ �Ȥ����ؿ��������򤦤ޤ����Ѥ��뤳�Ȥˤ��, �¼�Ū�ˤ�¿�༰�η׻��ϤۤȤ��ɬ�פʤ�, �������ä�C����Τ褦��, ¿�༰�׻��ε�ǽ�������Ƥ��ʤ��ץ�����ߥ󥰸���Ǥ��ñ��®���ץ�������񤯤��Ȥ��Ǥ���. ������ˡ�ˤĤ��ƽҤ٤�Τ��ܹƤ���Ū�Ǥ���. �ޤ�, ������ˡ�κǤ��ܼ�Ū����ʬ�Ͼ嵭�� $\tr\alpha$ �η׻��Ǥ���. �������äƻؿ��°ʳ��ˤ�, trace �η׻���ɬ�פʼ������˱��ѤǤ���. ���Τ褦�ʱ��Ѥ�1�ĤˤĤ���, ��5��ǽҤ٤�. �����, �����θ����������˴ؤ��뿷����ͽ�ۤ�ޤ�Ǥ���. �ʤ�, ��5������Ƥ�, ���潸��λ�忷���˹Ԥʤä��׻��η�̤Ǥ��뤳�Ȥ��Ǥꤷ�Ƥ���. 

\bigskip
\noindent{\large \S {\bf 2. }{\gt �� ��}}
%(section 2)

\bigskip
������Ǥ�ͭ���Τ˴ؤ������Ū�ʻ��¤��������. 
ͭ���� ${\bf F}_q$ ($q=p^k$)��, ���� ${\bf F}_p$ ���¿�༰ $X^q-X$ �κ����Τν���Ǥ���, ${\bf F}_p$ �� $k$ ������Ǥ���. ��ˡ�� ${\bf F}_q^\times$ �ϰ̿� $q-1$ �ν�󷲤Ǥ���. ${\bf F}_q^\times$ �������� $\gamma$ �� ${\bf F}_q$ ��{\gt ���ϸ�}�ȸƤ�. �ޤ�, ���ϸ��򺬤ˤ�� ${\bf F}_p$ ���¿�༰��, $X^q-X$ ���� $k$ ���δ���¿�༰�Ǥ��뤬, �����{\gt ����¿�༰}�ȸƤ�(����¿�༰��ɬ�����⸶��¿�༰�ǤϤʤ�. �º�, ${\bf F}_p$ ��� $k$ ������¿�༰�� 
$\displaystyle\frac1k\sum_{d|k}\mu(d)q^{\frac kd}$ �Ĥ��뤬, ���Τ���
 $\displaystyle\frac{\varphi(q-1)}k$ �Ĥ�����¿�༰�Ȥʤ�. ������, $\mu(d)$ �� M\"obius �δؿ�, $\varphi(n)$ �� Euler �δؿ�. $cf.$ MacWilliams-Sloane \cite{MS}, p.115. ���Τ�����λ���ˤĤ��ƤϤΤ��ۤɶ�����Ǵѻ�����). 

�ʲ�, ����¿�༰ $P(X)$ �Ȥ��κ� $\gamma$ ��1�ĸ��ꤷ, ${\bf F}_q^\times$ �θ���2�����ɽ����ˡ��Ƴ������. 

\bigskip
\noindent{\gt ����٤���ɽ��}

\medskip
��󷲤Ȥ��Ƥ�ɽ����, ${\bf F}_q^\times$ �γƸ��� $\gamma$ �Τ٤���ɽ����ˡ�Ǥ���:
$${\bf F}_q^\times=\{1, \gamma, \gamma^2, \cdots, \gamma^{q-2}\}.$$
����� ${\bf F}_q^\times$ �Τ٤���ɽ���ȸƤ֤��Ȥˤ���.

\bigskip
\noindent{\gt ����¿�༰ɽ��}

\medskip
$1, \gamma, \gamma^2, \cdots, \gamma^{k-1}$ �� ${\bf F}_p$ ��1����Ω�Ǥ��뤳�Ȥ���, ${\bf F}_q^\times$ �γƸ��� $\gamma$ �� $k-1$ ���ʲ���(${\bf F}_p$ ���)¿�༰��, ����1�̤��ɽ�����Ȥ��Ǥ���. ����� ${\bf F}_q^\times$ ��¿�༰ɽ���ȸƤ֤��Ȥˤ���. $\gamma^j$ ($0\leq j \leq q-2$) �Ȥ٤���ɽ��������Τ� $P(\gamma)$ �ǳ�ä�;�������, ���줬 $\gamma^j$ ��¿�༰ɽ���Ȥʤ�. 

\bigskip
\noindent{\gt ��} {\bf 2.1.} ${\bf F}_8.$

\medskip
$X^8-X$ �� ${\bf F}_2[X]$ �ˤ�����
$$X^8-X=X(X+1)(X^3+X^2+1)(X^3+X+1)$$
�Ȱ���ʬ�򤵤��. 3������¿�༰��2�Ĥ���($X^3+X^2+1$ �� $X^3+X+1$), �ɤ���⸶��¿�༰�Ǥ���. �㤨�� $P(X)=X^3+X+1$ �Ȥ�, ���κ���1�Ĥ� $\gamma$ �Ȥ����, 

\begin{center}
\begin{tabular}{rcr}
{�٤���ɽ��\ }&{\qquad}&{\ \ \ ¿�༰ɽ��}\\ \hline
 & & \\
$1=\gamma^0$ & {\qquad\qquad}& $1$\\
$\gamma\phantom{^0}$ & {\qquad}& $\gamma\phantom{\ +1}$\\
$\gamma^2$ & {\qquad}& $\gamma^2\phantom{\ +\gamma+1}$\\
$\gamma^3$ & {\qquad}& $\gamma^3=\ \phantom{\ \gamma^2+}\gamma+1$\\
$\gamma^4$ & {\qquad}& $\gamma^2+\gamma\ \phantom{+1}$\\
$\gamma^5$ & {\qquad}& $\gamma^3+\gamma^2=\ \gamma^2+\gamma+1$\\
$\gamma^6$ & {\qquad}& $\gamma^3+\gamma^2+\gamma=\ \gamma^2\ \phantom{+\gamma}+1$\\
\hline
\end{tabular}
\end{center}

\noindent{\gt ��} {\bf 2.2.}  ${\bf F}_9.$

\medskip
$X^9-X$ �� ${\bf F}_3[X]$ �ˤ�����
$$X^9-X=X(X-1)(X+1)(X^2+1)(X^2+X-1)(X^2-X-1)$$
�Ȱ���ʬ�򤵤��. 2������¿�༰��3�Ĥ��뤬, ���Τ��� $X^2+1$ �ϸ���¿�༰�ǤϤʤ�. �º�, $X^2+1$ �κ��� ${\bf F}_9^\times$ �ΰ̿�4����ʬ������������. ¾��2�Ĥϸ���¿�༰�Ǥ���. ������ $P(X)=X^2-X-1$ �Ȥ�, ���κ���1�Ĥ� $\gamma$ �Ȥ����, 

\begin{center}
\begin{tabular}{rcr}
{�٤���ɽ��\ }&{\qquad}&{\ \ \ ¿�༰ɽ��}\\ \hline
 & & \\
$1=\gamma^0$ & {\qquad\qquad}& $1$\\
$\gamma\phantom{^0}$ & {\qquad}& $\gamma\phantom{\ +1}$\\
$\gamma^2$ & {\qquad}& $\gamma^2=\ \ \gamma+1$\\
$\gamma^3$ & {\qquad}& $\gamma^2+\gamma=\ 2\gamma+1$\\
$\gamma^4$ & {\qquad}& $2\gamma^2+\gamma=\ \phantom{\ 2\gamma+}2$\\
$\gamma^5$ & {\qquad}& $2\gamma\phantom{\ +1}$\\
$\gamma^6$ & {\qquad}& $2\gamma^2=\ 2\gamma+2$\\
$\gamma^7$ & {\qquad}& $2\gamma^2+2\gamma=\ \ \gamma+2$\\
\hline
\end{tabular}
\end{center}


\bigskip
ͭ���ΤˤĤ��Ƥ��ܤ������Ȥ�, �㤨�� Lidl-Niederreiter \cite{LN}, MacWilliams-Sloane \cite{MS}, Chap.4, ʿ�� \cite{H1}, ��3�����򻲾�.

\bigskip
\noindent{\large \S {\bf 3. }{\bf trace} {\gt �Ȼؿ��¤η׻�}}
%(section 3)

\bigskip
ͭ���� ${\bf F}_q$ ($q=p^k$)�Ȥ��θ���¿�༰ $P(X)$, ���κ� $\gamma$ ��1�ĸ��ꤹ��. �� $\gamma^j$ ($0\leq j \leq q-2$) ��¿�༰ɽ���� $h_j(\gamma)$ �Ȥ���. ���ΤȤ�, ����ˤ��
\begin{eqnarray*}
\tr(\gamma^j)&=&\gamma^j+\gamma^{jp}+\cdots+\gamma^{jp^{k-1}}\\
             &=&h_j(\gamma)+h_{jp}(\gamma)+\cdots +h_{jp^{k-1}}(\gamma) 
\end{eqnarray*}
($h$ ��ź���� mod $q-1$ �Ǹ���). ���μ���, ���� $1, \gamma, \gamma^2, \cdots, \gamma^{k-1}$ ��ɽ���� ${\bf F}_p$ ��Υ٥��ȥ�Ȥߤ��, ���դ� $\tr(\gamma^j)$ �� ${\bf F}_p$ �θ�������, ���դˤ����� $\gamma$ ��1���ʾ�ι�Ϥ��٤�0 ($=p$ ���ܿ�)�Ǥʤ���Фʤ�ʤ�. ¨��, $\tr(\gamma^j)$ �η׻���ɬ�פʤΤ�, �� $h_j(\gamma)$ �����������Ǥ���:

\bigskip
\noindent{\gt �� ��} {\bf 3.1.}
$$\tr(\gamma^j)=h_j(0)+h_{jp}(0)+\cdots +h_{jp^{k-1}}(0).$$


\bigskip
\noindent{\gt ��} {\bf 3.2.} ${\bf F}_8.\quad$ 
������2.1�η׻����Ѥ��Ƽºݤˤ����Ĥ��θ��ˤĤ��� trace ����Ƥߤ��, ���Τ褦�ˤʤ�:
\begin{eqnarray*}
\tr(\gamma)&=&\gamma+\gamma^2+\gamma^4\\
               &=&\gamma+\gamma^2+(\gamma^2+\gamma)=2\gamma^2+2\gamma=0\\
  \\
\tr(\gamma^3)&=&\gamma^3+\gamma^6+\gamma^{12}\\
                 &=&\gamma^3+\gamma^6+\gamma^5\\
&=&(\gamma+1)+(\gamma^2+1)+(\gamma^2+\gamma+1)=2\gamma^2+2\gamma+3=1,
 \quad\mbox{etc.}
\end{eqnarray*}


\bigskip
\noindent{\gt ��} {\bf 3.3.} ${\bf F}_9.$
\begin{eqnarray*}
\tr(\gamma)  &=&\gamma+\gamma^3=\gamma+(2\gamma+1)=3\gamma+1=1\\
\tr(\gamma^2)&=&\gamma^2+\gamma^6=(\gamma+1)+(2\gamma+2)=3\gamma+3=0\\
\tr(\gamma^5)&=&\gamma^5+\gamma^{15}\\
                   &=&\gamma^5+\gamma^7=2\gamma+(\gamma+2)=3\gamma+2=2,
\quad\mbox{etc.}
\end{eqnarray*}


\bigskip
\noindent �ºݤΥץ������Ǥ�, �� $h_j(\gamma)$ ����������˵��Ƥ��������� x[$j$] ���ݴɤ�, 
$$\tr(\gamma^j)=\mbox{x[$j$]}+\mbox{x[$jp$ (mod $q-1$)]}+\cdots+\mbox{x[$jp^{k-1}$ (mod $q-1$)]}\quad(\mbox{mod }p)$$
�ˤ�äƷ׻�����Ф褤. 

���Τ���, ��ñ�Τ��� $S(f)$ �� ${\bf F}_q^\times$ ����¤Ǥ���Ȥ���:
$$S(f)=\sum_{x\in{\bf F}_q^\times}e\left(\tr f(x)\right).$$
$f(X)=a_mX^m+a_{m-1}X^{m-1}+\cdots +a_0\quad(a_j\in{\bf F}_q)$ �Ȥ���. �ؿ� tr ������������, $\tr f(X)$ �����ˤ�, �� $\tr(a_iX^i)$ �����, ���Ȥ�­���Ƥ��Ф褤. ���Τ褦�˸��ϸ� $\gamma$ ��Ȥ�, $a_i=\gamma^{n_i}$ �Ȥ�����, $X=\gamma^j$ �ΤȤ�, 
\begin{eqnarray*}
\tr(a_iX^i)=\tr(\gamma^{n_i}\gamma^{ij})
&=&\gamma^{n_i+ij}+\gamma^{p(n_i+ij)}+\cdots+\gamma^{p^{k-1}(n_i+ij)}\\
&=&\mbox{x[$n_i+ij$]$+$x[$p(n_i+ij)$]$+\cdots+$x[$p^{k-1}(n_i+ij)$]}\pmod p.
\end{eqnarray*}
������, �����ź���� mod $q-1$ �Ǹ���. ����ˤ�äƳ� $j\quad(0\leq j\leq q-2)$ ���Ф��� $\tr f(\gamma^j)$ ��׻���, 
$$t(r):=\sharp\left\{x\in{\bf F}_q \ |\  \tr f(x)=r\right\}
\quad(0\leq r\leq p-1).$$
������, 
$$S(f)=\sum_{r=0}^{p-1}t(r)e(r)$$
�Ȥʤ�. 

\bigskip
\noindent{\gt �� ��} {\bf 1.} ����Ǥⰷ����, $f(X)$ ��¿�༰�Ǥʤ��Ƥ�, �㤨�� $X^{-1}$ ���ι��ޤ�Ǥ��Ƥ�褤. �º�, Ǥ�դ� $x\in{\bf F}_q^\times$ ���Ф���, $x^{q-1}=1$ ������, $X^{-1}$ �Τ�����  $X^{q-2}$ �Ȥ����, ¿�༰�˵��夵���. 

\bigskip
\noindent{\gt �� ��} {\bf 2.} ��ˡ��ɸ $\chi:{\bf F}_q^\times\rightarrow {\bf C}^\times$ ��ޤ�ؿ���
$$S_\chi(f)=\sum_{x\in{\bf F}_q^\times}\chi(x)e\left(\tr f(x)\right)$$
��Ʊ�ͤ˷׻��Ǥ���. �º�, $\chi(\gamma^j)=\chi(\gamma)^j$ ������, ���ϸ� $\gamma$ �ιԤ��� $\chi(\gamma)$ ����Ƥ����Ф褤. ����, ���ʬ���������ƥץ�������ʣ���ˤʤ뤬, ���르�ꥺ��Ū�ˤ�, ��������������ʤ�. 

\bigskip
\noindent{\large \S {\bf 4. }{\gt ���� {\bf 1} $-$ {\bf Kloosterman }�¤η׻�}}
%(section 4)

\bigskip
������Ȥ��� Kloosterman ��
$$K(\alpha, \beta; q)
=\sum_{x\in{\bf F}_q^\times}
e\left(\tr(\alpha x+\beta x^{-1})\right)
 \quad (\alpha, \beta \in{\bf F}_q)$$
��׻�����. ����ޤǤε���Ǥ�����, $f(X)=\alpha X+\beta X^{-1}$ �Ȥʤ���Ǥ���. $\alpha$ �ޤ��� $\beta$ �� 0 �ΤȤ��� Gauss �¤˵��夵���Τ�, $a:=\alpha\beta\not=0$ �ΤȤ����ܼ�Ū�Ǥ���. ���ΤȤ�
$$K(\alpha, \beta; q)=K(a):=\sum_{x\in{\bf F}_q^\times}
e\left(\tr(x+ax^{-1})\right)$$
���ѷ������. ���Τ���, �� ${\bf F}_q$ ��1�ĸ��ꤷ, ���٤Ƥ� $a\in{\bf F}_q^\times$ ���Ф��� $K(a)$ ��׻����뤳�Ȥ�ͤ���. 

�ޤ� ${\bf F}_q$ �θ���¿�༰����Ƥ����ͤФʤ�ʤ���, ɮ�Ԥ� Lidl-Niederreiter \cite{LN} �����ο�ɽ�����Ѥ���. �ޤ�, �� $x\in{\bf F}_q^\times$ ��¿�༰ɽ���������������ˡ�ˤĤ��Ƥ�, ������Ǹ�˷Ǥ���ץ��������򻲾Ȥ��Ƥ�����������. 

����, $K(a)$ �η׻��Ǥ��뤬, �׻��Ρּ����ʤ��פ����̿���2���Ѱդ���:

\bigskip
\noindent{\gt �� ��} {\bf 4.1.}
$$K(\alpha, \beta; q)\in{\bf R}.$$


\bigskip
\noindent �������ä�, $e(y)$ �Τ����� $\cos \displaystyle\frac{2\pi y}p$ �Ƿ׻����ƽ�ʬ�Ǥ���. ����, 

\bigskip
\noindent{\gt �� ��} {\bf 4.2.} $q>p$ �ΤȤ�, $a, b\in{\bf F}_q^\times$�ˤĤ���, $a=b^\sigma$ �ʤ� $\sigma\in\Gal({\bf F}_q/{\bf F}_p)$ ��¸�ߤ���ʤ��,
$$K(a)=K(b).$$

\bigskip
\noindent ��ä�, ��ɽ�� ${\bf F}_q^\times/\Gal({\bf F}_q/{\bf F}_p)$ �γƸ� $a$ ���Ф��� $K(a)$ ��׻�����н�ʬ�Ǥ���. ������ɽ�Ϥ� {\bf cyclotomic coset}
$$C(j):=\{j, jp, jp^2, \cdots , jp^{k-1}\}\quad(\mbox{mod }q-1, 0\leq j\leq q-1)$$
��׻����뤳�ȤǤ狼��. $C(j)$ ��, �������ˤ狼��褦��, ���ϸ� $\gamma$ ���Ф��� $\gamma^j$ �ζ�������
 $$\gamma^j, \gamma^{jp}, \cdots, \gamma^{jp^{k-1}}$$
��ͤ�, ���λؿ��Τߤ���Ф����¤٤���ΤǤ���. �������ä�, $0\leq j\leq q-1$ �ʤ뤹�٤Ƥ� $j$ ���Ф��� $C(j)$ ��׻���, �ۤʤ� $C(j)$ ������ɽ����1�ĤŤļ��Ф��Ƥ�����, ���Ҥ���ɽ�Ϥ�狼�뤳�Ȥˤʤ�. 

\bigskip
\noindent{\gt �� }{\bf 4.2.} \ ${\bf F}_{16}.\quad$ 
���ξ��� cyclotomic cosets �򤹤٤Ƶ���ȼ��Τ褦�ˤʤ�:
$$\begin{array}{ll}
C(0)=\{0\} & C(5)=\{5, 10\}\\
C(1)=\{1, 2, 4, 8\}\qquad & C(7)=\{7, 14, 13, 11\}\\
C(3)=\{3, 6, 12, 9\} &   
\end{array}$$
�̾�ϳ� coset �κǾ�������ɽ���Ȥ�������, �����ν��� $R(q)$ �� coset representatives �Ȥ���. ������Ǥ�
$$R(16)=\{0,1,3,5,7\}.$$

\bigskip

\noindent ɮ�Ԥ�, cyclotomic cosets ����� coset representatives ��񤭽Ф��ץ��������̤˺��, �׻������Ѥ��Ƥ���. Cyclotomic coset �ˤĤ��ƾܤ�����, MacWilliams-Sloane \cite{MS}, p.104 �򻲾�. 

�ʾ�ˤ��, �� $i\in R(q)$ ���Ф�, 
\begin{eqnarray*}
K(\gamma^i)&=&\sum_{j=0}^{q-2}
              e\left(\tr(\gamma^j+\gamma^i\gamma^{-j})\right)\\
           &=&\sum_{j=0}^{q-2}
              e\left(\tr(\gamma^j+\gamma^{q+i-j-1})\right)\\
           &=&\sum_{r=0}^{p-1}t(r)\cos\displaystyle\frac{2\pi r}p
\end{eqnarray*}
��׻������, ���٤Ƥ� Kloosterman �¤���ޤ뤳�Ȥˤʤ�. 

\bigskip
\noindent{\gt �� }{\bf 4.3.}  1�Ĥ���Ȥ���, ${\bf F}_{81}$ ��� Kloosterman �¤򤹤٤Ƶ���ץ�������C����ˤ�äƽ񤤤Ƥߤ�. �������6��˷Ǥ����ɽ�����������Ȥ����Ѥ�����ΤȤۤ�Ʊ���Ǥ���. �ʤ�, ��ü�Υ��ߥ�����Ĥ��ο����Ϲ��ֹ�Ǥ���, �ץ������ΰ����ǤϤʤ�.

\bigskip
{\small
\begin{verbatim}
 1: /*   C����ˤ��ץ��������.   F_81 ���Kloosterman��   */
 2:
 3: #include <stdio.h>
 4: static short rep[ ]={0,1,2,4,5,7,8,10,11,13,14,16,17,20,22,
 5: 23,25,26,40,41,44,50,53};     /* �� coset representatives  */
 6: 
 7: main()
 8: {
 9:   int i, j, a, t[3];
10:   char coef[5], x[80];
11:   printf( "The Kloosterman sums over F_81.\n\n" );
12:   printf( "a: a root of a prim. polynomial x^4+x+2.\n\n" ); 
13:  /*  -------------------- ¿�༰ɽ���η׻� -----------------  */
14:   coef[0]=1;
15:   for( i=1; i<80; ++i )
16:   {
17:     for( j=4; j>0; --j )
18:     {
19:       coef[j]=coef[j-1];
20:     }
21:     coef[0]=0;
22:     if( coef[4]>0 )
23:     {
24:       coef[1]=(coef[1]+coef[4]*2)%3;
25:       coef[0]=(coef[0]+coef[4])%3;
26: /*    coef[4]=0;          	�� ������ɬ�פ����ºݤˤ����ס���*/
27:     }
28:     x[i]=coef[0];
29:   }
30:   x[0]=1;
31: /*  ----------------- Kloosterman�¤η׻� ------------------  */
32:   for( i=0; i<23; ++i )
33:   {
34:     t[0]=0; t[1]=0; t[2]=0;
35:     for( j=0; j<80; ++j )
36:     {
37:       a=( x[j]+x[(80+rep[i]-j)%80]
38:         +x[3*j%80]+x[3*(80+rep[i]-j)%80]
39:         +x[9*j%80]+x[9*(80+rep[i]-j)%80]
40:         +x[27*j%80]+x[27*(80+rep[i]-j)%80])%3;
41:       switch( a )
42:       {
43:         case 0: t[0]=++t[0]; break;
44:         case 1: t[1]=++t[1]; break;
45:         default: t[2]=++t[2]; break;
46:       }
47:     }
48:     printf( "K(a^%d)=%d\n", rep[i], t[0]-(t[1]+t[2])/2 );
49:   }
50: }
\end{verbatim}
}
\bigskip

\noindent {\gt �ץ�����������}

\medskip
\noindent ��4 - 5: ${\bf F}_{81}$ �� coset representatives ��ǡ����Ȥ������� rep[$i$] ���ݴ�. ��������Ҥ��̤�, �̤Υץ������ˤ���᤿. 

\medskip
\noindent ��9 - 10: �ѿ� i, j �ϥ����󥿡�. 2�ť롼�פǤ� j ����¦. t[$r$] ����ʸ��� $t(r)$ ��Ʊ��. coef[$i$] ��¿�༰ɽ���ˤ����� $\gamma^i\ (\gamma$ �ϸ��ϸ�)�η������ݴ�. x[$j$] �� $\gamma^j$ ��¿�༰ɽ�����������ݴ�. 

\medskip
\noindent ��14 - 30: ������ʬ�� ${\bf F}_{81}^\times$ �γƸ� $\gamma^j\ (0\leq j\leq 79)$ ��¿�༰ɽ������������, ���� x[$j$] �˵�Ͽ���Ƥ���. �ޤ�����¿�༰�Ȥ��� $X^4+X+2$ ���Ȥ�뤳�Ȥ����դ���. �����ƽ���ͤȤ��� $f(\gamma)=1$ �Ȥ����������ꤷ(��14), ����� $\gamma$ ��ݤ�(��17 - 21), $\gamma^4$ �ιब�����дط��� $\gamma^4+\gamma+2=0$ �ˤ�ä� reduction ����(��22 - 27). �ʤ�, ��26�� $\gamma^4$ �η�����0���᤹����Τ�Τ���, ���¤ˤ����פǤ���. �º�, j �Υ롼�פκǽ����ä��Ȥ�, ��19�ˤ�äƼ��θ��˹��������. 

\medskip
\noindent ��32 - 46: ���Υץ������Ρ����Ρפ�, Kloosterman �¤η׻��򤳤��ǹԤ�. �� $i$ ���Ф���, $K(\gamma^{\mbox{rep[$i$]}})$ ��׻�����. �ѿ� a �ˤ� $\tr(\gamma^{\mbox{x[$j$]}}+\gamma^{\mbox{rep[$i$]}}\gamma^{-\mbox{x[$j$]}})$ �����������(��37 - 40). ɸ����3������, �����ͤ�0, 1, 2�Τɤ줫�Ǥ���. ����򼡤� switch ʸ�ǽ��פ���(��41 - 46). ��̤� $\mbox{t[0]}\cos0+\mbox{t[1]}\cos(2\pi/3)+\mbox{t[2]}\cos(4\pi/3)=$ 
t[0]$-$(t[1]+t[2])/2 �Ǥ���, �����Ŭ����ɽ��������Ф褤. 

\bigskip

\noindent{\gt �� ��. } ɸ��2, 3�� Kloosterman �¤Ϥ��٤�ͭ�������Ǥ���, ɸ��5�� Kloosterman �¤Ϥ��٤�2���� ${\bf Q}(\sqrt 5)$ �������Ǥ��뤫��, ���Ϥ����Ū�ưפǤ���. ������, ɸ��7�ʾ�� Kloosterman �¤����­�Τ������ˡ�ɽ�������뤳�Ȥ�, �ޤ��̤β���Ǥ���. 
%�㤨��, ɸ��7�ξ��, 
%���Ѥ��� Kloosterman �¤�  ��ɽ�����Ȥ��Ǥ��뤬, ���δط����򤦤ޤ��������Ȥ�,% Maple V �ʤɤǤ��񤷤��褦�Ǥ���. 

\bigskip
\noindent{\large {\bf \S 5.} {\gt ���� {\bf 2} $-$ {\bf Hyperkloosterman }���η׻�}}
%(section 5)

\bigskip
����ޤǸ��Ƥ����褦��, ��3��ǽҤ٤���ˡ��, $\tr:{\bf F}_q\rightarrow{\bf F}_p$ �η׻����ܼ�Ū�Ǥ��뤫��, �㤨��, �������Τʤ���, {\gt �����}(trace code)�ȸƤФ��췲�����˴ؤ���׻��˱��ѤǤ���. ������, ${\bf F}_p$ ���������� $C$ �������Ǥ���Ȥ�, ���� ${\bf F}_q$ ���������� $\tilde C$ �ˤ�ä�
$$C=\{(\tr c_1, \tr c_2, \cdots , \tr c_n)\in{\bf F}_p^n\ |\ (c_1, c_2, \cdots , c_n)\in\tilde C\}$$
�Ȥ���������Ȥ��ˤ���. ���̤�, ��ʬ����ʬ�����������������Ȥʤ뤳�Ȥ��Τ��Ƥ���(Delsarte ������. $cf.$ MacWilliams-Sloane \cite{MS}, p.208). 
�����ζ�����Ȥ���, Hyperkloosterman ����������褦:

\medskip
\noindent {\gt �� �� }{\bf 5.1. } $m\in{\bf N}, m\geq2$, $q=p^r, r\geq2$ ���Ф���, Hyperkloosterman ��� $C_m(q)$ ��, ����
$$\varphi_m :{\bf F}_q^m\rightarrow {\bf F}_p^{(q-1)^{m-1}},$$
$$\varphi_m(\mbox{\boldmath $a$})=\{\Tr(\mbox{\boldmath $a,x$})\}_{\mbox{\boldmath $x$}\in({\bf F}_q^\times)^{m-1}}$$
�����Ȥ����������. ������, {\boldmath $a$}$=(a_1, a_2, \cdots , a_n)$, {\boldmath $x$}$=(x_1, x_2, \cdots , x_{n-1})$ ���Ф���
$$\Tr(\mbox{\boldmath $a,x$}):=\tr(a_1x_1+a_2x_2+\cdots a_{m-1}x_{m-1}+a_m(x_1x_2\cdots x_{m-1})^{-1}).$$
\medskip
\noindent ��ε�ˡ $\{\Tr(\mbox{\boldmath $a,x$})\}_{\mbox{\boldmath $x$}\in({\bf F}_q^\times)^{m-1}}$ �Ͼ����狼��Ť餤��, ���٤Ƥ� $\mbox{\boldmath $x$}\in({\bf F}_q^\times)^{m-1}$ ���Ф��� $\Tr(\mbox{\boldmath $a,x$})$ ��ͤ�, ����������¤٤� $(q-1)^{m-1}$ �����ԥ٥��ȥ�Ǥ���. 

�Ϥ���ˤ����������Ū�طʤˤĤ��ƽҤ٤Ƥ���. �ޤ�, C.M.Melas �ˤ�ä�, �С����ȡ����顼���Ф�������ǽ�Ϥ���ĤȤ��������椬�󾧤��줿( Melas ���, 1960ǯ). ���ʤ��, ${\bf F}_q$ �θ��ϸ� $\alpha$ ��Ȥ�, ��$\alpha$ �� ${\bf F}_p$ ��κǾ�¿�༰�� $M(x)$, $\alpha^{-1}$ �� ${\bf F}_p$ ��κǾ�¿�༰�� $M_-(x)$ �Ȥ���Ȥ�, $M(x)M_-(x)$ ������¿�༰�Ȥ��������Ǥ���($cf.$ Melas \cite{M}; Peterson \cite{P}, p.187; Wolfman \cite{W} ��). ���� Melas �����������Ȥ��Ƽ����˽ФƤ���Τ� Kloosterman ��� $C(q)$ �Ǥ���. ����Ͼ嵭 Hyperkloosterman ��������� $m=2$ �Ȥ������Ǥ���: $C(q)=C_2(q)$. Kloosterman ���ȸƤФ����ͳ��, ����� Hamming �Ť��� Kloosterman �� $K(\alpha, \beta)$ ��ɽ������뤳�Ȥˤ��. $K(a)$ �η��� Kloosterman �¤ˤĤ��Ƥ� $q\to\infty$ �ΤȤ��� $\{K(a)\ |\ a\in{\bf F}_q^\times\}$ ����ʬ�ۤ� Deligne-Katz ������($cf.$ Katz \cite{K}, Chap.13)�ˤ�äƵ��ҤǤ�, ����ˤ�ä� $C(q)$ �νŤ�ʬ�ۤ������뤳�Ȥ���, ��������Ԥ�������ܤ���Ƥ������Ǥ���. $C(q)$ �νŤ�ʬ�ۤ� $p=2$ �ξ�礬 Lachaud \cite{La}, $p$ �����̤ξ�礬 Hiramatsu \cite{H2} ���������Ƥ���. $C_m(q)$ �� $C(q)$ �ΰ��̲��Ȥ���, Chinen-Hiramatsu \cite{CH} �ˤ����ƹ������줿. $C_m(q)$ ��Hamming �Ť���, Hyperkloosterman ��
$$K_m(\mbox{\boldmath $a$};q)=\sum_{\mbox{\boldmath $x$}\in({\bf F}_q^\times)^{m-1}}e\left(\Tr(\mbox{\boldmath $a,x$})\right)$$
��ɽ�������. $m=2$ �ʤ��, $K_m(\mbox{\boldmath $a$};q)$ ����4���������� Kloosterman �¤Ȱ��פ��뤳�Ȥ��ưפˤ狼��.

�ܹƤΥ��르�ꥺ��� $C_m(q)$ ������ΰ���ɽ��׻����뤳�Ȥ��Ǥ���. �㤨��, $C_3(4)$ ������ϼ��Τ褦�ˤʤ�:

{\small
\begin{multicols}{6}

\noindent �Ť� 0: \\
000000000 \\
 \\
�Ť� 2: \\
000010001 \\
000100010 \\
000001100 \\
010001000 \\
100010000 \\
001100000 \\
001000010 \\
010000100 \\
100000001 \\
 \\
�Ť� 4: \\
011100100 \\
110001001 \\
101010010 \\
011001010 \\
101100001 \\
110010100 \\
100100011 \\
001001110 \\
010010101 \\
100001101 \\
010100110 \\
001010011 \\
100011100 \\
001110001 \\
010101010 \\
100110010 \\
010011001 \\
001101100 \\
000101110 \\
110000101 \\
101110000 \\
101000011 \\
011101000 \\
000011101 \\
110011000 \\
000110011 \\
011000110 \\
 \\
�Ť� 6: \\
110111010 \\
101001111 \\
101111100 \\
011010111 \\
110100111 \\
011111001 \\
000111111 \\
111010110 \\
001111101 \\
010111011 \\
111100101 \\
111001011 \\
100111110 \\
111111000 \\
111101001 \\
010110111 \\
100101111 \\
111011010 \\
001011111 \\
111110100 \\
111000111 \\
011011011 \\
110110110 \\
101101101 \\
011110101 \\
101011110 \\
110101011 \\
\end{multicols}
}
����ɽ���鼡�Τ��Ȥ��ѻ������: �Ť�2����1���ܤθ� 000010001 �κǸ�3�����Ƭ�ذܤ�, �ʹߤ��ˤ��餹�� 001000010 �Ȥʤ뤬, �������7���ܤθ��������. ����ˤ��θ��Ʊ������Ԥ�����4���ܤ� 010001000 �Ȥʤ�. ¾������˴ؤ��Ƥ�Ʊ�ͤ�, ������κǸ�3�����Ƭ�ذܤ�, �ʹߤ��ˤ��餷�ƤǤ����٥��ȥ�Ϥ�Ϥ�����ˤʤäƤ���. �����{\gt ����Ū}(quasi-cyclic)�ȸƤФ�������Ǥ���. ���Τˤ�, ��� $C$ ��Ĺ�� $s$ �� basic block ���Ľ������Ǥ���Ȥ�, 
$$c_1c_2\cdots c_n \in C \Rightarrow c_{n-s+1}\cdots c_nc_1c_2 \cdots c_{n-s}\in C$$
��Ǥ�դ� $c_1c_2\cdots c_n \in C$ ���Ф�������Ω�Ĥ��ȤǤ���. �����Ǥϴ�ñ�ˡ�$s$-����Ū�פȸƤ֤��Ȥˤ��褦. $s=1$ �ξ�礬�̾�ν�����Ǥ���. ���Ĺ�ξ����ʤ����Ĥ��� $C_m(2^r)$ ��׻������Ȥ���, ���η�̤�������

\bigskip
\begin{center}
{\small
\protect\renewcommand{\arraystretch}{1.2}
\begin{tabular}{c||c|c|c|c|c|c}
$C_m(2^r)$ & ���Ĺ($n$) & ����($k$) & �Ǿ���Υ($d$) & $k/n$ & $d/n$ & �����\\ \hline
$C_3(2^2)$ &  9  &  6  &  2  & 0.67 & 0.22 &  3-����Ū \\
$C_4(2^2)$ & 27  &  8  &  8  & 0.30 & 0.30 &  9-����Ū \\
$C_5(2^2)$ & 81  & 10  & 30  & 0.12 & 0.37 & 27-����Ū \\
 \  & \  & \phantom{9} & \  & \  & \  & \ \\
$C_3(2^3)$ & 49  &  9  & 16  & 0.18 & 0.33 &  7-����Ū \\
\end{tabular}
}
\end{center}

\bigskip
\noindent ������, 

\bigskip
\noindent{\gt ͽ �� }{\bf 5.2. } $C_m(q)$ �� $(q-1)^{m-2}$-�������Ǥ���. 

\bigskip
\noindent $m=2$ �ξ��, ¨��2�ѿ��λؿ��¤Ǥ��� Kloosterman �¤���Ϣ���� $C_2(q)=C(q)$ �Ͻ�����Ǥ��ä��櫓����, $m=2$ �ʤ�Ĥͤ� $(q-1)^{m-2}=1$ ������, ���ξ�������Ω�äƤ���. �������ǤϤޤ�ͽ�ۤ���, �ؿ��¤��ѿ������䤷�����Ȥ�, ���� cyclic shift �η�������䤹�Ȥ�����̤�⤿�餷����Τȹͤ�����. 

Hyperkloosterman ���˴ؤ��Ƥ�̤��������ʬ��¿��, ��ǽ�̤�ޤ��, ���߸�����Ǥ���. 

\bigskip
\noindent{\large {\bf \S 6.} {\gt �� ɽ}}
%(section 6)

\bigskip
ɸ����2, 3, 5��, �����Ĥ����Τξ�� Kloosterman �¤�ɽ, ����� cyclotomic cosets ��ɽ��Ǥ���. ��Ͽ�����Τ�, ɸ��2�� ${\bf F}_4$ ���� ${\bf F}_{1024}$ �ޤ�, ɸ��3�� ${\bf F}_9$ ���� ${\bf F}_{729}$ �ޤ�, ɸ��5�� ${\bf F}_{25}$ ���� ${\bf F}_{625}$ �ޤǤǤ���. �׻����Ѥ�������¿�༰�ϼ����̤�Ǥ���:
\begin{multicols}{2}

\noindent${\bf F}_4 \cdots  X^2+X+1$\\
${\bf F}_8 \cdots  X^3+X+1$\\
${\bf F}_{16} \cdots X^4+X+1$\\
${\bf F}_{32} \cdots X^5+X^2+1$\\
${\bf F}_{64} \cdots X^6+X+1$\\
${\bf F}_{128} \cdots X^7+X+1$\\
${\bf F}_{256} \cdots X^8+X^4+X^3+X^2+1$\\
${\bf F}_{512} \cdots X^9+X^4+1$\\
${\bf F}_{1024} \cdots X^{10}+X^3+1$\\
${\bf F}_9 \cdots X^2+X+2$\\
${\bf F}_{27} \cdots X^3+2X+1$\\
${\bf F}_{81} \cdots X^4+X+2$\\
${\bf F}_{243} \cdots X^5+2X+1$\\
${\bf F}_{729} \cdots X^6+X+2$\\
$ \phantom{\cdots} $\\
${\bf F}_{25} \cdots X^2+X+2$\\
${\bf F}_{125} \cdots X^3+4X+2$\\
${\bf F}_{625} \cdots X^4+X^3+2X^2+2$
\end{multicols}
ɽ��Ǥ�, �嵭��¿�༰�κ���1��(����ޤ� $\gamma$ �Ƚ񤤤Ƥ������)�� $a$ ��ɽ���Ƥ���ΤǤ����դ�����������. �ޤ�, ��4��ǽҤ٤��Ȥ���, $\Gal({\bf F}_q/{\bf F}_p)$ �κ��Ѥǰܤꤢ��ʤ����Τߤ��Ф��Ʒ׻���ԤʤäƤ���. �������ä�, Galois ���κ��Ѥǰܤꤢ������Ĵ�٤�ˤ�, �Ǹ�� cyclotomic cosets ��ɽ�򻲾Ȥ��Ƥ����������Ȥˤʤ�. 

\bigskip
\noindent{\gt �� {\bf 6.1. }} ${\bf F}_8. \quad$ ��ɽ�ˤ���, ${\bf F}_8$ ��� Kloosterman �¤�
\begin{eqnarray*}
K(1)&=&-5\\
K(a)&=&-1\\
K(a^3)&=&3.
\end{eqnarray*}
¾�θ����Ф����ͤ�, cyclotomic cosets ��ɽ
\begin{eqnarray*}
C(0)&=&\{0\}\\
C(1)&=&\{1, 2, 4\}\\
C(3)&=&\{3, 5, 6\}
\end{eqnarray*}
�ˤ��, 
\begin{eqnarray*}
K(a)&=&K(a^2)=K(a^4)=-1\\
K(a^3)&=&K(a^5)=K(a^6)=3
\end{eqnarray*}
�Ȥ��Ƶ�����. 

\medskip
\begin{center}
{\gt ɸ��{\bf 2}�� {\bf Kloosterman} ��}
\end{center}
\begin{multicols}{5}
{\scriptsize
\noindent ${\bf F}_4$

\medskip
\noindent $K(a^0)=3$\\
$K(a^1)=-1$\\

\bigskip
\noindent ${\bf F}_8$

\medskip
\noindent $K(a^0)=-5$\\
$K(a^1)=-1$\\
$K(a^3)=3$\\

\bigskip
\noindent ${\bf F}_{16}$

\medskip
\noindent $K(a^0)=-1$\\
$K(a^1)=-1$\\
$K(a^3)=3$\\
$K(a^5)=7$\\
$K(a^7)=-5$\\

\bigskip
\noindent ${\bf F}_{32}$

\medskip
\noindent $K(a^{0})=11$\\
$K(a^{1})=-1$\\
$K(a^{3})=3$\\
$K(a^{5})=-5$\\
$K(a^{7})=-9$\\
$K(a^{11})=3$\\
$K(a^{15})=7$\\

\bigskip
\noindent ${\bf F}_{64}$

\medskip
\noindent $K(a^{0})=-9$\\
$K(a^{1})=-1$\\
$K(a^{3})=-9$\\
$K(a^{5})=11$\\
$K(a^{7})=-1$\\
$K(a^{9})=7$\\
$K(a^{11})=-5$\\
$K(a^{13})=7$\\
$K(a^{15})=3$\\
$K(a^{21})=11$\\
$K(a^{23})=-13$\\
$K(a^{27})=15$\\
$K(a^{31})=-5$\\

\bigskip
\noindent ${\bf F}_{128}$

\medskip
\noindent $K(a^{0})=-13$\\
$K(a^{1})=-1$\\
$K(a^{3})=-9$\\
$K(a^{5})=15$\\
$K(a^{7})=-13$\\
$K(a^{9})=-9$\\
$K(a^{11})=15$\\
$K(a^{13})=19$\\
$K(a^{15})=-9$\\
$K(a^{19})=3$\\
$K(a^{21})=11$\\
$K(a^{23})=7$\\
$K(a^{27})=3$\\
$K(a^{29})=-1$\\
$K(a^{31})=11$\\
$K(a^{43})=-21$\\
$K(a^{47})=3$\\
$K(a^{55})=-17$\\
$K(a^{63})=-5$\\

\bigskip
\noindent ${\bf F}_{256}$

\medskip
\noindent $K(a^{0})=31$\\
$K(a^{1})=-17$\\
$K(a^{3})=-9$\\
$K(a^{5})=19$\\
$K(a^{7})=7$\\
$K(a^{9})=27$\\
$K(a^{11})=-29$\\
$K(a^{13})=-1$\\
$K(a^{15})=3$\\
$K(a^{17})=31$\\
$K(a^{19})=7$\\
$K(a^{21})=-5$\\
$K(a^{23})=-25$\\
$K(a^{25})=-17$\\
$K(a^{27})=15$\\
$K(a^{29})=-5$\\
$K(a^{31})=-1$\\
$K(a^{37})=23$\\
$K(a^{39})=-13$\\
$K(a^{43})=19$\\
$K(a^{45})=15$\\
$K(a^{47})=11$\\
$K(a^{51})=23$\\
$K(a^{53})=-5$\\
$K(a^{55})=-5$\\
$K(a^{59})=-25$\\
$K(a^{61})=-13$\\
$K(a^{63})=3$\\
$K(a^{85})=-17$\\
$K(a^{87})=-21$\\
$K(a^{91})=-13$\\
$K(a^{95})=11$\\
$K(a^{111})=-9$\\
$K(a^{119})=7$\\
$K(a^{127})=23$\\

\bigskip
\noindent ${\bf F}_{512}$

\medskip
\noindent $K(a^{0})=-5$\\
$K(a^{1})=-1$\\
$K(a^{3})=-33$\\
$K(a^{5})=43$\\
$K(a^{7})=7$\\
$K(a^{9})=27$\\
$K(a^{11})=15$\\
$K(a^{13})=15$\\
$K(a^{15})=-37$\\
$K(a^{17})=39$\\
$K(a^{19})=-13$\\
$K(a^{21})=-33$\\
$K(a^{23})=19$\\
$K(a^{25})=-13$\\
$K(a^{27})=19$\\
$K(a^{29})=27$\\
$K(a^{31})=-33$\\
$K(a^{35})=27$\\
$K(a^{37})=-17$\\
$K(a^{39})=3$\\
$K(a^{41})=-5$\\
$K(a^{43})=3$\\
$K(a^{45})=-17$\\
$K(a^{47})=-21$\\
$K(a^{51})=3$\\
$K(a^{53})=-25$\\
$K(a^{55})=-9$\\
$K(a^{57})=-41$\\
$K(a^{59})=7$\\
$K(a^{61})=-1$\\
$K(a^{63})=15$\\
$K(a^{73})=23$\\
$K(a^{75})=-13$\\
$K(a^{77})=11$\\
$K(a^{79})=35$\\
$K(a^{83})=-37$\\
$K(a^{85})=-9$\\
$K(a^{87})=23$\\
$K(a^{91})=-33$\\
$K(a^{93})=-9$\\
$K(a^{95})=-29$\\
$K(a^{103})=23$\\
$K(a^{107})=-9$\\
$K(a^{109})=-5$\\
$K(a^{111})=27$\\
$K(a^{117})=11$\\
$K(a^{119})=31$\\
$K(a^{123})=7$\\
$K(a^{125})=39$\\
$K(a^{127})=15$\\
$K(a^{171})=3$\\
$K(a^{175})=3$\\
$K(a^{183})=-21$\\
$K(a^{187})=-29$\\
$K(a^{191})=-17$\\
$K(a^{219})=-45$\\
$K(a^{223})=-21$\\
$K(a^{239})=27$\\
$K(a^{255})=15$\\

\bigskip
\noindent ${\bf F}_{1024}$

\medskip
\noindent $K(a^{0})=-57$\\
$K(a^{1})=-1$\\
$K(a^{3})=-33$\\
$K(a^{5})=7$\\
$K(a^{7})=-13$\\
$K(a^{9})=-1$\\
$K(a^{11})=31$\\
$K(a^{13})=47$\\
$K(a^{15})=39$\\
$K(a^{17})=-61$\\
$K(a^{19})=55$\\
$K(a^{21})=51$\\
$K(a^{23})=-1$\\
$K(a^{25})=-25$\\
$K(a^{27})=3$\\
$K(a^{29})=7$\\
$K(a^{31})=-5$\\
$K(a^{33})=55$\\
$K(a^{35})=-5$\\
$K(a^{37})=35$\\
$K(a^{39})=-17$\\
$K(a^{41})=55$\\
$K(a^{43})=7$\\
$K(a^{45})=-37$\\
$K(a^{47})=11$\\
$K(a^{49})=35$\\
$K(a^{51})=-9$\\
$K(a^{53})=-25$\\
$K(a^{55})=19$\\
$K(a^{57})=-21$\\
$K(a^{59})=19$\\
$K(a^{61})=19$\\
$K(a^{63})=-21$\\
$K(a^{69})=-9$\\
$K(a^{71})=-5$\\
$K(a^{73})=-29$\\
$K(a^{75})=39$\\
$K(a^{77})=-25$\\
$K(a^{79})=-49$\\
$K(a^{83})=35$\\
$K(a^{85})=-41$\\
$K(a^{87})=15$\\
$K(a^{89})=-17$\\
$K(a^{91})=43$\\
$K(a^{93})=-45$\\
$K(a^{95})=-17$\\
$K(a^{99})=39$\\
$K(a^{101})=31$\\
$K(a^{103})=-29$\\
$K(a^{105})=3$\\
$K(a^{107})=43$\\
$K(a^{109})=31$\\
$K(a^{111})=15$\\
$K(a^{115})=-41$\\
$K(a^{117})=-1$\\
$K(a^{119})=-53$\\
$K(a^{121})=11$\\
$K(a^{123})=-45$\\
$K(a^{125})=-37$\\
$K(a^{127})=-49$\\
$K(a^{147})=15$\\
$K(a^{149})=-37$\\
$K(a^{151})=-13$\\
$K(a^{155})=31$\\
$K(a^{157})=-41$\\
$K(a^{159})=27$\\
$K(a^{165})=15$\\
$K(a^{167})=-17$\\
$K(a^{171})=27$\\
$K(a^{173})=59$\\
$K(a^{175})=-29$\\
$K(a^{179})=11$\\
$K(a^{181})=-13$\\
$K(a^{183})=-45$\\
$K(a^{187})=-53$\\
$K(a^{189})=15$\\
$K(a^{191})=-29$\\
$K(a^{205})=-1$\\
$K(a^{207})=19$\\
$K(a^{213})=39$\\
$K(a^{215})=-25$\\
$K(a^{219})=-57$\\
$K(a^{221})=35$\\
$K(a^{223})=-37$\\
$K(a^{231})=55$\\
$K(a^{235})=11$\\
$K(a^{237})=51$\\
$K(a^{239})=23$\\
$K(a^{245})=47$\\
$K(a^{247})=11$\\
$K(a^{251})=59$\\
$K(a^{253})=23$\\
$K(a^{255})=35$\\
$K(a^{341})=-61$\\
$K(a^{343})=19$\\
$K(a^{347})=-41$\\
$K(a^{351})=-53$\\
$K(a^{363})=63$\\
$K(a^{367})=-25$\\
$K(a^{375})=-33$\\
$K(a^{379})=-5$\\
$K(a^{383})=-25$\\
$K(a^{439})=-13$\\
$K(a^{447})=3$\\
$K(a^{479})=7$\\
$K(a^{495})=-17$\\
$K(a^{511})=-1$\\
}
\end{multicols}

\begin{center}
{\gt ɸ��{\bf 3}�� {\bf Kloosterman} ��}
\end{center}

\begin{multicols}{5}

{\scriptsize
\noindent ${\bf F}_{9}$

\medskip
\noindent $K(a^{0})=5$\\
$K(a^{1})=-4$\\
$K(a^{2})=-1$\\
$K(a^{4})=2$\\
$K(a^{5})=2$\\

\bigskip
\noindent ${\bf F}_{27}$

\medskip
\noindent $K(a^{0})=8$\\
$K(a^{1})=8$\\
$K(a^{2})=-4$\\
$K(a^{4})=5$\\
$K(a^{5})=2$\\
$K(a^{7})=-4$\\
$K(a^{8})=-7$\\
$K(a^{13})=-10$\\
$K(a^{14})=-1$\\
$K(a^{17})=2$\\

\bigskip
\noindent ${\bf F}_{81}$

\medskip
\noindent $K(a^{0})=-7$\\
$K(a^{1})=-10$\\
$K(a^{2})=-10$\\
$K(a^{4})=2$\\
$K(a^{5})=8$\\
$K(a^{7})=14$\\
$K(a^{8})=-7$\\
$K(a^{10})=2$\\
$K(a^{11})=2$\\
$K(a^{13})=-4$\\
$K(a^{14})=-1$\\
$K(a^{16})=5$\\
$K(a^{17})=-4$\\
$K(a^{20})=17$\\
$K(a^{22})=-13$\\
$K(a^{23})=2$\\
$K(a^{25})=-10$\\
$K(a^{26})=11$\\
$K(a^{40})=14$\\
$K(a^{41})=8$\\
$K(a^{44})=5$\\
$K(a^{50})=14$\\
$K(a^{53})=-16$\\

\bigskip
\noindent ${\bf F}_{243}$

\medskip
\noindent $K(a^{0})=-31$\\
$K(a^{1})=26$\\
$K(a^{2})=8$\\
$K(a^{4})=20$\\
$K(a^{5})=2$\\
$K(a^{7})=8$\\
$K(a^{8})=11$\\
$K(a^{10})=5$\\
$K(a^{11})=-28$\\
$K(a^{13})=14$\\
$K(a^{14})=-4$\\
$K(a^{16})=-25$\\
$K(a^{17})=2$\\
$K(a^{19})=-22$\\
$K(a^{20})=-19$\\
$K(a^{22})=-4$\\
$K(a^{23})=-4$\\
$K(a^{25})=-10$\\
$K(a^{26})=23$\\
$K(a^{31})=20$\\
$K(a^{32})=-13$\\
$K(a^{34})=17$\\
$K(a^{35})=-4$\\
$K(a^{38})=-1$\\
$K(a^{40})=-4$\\
$K(a^{41})=8$\\
$K(a^{43})=-4$\\
$K(a^{44})=-19$\\
$K(a^{47})=26$\\
$K(a^{49})=-16$\\
$K(a^{50})=-16$\\
$K(a^{52})=-7$\\
$K(a^{53})=-10$\\
$K(a^{61})=14$\\
$K(a^{62})=29$\\
$K(a^{67})=14$\\
$K(a^{68})=-28$\\
$K(a^{70})=-1$\\
$K(a^{71})=8$\\
$K(a^{76})=20$\\
$K(a^{77})=20$\\
$K(a^{79})=-22$\\
$K(a^{80})=-7$\\
$K(a^{121})=2$\\
$K(a^{122})=-1$\\
$K(a^{125})=14$\\
$K(a^{131})=-16$\\
$K(a^{134})=11$\\
$K(a^{152})=-13$\\
$K(a^{161})=-16$\\

\bigskip
\noindent ${\bf F}_{729}$

\medskip
\noindent $K(a^{0})=-10$\\
$K(a^{1})=-28$\\
$K(a^{2})=-10$\\
$K(a^{4})=-10$\\
$K(a^{5})=2$\\
$K(a^{7})=44$\\
$K(a^{8})=26$\\
$K(a^{10})=-13$\\
$K(a^{11})=20$\\
$K(a^{13})=26$\\
$K(a^{14})=-10$\\
$K(a^{16})=2$\\
$K(a^{17})=-34$\\
$K(a^{19})=-10$\\
$K(a^{20})=41$\\
$K(a^{22})=-1$\\
$K(a^{23})=20$\\
$K(a^{25})=38$\\
$K(a^{26})=14$\\
$K(a^{28})=50$\\
$K(a^{29})=20$\\
$K(a^{31})=14$\\
$K(a^{32})=-31$\\
$K(a^{34})=11$\\
$K(a^{35})=-4$\\
$K(a^{37})=8$\\
$K(a^{38})=-34$\\
$K(a^{40})=41$\\
$K(a^{41})=-4$\\
$K(a^{43})=-4$\\
$K(a^{44})=2$\\
$K(a^{46})=35$\\
$K(a^{47})=-22$\\
$K(a^{49})=2$\\
$K(a^{50})=-1$\\
$K(a^{52})=-7$\\
$K(a^{53})=38$\\
$K(a^{56})=5$\\
$K(a^{58})=-1$\\
$K(a^{59})=2$\\
$K(a^{61})=-22$\\
$K(a^{62})=-13$\\
$K(a^{64})=5$\\
$K(a^{65})=-4$\\
$K(a^{67})=26$\\
$K(a^{68})=-31$\\
$K(a^{70})=35$\\
$K(a^{71})=-40$\\
$K(a^{73})=20$\\
$K(a^{74})=-34$\\
$K(a^{76})=29$\\
$K(a^{77})=14$\\
$K(a^{79})=44$\\
$K(a^{80})=17$\\
$K(a^{91})=-46$\\
$K(a^{92})=-22$\\
$K(a^{94})=11$\\
$K(a^{95})=-40$\\
$K(a^{97})=-34$\\
$K(a^{98})=-37$\\
$K(a^{101})=-10$\\
$K(a^{103})=20$\\
$K(a^{104})=5$\\
$K(a^{106})=-49$\\
$K(a^{107})=14$\\
$K(a^{112})=53$\\
$K(a^{113})=-40$\\
$K(a^{115})=14$\\
$K(a^{116})=26$\\
$K(a^{119})=2$\\
$K(a^{121})=50$\\
$K(a^{122})=35$\\
$K(a^{124})=17$\\
$K(a^{125})=50$\\
$K(a^{128})=-43$\\
$K(a^{130})=47$\\
$K(a^{131})=-4$\\
$K(a^{133})=-34$\\
$K(a^{134})=14$\\
$K(a^{140})=38$\\
$K(a^{142})=-25$\\
$K(a^{143})=-16$\\
$K(a^{146})=23$\\
$K(a^{148})=-19$\\
$K(a^{149})=26$\\
$K(a^{151})=-22$\\
$K(a^{152})=-7$\\
$K(a^{155})=-10$\\
$K(a^{157})=-34$\\
$K(a^{158})=26$\\
$K(a^{160})=-46$\\
$K(a^{161})=-40$\\
$K(a^{182})=26$\\
$K(a^{184})=5$\\
$K(a^{185})=-16$\\
$K(a^{187})=26$\\
$K(a^{188})=-43$\\
$K(a^{196})=50$\\
$K(a^{197})=-10$\\
$K(a^{202})=-22$\\
$K(a^{203})=8$\\
$K(a^{205})=-22$\\
$K(a^{206})=-13$\\
$K(a^{211})=-46$\\
$K(a^{212})=29$\\
$K(a^{214})=-25$\\
$K(a^{215})=-52$\\
$K(a^{224})=38$\\
$K(a^{229})=32$\\
$K(a^{230})=-37$\\
$K(a^{232})=-22$\\
$K(a^{233})=32$\\
$K(a^{238})=-1$\\
$K(a^{239})=8$\\
$K(a^{241})=20$\\
$K(a^{242})=-25$\\
$K(a^{364})=-46$\\
$K(a^{365})=-28$\\
$K(a^{368})=-19$\\
$K(a^{374})=38$\\
$K(a^{377})=8$\\
$K(a^{392})=29$\\
$K(a^{395})=-16$\\
$K(a^{401})=-46$\\
$K(a^{404})=29$\\
$K(a^{455})=44$\\
$K(a^{458})=23$\\
$K(a^{476})=-10$\\
$K(a^{485})=-16$\\
}
\end{multicols}

\begin{center}
{\gt ɸ��{\bf 5}�� {\bf Kloosterman} ��}
\end{center}
\begin{multicols}{4}

{\scriptsize
\noindent  ${\bf F}_{25}$

\medskip
\noindent $K(a^0)=\displaystyle{\frac{13}2+\frac{3}2\sqrt 5}$\\
$K(a^1)=\displaystyle{-1+3\sqrt 5}$\\
$K(a^{2})=\displaystyle{\frac{3}2+\frac{3}2\sqrt 5}$\\
$K(a^{3})=\displaystyle{-6}$\\
$K(a^{4})=\displaystyle{-\frac{7}2-\frac{1}2\sqrt 5}$\\
$K(a^{6})=\displaystyle{4-2\sqrt 5}$\\
$K(a^{7})=\displaystyle{-1+\sqrt 5}$\\
$K(a^{8})=\displaystyle{-\frac{7}2+\frac{1}2\sqrt 5}$\\
$K(a^{9})=\displaystyle{4}$\\
$K(a^{12})=\displaystyle{\frac{13}2-\frac{3}2\sqrt 5}$\\
$K(a^{13})=\displaystyle{-1-3\sqrt 5}$\\
$K(a^{14})=\displaystyle{\frac{3}2-\frac{3}2\sqrt 5}$\\
$K(a^{18})=\displaystyle{4+2\sqrt 5}$\\
$K(a^{19})=\displaystyle{-1-\sqrt 5}$\\

\bigskip
\noindent ${\bf F}_{125}$

\medskip
\noindent $K(a^0)=\displaystyle{ -\frac{27}2+\frac{7}2\sqrt 5}$\\
$K(a^1)=\displaystyle{ 9-5\sqrt 5}$\\
$K(a^{2})=\displaystyle{ -6+4\sqrt 5}$\\
$K(a^{3})=\displaystyle{ -16-2\sqrt 5}$\\
$K(a^{4})=\displaystyle{ \frac{33}2+\frac{3}2\sqrt 5}$\\
$K(a^{6})=\displaystyle{ \frac{13}2+\frac{1}2\sqrt 5}$\\
$K(a^{7})=\displaystyle{ -6+2\sqrt 5}$\\
$K(a^{8})=\displaystyle{ -\frac{7}2+\frac{11}2\sqrt 5}$\\
$K(a^{9})=\displaystyle{ -11+\sqrt 5}$\\
$K(a^{11})=\displaystyle{ -6}$\\
$K(a^{12})=\displaystyle{ \frac{3}2-\frac{1}2\sqrt 5}$\\
$K(a^{13})=\displaystyle{ -16+2\sqrt 5}$\\
$K(a^{14})=\displaystyle{ -\frac{7}2-\frac{11}2\sqrt 5}$\\
$K(a^{16})=\displaystyle{ \frac{3}2-\frac{9}2\sqrt 5}$\\
$K(a^{17})=\displaystyle{ -6-6\sqrt 5}$\\
$K(a^{18})=\displaystyle{ \frac{3}2+\frac{9}2\sqrt 5}$\\
$K(a^{19})=\displaystyle{ 4-2\sqrt 5}$\\
$K(a^{21})=\displaystyle{ 9-\sqrt 5}$\\
$K(a^{22})=\displaystyle{ 4-2\sqrt 5}$\\
$K(a^{23})=\displaystyle{ -6+6\sqrt 5}$\\
$K(a^{24})=\displaystyle{ -\frac{17}2-\frac{5}2\sqrt 5}$\\
$K(a^{31})=\displaystyle{ -1-7\sqrt 5}$\\
$K(a^{32})=\displaystyle{ \frac{3}2+\frac{15}2\sqrt 5}$\\
$K(a^{33})=\displaystyle{ 4+2\sqrt 5}$\\
$K(a^{34})=\displaystyle{ \frac{3}2-\frac{7}2\sqrt 5}$\\
$K(a^{37})=\displaystyle{ 4+4\sqrt 5}$\\
$K(a^{38})=\displaystyle{ \frac{33}2-\frac{3}2\sqrt 5}$\\
$K(a^{39})=\displaystyle{ -11-\sqrt 5}$\\
$K(a^{42})=\displaystyle{ -\frac{17}2+\frac{5}2\sqrt 5}$\\
$K(a^{43})=\displaystyle{ 9+\sqrt 5}$\\
$K(a^{44})=\displaystyle{ \frac{3}2+\frac{7}2\sqrt 5}$\\
$K(a^{47})=\displaystyle{ 9-\sqrt 5}$\\
$K(a^{48})=\displaystyle{ 4+2\sqrt 5}$\\
$K(a^{49})=\displaystyle{ 9+\sqrt 5}$\\
$K(a^{62})=\displaystyle{ -\frac{27}2-\frac{7}2\sqrt 5}$\\
$K(a^{63})=\displaystyle{ 9+5\sqrt 5}$\\
$K(a^{64})=\displaystyle{ -6-4\sqrt 5}$\\
$K(a^{68})=\displaystyle{ \frac{13}2-\frac{1}2\sqrt 5}$\\
$K(a^{69})=\displaystyle{ -6-2\sqrt 5}$\\
$K(a^{73})=\displaystyle{ -6}$\\
$K(a^{74})=\displaystyle{ \frac{3}2+\frac{1}2\sqrt 5}$\\
$K(a^{93})=\displaystyle{ -1+7\sqrt 5}$\\
$K(a^{94})=\displaystyle{ \frac{3}2-\frac{15}2\sqrt 5}$\\
$K(a^{99})=\displaystyle{ 4-4\sqrt 5}$\\

\bigskip
\noindent ${\bf F}_{625}$

\medskip
\noindent $K(a^0)=\displaystyle{-\frac{7}2-\frac{39}2\sqrt 5}$\\
$K(a^1)=\displaystyle{-26-2\sqrt 5}$\\
$K(a^{2})=\displaystyle{\frac{23}2+\frac{23}2\sqrt 5}$\\
$K(a^{3})=\displaystyle{4-20\sqrt 5}$\\
$K(a^{4})=\displaystyle{\frac{53}2-\frac{13}2\sqrt 5}$\\
$K(a^{6})=\displaystyle{-6+12\sqrt 5}$\\
$K(a^{7})=\displaystyle{-16+12\sqrt 5}$\\
$K(a^{8})=\displaystyle{-\frac{47}2-\frac{1}2\sqrt 5}$\\
$K(a^{9})=\displaystyle{-26+8\sqrt 5}$\\
$K(a^{11})=\displaystyle{4+6\sqrt 5}$\\
$K(a^{12})=\displaystyle{\frac{33}2+\frac{9}2\sqrt 5}$\\
$K(a^{13})=\displaystyle{34}$\\
$K(a^{14})=\displaystyle{-\frac{37}2+\frac{11}2\sqrt 5}$\\
$K(a^{16})=\displaystyle{\frac{33}2-\frac{15}2\sqrt 5}$\\
$K(a^{17})=\displaystyle{14+10\sqrt 5}$\\
$K(a^{18})=\displaystyle{\frac{3}2+\frac{3}2\sqrt 5}$\\
$K(a^{19})=\displaystyle{14+6\sqrt 5}$\\
$K(a^{21})=\displaystyle{-16+12\sqrt 5}$\\
$K(a^{22})=\displaystyle{-\frac{17}2+\frac{1}2\sqrt 5}$\\
$K(a^{23})=\displaystyle{-31+\sqrt 5}$\\
$K(a^{24})=\displaystyle{-\frac{67}2-\frac{3}2\sqrt 5}$\\
$K(a^{26})=\displaystyle{4-6\sqrt 5}$\\
$K(a^{27})=\displaystyle{-1+13\sqrt 5}$\\
$K(a^{28})=\displaystyle{-\frac{47}2+\frac{9}2\sqrt 5}$\\
$K(a^{29})=\displaystyle{4+6\sqrt 5}$\\
$K(a^{31})=\displaystyle{9+15\sqrt 5}$\\
$K(a^{32})=\displaystyle{-16+2\sqrt 5}$\\
$K(a^{33})=\displaystyle{-1+\sqrt 5}$\\
$K(a^{34})=\displaystyle{4-14\sqrt 5}$\\
$K(a^{36})=\displaystyle{\frac{13}2+\frac{19}2\sqrt 5}$\\
$K(a^{37})=\displaystyle{-6+12\sqrt 5}$\\
$K(a^{38})=\displaystyle{-\frac{37}2-\frac{11}2\sqrt 5}$\\
$K(a^{39})=\displaystyle{-26}$\\
$K(a^{41})=\displaystyle{-26+4\sqrt 5}$\\
$K(a^{42})=\displaystyle{\frac{23}2-\frac{9}2\sqrt 5}$\\
$K(a^{43})=\displaystyle{-16+14\sqrt 5}$\\
$K(a^{44})=\displaystyle{-\frac{7}2+\frac{17}2\sqrt 5}$\\
$K(a^{46})=\displaystyle{\frac{3}2-\frac{27}2\sqrt 5}$\\
$K(a^{47})=\displaystyle{19-3\sqrt 5}$\\
$K(a^{48})=\displaystyle{\frac{33}2-\frac{9}2\sqrt 5}$\\
$K(a^{49})=\displaystyle{-1-3\sqrt 5}$\\
$K(a^{52})=\displaystyle{\frac{73}2-\frac{9}2\sqrt 5}$\\
$K(a^{53})=\displaystyle{19-3\sqrt 5}$\\
$K(a^{54})=\displaystyle{\frac{23}2-\frac{17}2\sqrt 5}$\\
$K(a^{56})=\displaystyle{-\frac{47}2-\frac{15}2\sqrt 5}$\\
$K(a^{57})=\displaystyle{-11+9\sqrt 5}$\\
$K(a^{58})=\displaystyle{\frac{43}2+\frac{21}2\sqrt 5}$\\
$K(a^{59})=\displaystyle{39-3\sqrt 5}$\\
$K(a^{61})=\displaystyle{9-9\sqrt 5}$\\
$K(a^{62})=\displaystyle{-16-6\sqrt 5}$\\
$K(a^{63})=\displaystyle{4+20\sqrt 5}$\\
$K(a^{64})=\displaystyle{-\frac{47}2+\frac{1}2\sqrt 5}$\\
$K(a^{66})=\displaystyle{\frac{3}2-\frac{3}2\sqrt 5}$\\
$K(a^{67})=\displaystyle{-31-\sqrt 5}$\\
$K(a^{68})=\displaystyle{-\frac{47}2-\frac{9}2\sqrt 5}$\\
$K(a^{69})=\displaystyle{-1-\sqrt 5}$\\
$K(a^{71})=\displaystyle{-16-14\sqrt 5}$\\
$K(a^{72})=\displaystyle{\frac{33}2+\frac{9}2\sqrt 5}$\\
$K(a^{73})=\displaystyle{19+3\sqrt 5}$\\
$K(a^{74})=\displaystyle{\frac{43}2-\frac{21}2\sqrt 5}$\\
$K(a^{78})=\displaystyle{14}$\\
$K(a^{79})=\displaystyle{4-4\sqrt 5}$\\
$K(a^{81})=\displaystyle{19-\sqrt 5}$\\
$K(a^{82})=\displaystyle{-\frac{37}2-\frac{3}2\sqrt 5}$\\
$K(a^{83})=\displaystyle{4+4\sqrt 5}$\\
$K(a^{84})=\displaystyle{\frac{13}2-\frac{27}2\sqrt 5}$\\
$K(a^{86})=\displaystyle{-\frac{77}2+\frac{5}2\sqrt 5}$\\
$K(a^{87})=\displaystyle{9-3\sqrt 5}$\\
$K(a^{88})=\displaystyle{\frac{33}2+\frac{15}2\sqrt 5}$\\
$K(a^{89})=\displaystyle{-26-4\sqrt 5}$\\
$K(a^{91})=\displaystyle{-6}$\\
$K(a^{92})=\displaystyle{-\frac{27}2-\frac{21}2\sqrt 5}$\\
$K(a^{93})=\displaystyle{19+\sqrt 5}$\\
$K(a^{94})=\displaystyle{\frac{23}2-\frac{1}2\sqrt 5}$\\
$K(a^{96})=\displaystyle{\frac{13}2-\frac{3}2\sqrt 5}$\\
$K(a^{97})=\displaystyle{-16-10\sqrt 5}$\\
$K(a^{98})=\displaystyle{-\frac{37}2+\frac{3}2\sqrt 5}$\\
$K(a^{99})=\displaystyle{-1+9\sqrt 5}$\\
$K(a^{104})=\displaystyle{\frac{73}2-\frac{7}2\sqrt 5}$\\
$K(a^{106})=\displaystyle{\frac{3}2-\frac{21}2\sqrt 5}$\\
$K(a^{107})=\displaystyle{-21-3\sqrt 5}$\\
$K(a^{108})=\displaystyle{\frac{13}2+\frac{27}2\sqrt 5}$\\
$K(a^{109})=\displaystyle{-21-9\sqrt 5}$\\
$K(a^{111})=\displaystyle{-11+\sqrt 5}$\\
$K(a^{112})=\displaystyle{24+6\sqrt 5}$\\
$K(a^{113})=\displaystyle{14-10\sqrt 5}$\\
$K(a^{114})=\displaystyle{\frac{23}2+\frac{9}2\sqrt 5}$\\
$K(a^{116})=\displaystyle{-\frac{27}2+\frac{21}2\sqrt 5}$\\
$K(a^{117})=\displaystyle{24}$\\
$K(a^{118})=\displaystyle{-\frac{77}2-\frac{5}2\sqrt 5}$\\
$K(a^{119})=\displaystyle{19-7\sqrt 5}$\\
$K(a^{121})=\displaystyle{-11+3\sqrt 5}$\\
$K(a^{122})=\displaystyle{\frac{23}2-\frac{15}2\sqrt 5}$\\
$K(a^{123})=\displaystyle{9+3\sqrt 5}$\\
$K(a^{124})=\displaystyle{-\frac{7}2+\frac{15}2\sqrt 5}$\\
$K(a^{156})=\displaystyle{14+16\sqrt 5}$\\
$K(a^{157})=\displaystyle{-1-9\sqrt 5}$\\
$K(a^{158})=\displaystyle{\frac{23}2+\frac{1}2\sqrt 5}$\\
$K(a^{159})=\displaystyle{-6}$\\
$K(a^{162})=\displaystyle{-\frac{17}2-\frac{7}2\sqrt 5}$\\
$K(a^{163})=\displaystyle{14-6\sqrt 5}$\\
$K(a^{164})=\displaystyle{-\frac{7}2-\frac{17}2\sqrt 5}$\\
$K(a^{167})=\displaystyle{19+7\sqrt 5}$\\
$K(a^{168})=\displaystyle{\frac{13}2+\frac{3}2\sqrt 5}$\\
$K(a^{169})=\displaystyle{-36}$\\
$K(a^{172})=\displaystyle{-6}$\\
$K(a^{173})=\displaystyle{-16+10\sqrt 5}$\\
$K(a^{174})=\displaystyle{\frac{43}2+\frac{11}2\sqrt 5}$\\
$K(a^{182})=\displaystyle{44-2\sqrt 5}$\\
$K(a^{183})=\displaystyle{-1-9\sqrt 5}$\\
$K(a^{184})=\displaystyle{-\frac{7}2+\frac{33}2\sqrt 5}$\\
$K(a^{187})=\displaystyle{14+10\sqrt 5}$\\
$K(a^{188})=\displaystyle{\frac{53}2+\frac{13}2\sqrt 5}$\\
$K(a^{189})=\displaystyle{-26-8\sqrt 5}$\\
$K(a^{192})=\displaystyle{-\frac{67}2+\frac{3}2\sqrt 5}$\\
$K(a^{193})=\displaystyle{4-6\sqrt 5}$\\
$K(a^{194})=\displaystyle{4+14\sqrt 5}$\\
$K(a^{197})=\displaystyle{-1+3\sqrt 5}$\\
$K(a^{198})=\displaystyle{\frac{23}2+\frac{17}2\sqrt 5}$\\
$K(a^{199})=\displaystyle{39+3\sqrt 5}$\\
$K(a^{208})=\displaystyle{\frac{73}2+\frac{7}2\sqrt 5}$\\
$K(a^{209})=\displaystyle{-21+9\sqrt 5}$\\
$K(a^{212})=\displaystyle{-\frac{7}2-\frac{15}2\sqrt 5}$\\
$K(a^{213})=\displaystyle{-16-12\sqrt 5}$\\
$K(a^{214})=\displaystyle{\frac{3}2+\frac{27}2\sqrt 5}$\\
$K(a^{217})=\displaystyle{-11-3\sqrt 5}$\\
$K(a^{218})=\displaystyle{\frac{3}2+\frac{21}2\sqrt 5}$\\
$K(a^{219})=\displaystyle{-6}$\\
$K(a^{222})=\displaystyle{\frac{43}2-\frac{11}2\sqrt 5}$\\
$K(a^{223})=\displaystyle{-21+3\sqrt 5}$\\
$K(a^{224})=\displaystyle{-\frac{7}2-\frac{33}2\sqrt 5}$\\
$K(a^{234})=\displaystyle{34}$\\
$K(a^{237})=\displaystyle{29-7\sqrt 5}$\\
$K(a^{238})=\displaystyle{-\frac{17}2-\frac{1}2\sqrt 5}$\\
$K(a^{239})=\displaystyle{19+3\sqrt 5}$\\
$K(a^{242})=\displaystyle{\frac{23}2+\frac{15}2\sqrt 5}$\\
$K(a^{243})=\displaystyle{-11-\sqrt 5}$\\
$K(a^{244})=\displaystyle{-6}$\\
$K(a^{247})=\displaystyle{4}$\\
$K(a^{248})=\displaystyle{24-6\sqrt 5}$\\
$K(a^{249})=\displaystyle{29+7\sqrt 5}$\\
$K(a^{312})=\displaystyle{-\frac{7}2+\frac{39}2\sqrt 5}$\\
$K(a^{313})=\displaystyle{-26+2\sqrt 5}$\\
$K(a^{314})=\displaystyle{\frac{23}2-\frac{23}2\sqrt 5}$\\
$K(a^{318})=\displaystyle{-6-12\sqrt 5}$\\
$K(a^{319})=\displaystyle{-16-12\sqrt 5}$\\
$K(a^{323})=\displaystyle{4-6\sqrt 5}$\\
$K(a^{324})=\displaystyle{\frac{33}2-\frac{9}2\sqrt 5}$\\
$K(a^{338})=\displaystyle{4+6\sqrt 5}$\\
$K(a^{339})=\displaystyle{-1-13\sqrt 5}$\\
$K(a^{343})=\displaystyle{9-15\sqrt 5}$\\
$K(a^{344})=\displaystyle{-16-2\sqrt 5}$\\
$K(a^{348})=\displaystyle{\frac{13}2-\frac{19}2\sqrt 5}$\\
$K(a^{349})=\displaystyle{-6-12\sqrt 5}$\\
$K(a^{364})=\displaystyle{\frac{73}2+\frac{9}2\sqrt 5}$\\
$K(a^{368})=\displaystyle{-\frac{47}2+\frac{15}2\sqrt 5}$\\
$K(a^{369})=\displaystyle{-11-9\sqrt 5}$\\
$K(a^{373})=\displaystyle{9+9\sqrt 5}$\\
$K(a^{374})=\displaystyle{-16+6\sqrt 5}$\\
$K(a^{468})=\displaystyle{14-16\sqrt 5}$\\
$K(a^{469})=\displaystyle{-1+9\sqrt 5}$\\
$K(a^{474})=\displaystyle{-\frac{17}2+\frac{7}2\sqrt 5}$\\
$K(a^{494})=\displaystyle{44+2\sqrt 5}$\\
$K(a^{499})=\displaystyle{14-10\sqrt 5}$\\
}
\end{multicols}

\newpage
\begin{center}
{\bf Cyclotomic Cosets}
\end{center}

\medskip
\begin{multicols}{3}
{\tiny
\noindent ${\bf F}_{4}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 2\}$\\
�� There are 2 cosets.

\bigskip
\noindent ${\bf F}_{8}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 2, 4\}$\\
$C(3)=\{3, 5, 6\}$\\
�� There are 3 cosets.

\bigskip
\noindent ${\bf F}_{16}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 2, 4, 8\}$\\
$C(3)=\{3, 6, 9, 12\}$\\
$C(5)=\{5, 10\}$\\
$C(7)=\{7, 11, 13, 14\}$\\
�� There are 5 cosets.

\bigskip
\noindent ${\bf F}_{32}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 2, 4, 8, 16\}$\\
$C(3)=\{3, 6, 12, 17, 24\}$\\
$C(5)=\{5, 9, 10, 18, 20\}$\\
$C(7)=\{7, 14, 19, 25, 28\}$\\
$C(11)=\{11, 13, 21, 22, 26\}$\\
$C(15)=\{15, 23, 27, 29, 30\}$\\
�� There are 7 cosets.

\bigskip
\noindent ${\bf F}_{64}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 2, 4, 8, 16, 32\}$\\
$C(3)=\{3, 6, 12, 24, 33, 48\}$\\
$C(5)=\{5, 10, 17, 20, 34, 40\}$\\
$C(7)=\{7, 14, 28, 35, 49, 56\}$\\
$C(9)=\{9, 18, 36\}$\\
$C(11)=\{11, 22, 25, 37, 44, 50\}$\\
$C(13)=\{13, 19, 26, 38, 41, 52\}$\\
$C(15)=\{15, 30, 39, 51, 57, 60\}$\\
$C(21)=\{21, 42\}$\\
$C(23)=\{23, 29, 43, 46, 53, 58\}$\\
$C(27)=\{27, 45, 54\}$\\
$C(31)=\{31, 47, 55, 59, 61, 62\}$\\
�� There are 13 cosets.

\bigskip
\noindent ${\bf F}_{128}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 2, 4, 8, 16, 32, 64\}$\\
$C(3)=\{3, 6, 12, 24, 48, 65, 96\}$\\
$C(5)=\{5, 10, 20, 33, 40, 66, 80\}$\\
$C(7)=\{7, 14, 28, 56, 67, 97, 112\}$\\
$C(9)=\{9, 17, 18, 34, 36, 68, 72\}$\\
$C(11)=\{11, 22, 44, 49, 69, 88, 98\}$\\
$C(13)=\{13, 26, 35, 52, 70, 81, 104\}$\\
$C(15)=\{15, 30, 60, 71, 99, 113, 120\}$\\
$C(19)=\{19, 25, 38, 50, 73, 76, 100\}$\\
$C(21)=\{21, 37, 41, 42, 74, 82, 84\}$\\
$C(23)=\{23, 46, 57, 75, 92, 101, 114\}$\\
$C(27)=\{27, 51, 54, 77, 89, 102, 108\}$\\
$C(29)=\{29, 39, 58, 78, 83, 105, 116\}$\\
$C(31)=\{31, 62, 79, 103, 115, 121, 124\}$\\
$C(43)=\{43, 45, 53, 85, 86, 90, 106\}$\\
$C(47)=\{47, 61, 87, 94, 107, 117, 122\}$\\
$C(55)=\{55, 59, 91, 93, 109, 110, 118\}$\\
$C(63)=\{63, 95, 111, 119, 123, 125, 126\}$\\
�� There are 19 cosets.
}
\end{multicols}

\begin{multicols}{2}
{\tiny
\noindent ${\bf F}_{256}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 2, 4, 8, 16, 32, 64, 128\}$\\
$C(3)=\{3, 6, 12, 24, 48, 96, 129, 192\}$\\
$C(5)=\{5, 10, 20, 40, 65, 80, 130, 160\}$\\
$C(7)=\{7, 14, 28, 56, 112, 131, 193, 224\}$\\
$C(9)=\{9, 18, 33, 36, 66, 72, 132, 144\}$\\
$C(11)=\{11, 22, 44, 88, 97, 133, 176, 194\}$\\
$C(13)=\{13, 26, 52, 67, 104, 134, 161, 208\}$\\
$C(15)=\{15, 30, 60, 120, 135, 195, 225, 240\}$\\
$C(17)=\{17, 34, 68, 136\}$\\
$C(19)=\{19, 38, 49, 76, 98, 137, 152, 196\}$\\
$C(21)=\{21, 42, 69, 81, 84, 138, 162, 168\}$\\
$C(23)=\{23, 46, 92, 113, 139, 184, 197, 226\}$\\
$C(25)=\{25, 35, 50, 70, 100, 140, 145, 200\}$\\
$C(27)=\{27, 54, 99, 108, 141, 177, 198, 216\}$\\
$C(29)=\{29, 58, 71, 116, 142, 163, 209, 232\}$\\
$C(31)=\{31, 62, 124, 143, 199, 227, 241, 248\}$\\
$C(37)=\{37, 41, 73, 74, 82, 146, 148, 164\}$\\
$C(39)=\{39, 57, 78, 114, 147, 156, 201, 228\}$\\
$C(43)=\{43, 86, 89, 101, 149, 172, 178, 202\}$\\
$C(45)=\{45, 75, 90, 105, 150, 165, 180, 210\}$\\
$C(47)=\{47, 94, 121, 151, 188, 203, 229, 242\}$\\
$C(51)=\{51, 102, 153, 204\}$\\
$C(53)=\{53, 77, 83, 106, 154, 166, 169, 212\}$\\
$C(55)=\{55, 110, 115, 155, 185, 205, 220, 230\}$\\
$C(59)=\{59, 103, 118, 157, 179, 206, 217, 236\}$\\
$C(61)=\{61, 79, 122, 158, 167, 211, 233, 244\}$\\
$C(63)=\{63, 126, 159, 207, 231, 243, 249, 252\}$\\
$C(85)=\{85, 170\}$\\
$C(87)=\{87, 93, 117, 171, 174, 186, 213, 234\}$\\
$C(91)=\{91, 107, 109, 173, 181, 182, 214, 218\}$\\
$C(95)=\{95, 125, 175, 190, 215, 235, 245, 250\}$\\
$C(111)=\{111, 123, 183, 189, 219, 222, 237, 246\}$\\
$C(119)=\{119, 187, 221, 238\}$\\
$C(127)=\{127, 191, 223, 239, 247, 251, 253, 254\}$\\
�� There are 35 cosets.

\bigskip
\noindent ${\bf F}_{512}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 2, 4, 8, 16, 32, 64, 128, 256\}$\\
$C(3)=\{3, 6, 12, 24, 48, 96, 192, 257, 384\}$\\
$C(5)=\{5, 10, 20, 40, 80, 129, 160, 258, 320\}$\\
$C(7)=\{7, 14, 28, 56, 112, 224, 259, 385, 448\}$\\
$C(9)=\{9, 18, 36, 65, 72, 130, 144, 260, 288\}$\\
$C(11)=\{11, 22, 44, 88, 176, 193, 261, 352, 386\}$\\
$C(13)=\{13, 26, 52, 104, 131, 208, 262, 321, 416\}$\\
$C(15)=\{15, 30, 60, 120, 240, 263, 387, 449, 480\}$\\
$C(17)=\{17, 33, 34, 66, 68, 132, 136, 264, 272\}$\\
$C(19)=\{19, 38, 76, 97, 152, 194, 265, 304, 388\}$\\
$C(21)=\{21, 42, 84, 133, 161, 168, 266, 322, 336\}$\\
$C(23)=\{23, 46, 92, 184, 225, 267, 368, 389, 450\}$\\
$C(25)=\{25, 50, 67, 100, 134, 200, 268, 289, 400\}$\\
$C(27)=\{27, 54, 108, 195, 216, 269, 353, 390, 432\}$\\
$C(29)=\{29, 58, 116, 135, 232, 270, 323, 417, 464\}$\\
$C(31)=\{31, 62, 124, 248, 271, 391, 451, 481, 496\}$\\
$C(35)=\{35, 49, 70, 98, 140, 196, 273, 280, 392\}$\\
$C(37)=\{37, 74, 81, 137, 148, 162, 274, 296, 324\}$\\
$C(39)=\{39, 78, 113, 156, 226, 275, 312, 393, 452\}$\\
$C(41)=\{41, 69, 82, 138, 145, 164, 276, 290, 328\}$\\
$C(43)=\{43, 86, 172, 177, 197, 277, 344, 354, 394\}$\\
$C(45)=\{45, 90, 139, 180, 209, 278, 325, 360, 418\}$\\
$C(47)=\{47, 94, 188, 241, 279, 376, 395, 453, 482\}$\\
$C(51)=\{51, 99, 102, 198, 204, 281, 305, 396, 408\}$\\
$C(53)=\{53, 106, 141, 163, 212, 282, 326, 337, 424\}$\\
$C(55)=\{55, 110, 220, 227, 283, 369, 397, 440, 454\}$\\
$C(57)=\{57, 71, 114, 142, 228, 284, 291, 401, 456\}$\\
$C(59)=\{59, 118, 199, 236, 285, 355, 398, 433, 472\}$\\
$C(61)=\{61, 122, 143, 244, 286, 327, 419, 465, 488\}$\\
$C(63)=\{63, 126, 252, 287, 399, 455, 483, 497, 504\}$\\
$C(73)=\{73, 146, 292\}$\\
$C(75)=\{75, 89, 150, 178, 201, 293, 300, 356, 402\}$\\
$C(77)=\{77, 105, 147, 154, 210, 294, 308, 329, 420\}$\\
$C(79)=\{79, 121, 158, 242, 295, 316, 403, 457, 484\}$\\
$C(83)=\{83, 101, 153, 166, 202, 297, 306, 332, 404\}$\\
$C(85)=\{85, 149, 165, 169, 170, 298, 330, 338, 340\}$\\
$C(87)=\{87, 174, 185, 229, 299, 348, 370, 405, 458\}$\\
$C(91)=\{91, 182, 203, 217, 301, 357, 364, 406, 434\}$\\
$C(93)=\{93, 151, 186, 233, 302, 331, 372, 421, 466\}$\\
$C(95)=\{95, 190, 249, 303, 380, 407, 459, 485, 498\}$\\
$C(103)=\{103, 115, 206, 230, 307, 313, 409, 412, 460\}$\\
$C(107)=\{107, 179, 205, 214, 309, 345, 358, 410, 428\}$\\
$C(109)=\{109, 155, 211, 218, 310, 333, 361, 422, 436\}$\\
$C(111)=\{111, 222, 243, 311, 377, 411, 444, 461, 486\}$\\
$C(117)=\{117, 157, 167, 234, 314, 334, 339, 425, 468\}$\\
$C(119)=\{119, 231, 238, 315, 371, 413, 441, 462, 476\}$\\
$C(123)=\{123, 207, 246, 317, 359, 414, 435, 473, 492\}$\\
$C(125)=\{125, 159, 250, 318, 335, 423, 467, 489, 500\}$\\
$C(127)=\{127, 254, 319, 415, 463, 487, 499, 505, 508\}$\\
$C(171)=\{171, 173, 181, 213, 341, 342, 346, 362, 426\}$\\
$C(175)=\{175, 189, 245, 343, 350, 378, 427, 469, 490\}$\\
$C(183)=\{183, 221, 235, 347, 366, 373, 429, 442, 470\}$\\
$C(187)=\{187, 215, 237, 349, 363, 374, 430, 437, 474\}$\\
$C(191)=\{191, 253, 351, 382, 431, 471, 491, 501, 506\}$\\
$C(219)=\{219, 365, 438\}$\\
$C(223)=\{223, 251, 367, 381, 439, 446, 475, 493, 502\}$\\
$C(239)=\{239, 247, 375, 379, 443, 445, 477, 478, 494\}$\\
$C(255)=\{255, 383, 447, 479, 495, 503, 507, 509, 510\}$\\
�� There are 59 cosets.

\bigskip
\noindent ${\bf F}_{1024}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 2, 4, 8, 16, 32, 64, 128, 256, 512\}$\\
$C(3)=\{3, 6, 12, 24, 48, 96, 192, 384, 513, 768\}$\\
$C(5)=\{5, 10, 20, 40, 80, 160, 257, 320, 514, 640\}$\\
$C(7)=\{7, 14, 28, 56, 112, 224, 448, 515, 769, 896\}$\\
$C(9)=\{9, 18, 36, 72, 129, 144, 258, 288, 516, 576\}$\\
$C(11)=\{11, 22, 44, 88, 176, 352, 385, 517, 704, 770\}$\\
$C(13)=\{13, 26, 52, 104, 208, 259, 416, 518, 641, 832\}$\\
$C(15)=\{15, 30, 60, 120, 240, 480, 519, 771, 897, 960\}$\\
$C(17)=\{17, 34, 65, 68, 130, 136, 260, 272, 520, 544\}$\\
$C(19)=\{19, 38, 76, 152, 193, 304, 386, 521, 608, 772\}$\\
$C(21)=\{21, 42, 84, 168, 261, 321, 336, 522, 642, 672\}$\\
$C(23)=\{23, 46, 92, 184, 368, 449, 523, 736, 773, 898\}$\\
$C(25)=\{25, 50, 100, 131, 200, 262, 400, 524, 577, 800\}$\\
$C(27)=\{27, 54, 108, 216, 387, 432, 525, 705, 774, 864\}$\\
$C(29)=\{29, 58, 116, 232, 263, 464, 526, 643, 833, 928\}$\\
$C(31)=\{31, 62, 124, 248, 496, 527, 775, 899, 961, 992\}$\\
$C(33)=\{33, 66, 132, 264, 528\}$\\
$C(35)=\{35, 70, 97, 140, 194, 280, 388, 529, 560, 776\}$\\
$C(37)=\{37, 74, 148, 161, 265, 296, 322, 530, 592, 644\}$\\
$C(39)=\{39, 78, 156, 225, 312, 450, 531, 624, 777, 900\}$\\
$C(41)=\{41, 82, 133, 164, 266, 289, 328, 532, 578, 656\}$\\
$C(43)=\{43, 86, 172, 344, 353, 389, 533, 688, 706, 778\}$\\
$C(45)=\{45, 90, 180, 267, 360, 417, 534, 645, 720, 834\}$\\
$C(47)=\{47, 94, 188, 376, 481, 535, 752, 779, 901, 962\}$\\
$C(49)=\{49, 67, 98, 134, 196, 268, 392, 536, 545, 784\}$\\
$C(51)=\{51, 102, 195, 204, 390, 408, 537, 609, 780, 816\}$\\
$C(53)=\{53, 106, 212, 269, 323, 424, 538, 646, 673, 848\}$\\
$C(55)=\{55, 110, 220, 440, 451, 539, 737, 781, 880, 902\}$\\
$C(57)=\{57, 114, 135, 228, 270, 456, 540, 579, 801, 912\}$\\
$C(59)=\{59, 118, 236, 391, 472, 541, 707, 782, 865, 944\}$\\
$C(61)=\{61, 122, 244, 271, 488, 542, 647, 835, 929, 976\}$\\
$C(63)=\{63, 126, 252, 504, 543, 783, 903, 963, 993, 1008\}$\\
$C(69)=\{69, 81, 138, 162, 273, 276, 324, 546, 552, 648\}$\\
$C(71)=\{71, 113, 142, 226, 284, 452, 547, 568, 785, 904\}$\\
$C(73)=\{73, 137, 145, 146, 274, 290, 292, 548, 580, 584\}$\\
$C(75)=\{75, 150, 177, 300, 354, 393, 549, 600, 708, 786\}$\\
$C(77)=\{77, 154, 209, 275, 308, 418, 550, 616, 649, 836\}$\\
$C(79)=\{79, 158, 241, 316, 482, 551, 632, 787, 905, 964\}$\\
$C(83)=\{83, 166, 197, 305, 332, 394, 553, 610, 664, 788\}$\\
$C(85)=\{85, 170, 277, 325, 337, 340, 554, 650, 674, 680\}$\\
$C(87)=\{87, 174, 348, 369, 453, 555, 696, 738, 789, 906\}$\\
$C(89)=\{89, 139, 178, 278, 356, 401, 556, 581, 712, 802\}$\\
$C(91)=\{91, 182, 364, 395, 433, 557, 709, 728, 790, 866\}$\\
$C(93)=\{93, 186, 279, 372, 465, 558, 651, 744, 837, 930\}$\\
$C(95)=\{95, 190, 380, 497, 559, 760, 791, 907, 965, 994\}$\\
$C(99)=\{99, 198, 396, 561, 792\}$\\
$C(101)=\{101, 163, 202, 281, 326, 404, 562, 593, 652, 808\}$\\
$C(103)=\{103, 206, 227, 412, 454, 563, 625, 793, 824, 908\}$\\
$C(105)=\{105, 141, 210, 282, 291, 420, 564, 582, 657, 840\}$\\
$C(107)=\{107, 214, 355, 397, 428, 565, 689, 710, 794, 856\}$\\
$C(109)=\{109, 218, 283, 419, 436, 566, 653, 721, 838, 872\}$\\
$C(111)=\{111, 222, 444, 483, 567, 753, 795, 888, 909, 966\}$\\
$C(115)=\{115, 199, 230, 398, 460, 569, 611, 796, 817, 920\}$\\
$C(117)=\{117, 234, 285, 327, 468, 570, 654, 675, 849, 936\}$\\
$C(119)=\{119, 238, 455, 476, 571, 739, 797, 881, 910, 952\}$\\
$C(121)=\{121, 143, 242, 286, 484, 572, 583, 803, 913, 968\}$\\
$C(123)=\{123, 246, 399, 492, 573, 711, 798, 867, 945, 984\}$\\
$C(125)=\{125, 250, 287, 500, 574, 655, 839, 931, 977, 1000\}$\\
$C(127)=\{127, 254, 508, 575, 799, 911, 967, 995, 1009, 1016\}$\\
$C(147)=\{147, 153, 201, 294, 306, 402, 585, 588, 612, 804\}$\\
$C(149)=\{149, 169, 293, 298, 329, 338, 586, 596, 658, 676\}$\\
$C(151)=\{151, 185, 302, 370, 457, 587, 604, 740, 805, 914\}$\\
$C(155)=\{155, 217, 310, 403, 434, 589, 620, 713, 806, 868\}$\\
$C(157)=\{157, 233, 295, 314, 466, 590, 628, 659, 841, 932\}$\\
$C(159)=\{159, 249, 318, 498, 591, 636, 807, 915, 969, 996\}$\\
$C(165)=\{165, 297, 330, 594, 660\}$\\
$C(167)=\{167, 229, 313, 334, 458, 595, 626, 668, 809, 916\}$\\
$C(171)=\{171, 342, 345, 357, 405, 597, 684, 690, 714, 810\}$\\
$C(173)=\{173, 299, 346, 361, 421, 598, 661, 692, 722, 842\}$\\
$C(175)=\{175, 350, 377, 485, 599, 700, 754, 811, 917, 970\}$\\
$C(179)=\{179, 203, 358, 406, 409, 601, 613, 716, 812, 818\}$\\
$C(181)=\{181, 301, 331, 362, 425, 602, 662, 677, 724, 850\}$\\
$C(183)=\{183, 366, 441, 459, 603, 732, 741, 813, 882, 918\}$\\
$C(187)=\{187, 374, 407, 473, 605, 715, 748, 814, 869, 946\}$\\
$C(189)=\{189, 303, 378, 489, 606, 663, 756, 843, 933, 978\}$\\
$C(191)=\{191, 382, 505, 607, 764, 815, 919, 971, 997, 1010\}$\\
$C(205)=\{205, 211, 307, 410, 422, 614, 617, 665, 820, 844\}$\\
$C(207)=\{207, 243, 414, 486, 615, 633, 819, 828, 921, 972\}$\\
$C(213)=\{213, 309, 333, 339, 426, 618, 666, 678, 681, 852\}$\\
$C(215)=\{215, 371, 430, 461, 619, 697, 742, 821, 860, 922\}$\\
$C(219)=\{219, 411, 435, 438, 621, 717, 729, 822, 870, 876\}$\\
$C(221)=\{221, 311, 442, 467, 622, 667, 745, 845, 884, 934\}$\\
$C(223)=\{223, 446, 499, 623, 761, 823, 892, 923, 973, 998\}$\\
$C(231)=\{231, 462, 627, 825, 924\}$\\
$C(235)=\{235, 359, 413, 470, 629, 691, 718, 826, 857, 940\}$\\
$C(237)=\{237, 315, 423, 474, 630, 669, 723, 846, 873, 948\}$\\
$C(239)=\{239, 478, 487, 631, 755, 827, 889, 925, 956, 974\}$\\
$C(245)=\{245, 317, 335, 490, 634, 670, 679, 851, 937, 980\}$\\
$C(247)=\{247, 463, 494, 635, 743, 829, 883, 926, 953, 988\}$\\
$C(251)=\{251, 415, 502, 637, 719, 830, 871, 947, 985, 1004\}$\\
$C(253)=\{253, 319, 506, 638, 671, 847, 935, 979, 1001, 1012\}$\\
$C(255)=\{255, 510, 639, 831, 927, 975, 999, 1011, 1017, 1020\}$\\
$C(341)=\{341, 682\}$\\
$C(343)=\{343, 349, 373, 469, 683, 686, 698, 746, 853, 938\}$\\
$C(347)=\{347, 365, 427, 437, 685, 694, 725, 730, 854, 874\}$\\
$C(351)=\{351, 381, 501, 687, 702, 762, 855, 939, 981, 1002\}$\\
$C(363)=\{363, 429, 693, 726, 858\}$\\
$C(367)=\{367, 445, 491, 695, 734, 757, 859, 890, 941, 982\}$\\
$C(375)=\{375, 471, 477, 699, 747, 750, 861, 885, 942, 954\}$\\
$C(379)=\{379, 431, 493, 701, 727, 758, 862, 875, 949, 986\}$\\
$C(383)=\{383, 509, 703, 766, 863, 943, 983, 1003, 1013, 1018\}$\\
$C(439)=\{439, 443, 475, 731, 733, 749, 877, 878, 886, 950\}$\\
$C(447)=\{447, 507, 735, 765, 879, 894, 951, 987, 1005, 1014\}$\\
$C(479)=\{479, 503, 751, 763, 887, 893, 955, 958, 989, 1006\}$\\
$C(495)=\{495, 759, 891, 957, 990\}$\\
$C(511)=\{511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022\}$\\
�� There are 107 cosets.
}
\end{multicols}


\begin{multicols}{3}
{\tiny
\noindent ${\bf F}_{9}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 3\}$\\
$C(2)=\{2, 6\}$\\
$C(4)=\{4\}$\\
$C(5)=\{5, 7\}$\\
�� There are 5 cosets.

\bigskip
\noindent ${\bf F}_{27}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 3, 9\}$\\
$C(2)=\{2, 6, 18\}$\\
$C(4)=\{4, 10, 12\}$\\
$C(5)=\{5, 15, 19\}$\\
$C(7)=\{7, 11, 21\}$\\
$C(8)=\{8, 20, 24\}$\\
$C(13)=\{13\}$\\
$C(14)=\{14, 16, 22\}$\\
$C(17)=\{17, 23, 25\}$\\
�� There are 10 cosets.

\bigskip
\noindent ${\bf F}_{81}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 3, 9, 27\}$\\
$C(2)=\{2, 6, 18, 54\}$\\
$C(4)=\{4, 12, 28, 36\}$\\
$C(5)=\{5, 15, 45, 55\}$\\
$C(7)=\{7, 21, 29, 63\}$\\
$C(8)=\{8, 24, 56, 72\}$\\
$C(10)=\{10, 30\}$\\
$C(11)=\{11, 19, 33, 57\}$\\
$C(13)=\{13, 31, 37, 39\}$\\
$C(14)=\{14, 42, 46, 58\}$\\
$C(16)=\{16, 32, 48, 64\}$\\
$C(17)=\{17, 51, 59, 73\}$\\
$C(20)=\{20, 60\}$\\
$C(22)=\{22, 34, 38, 66\}$\\
$C(23)=\{23, 47, 61, 69\}$\\
$C(25)=\{25, 35, 65, 75\}$\\
$C(26)=\{26, 62, 74, 78\}$\\
$C(40)=\{40\}$\\
$C(41)=\{41, 43, 49, 67\}$\\
$C(44)=\{44, 52, 68, 76\}$\\
$C(50)=\{50, 70\}$\\
$C(53)=\{53, 71, 77, 79\}$\\
�� There are 23 cosets.

\bigskip
\bigskip
\noindent ${\bf F}_{243}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 3, 9, 27, 81\}$\\
$C(2)=\{2, 6, 18, 54, 162\}$\\
$C(4)=\{4, 12, 36, 82, 108\}$\\
$C(5)=\{5, 15, 45, 135, 163\}$\\
$C(7)=\{7, 21, 63, 83, 189\}$\\
$C(8)=\{8, 24, 72, 164, 216\}$\\
$C(10)=\{10, 28, 30, 84, 90\}$\\
$C(11)=\{11, 33, 55, 99, 165\}$\\
$C(13)=\{13, 39, 85, 109, 117\}$\\
$C(14)=\{14, 42, 126, 136, 166\}$\\
$C(16)=\{16, 48, 86, 144, 190\}$\\
$C(17)=\{17, 51, 153, 167, 217\}$\\
$C(19)=\{19, 29, 57, 87, 171\}$\\
$C(20)=\{20, 56, 60, 168, 180\}$\\
$C(22)=\{22, 66, 88, 110, 198\}$\\
$C(23)=\{23, 69, 137, 169, 207\}$\\
$C(25)=\{25, 75, 89, 191, 225\}$\\
$C(26)=\{26, 78, 170, 218, 234\}$\\
$C(31)=\{31, 37, 91, 93, 111\}$\\
$C(32)=\{32, 46, 96, 138, 172\}$\\
$C(34)=\{34, 64, 92, 102, 192\}$\\
$C(35)=\{35, 73, 105, 173, 219\}$\\
$C(38)=\{38, 58, 100, 114, 174\}$\\
$C(40)=\{40, 94, 112, 118, 120\}$\\
$C(41)=\{41, 123, 127, 139, 175\}$\\
$C(43)=\{43, 95, 129, 145, 193\}$\\
$C(44)=\{44, 132, 154, 176, 220\}$\\
$C(47)=\{47, 59, 141, 177, 181\}$\\
$C(49)=\{49, 97, 113, 147, 199\}$\\
$C(50)=\{50, 140, 150, 178, 208\}$\\
$C(52)=\{52, 98, 156, 194, 226\}$\\
$C(53)=\{53, 159, 179, 221, 235\}$\\
$C(61)=\{61, 65, 101, 183, 195\}$\\
$C(62)=\{62, 74, 182, 186, 222\}$\\
$C(67)=\{67, 103, 115, 119, 201\}$\\
$C(68)=\{68, 128, 142, 184, 204\}$\\
$C(70)=\{70, 104, 146, 196, 210\}$\\
$C(71)=\{71, 155, 185, 213, 223\}$\\
$C(76)=\{76, 106, 116, 200, 228\}$\\
$C(77)=\{77, 143, 187, 209, 231\}$\\
$C(79)=\{79, 107, 197, 227, 237\}$\\
$C(80)=\{80, 188, 224, 236, 240\}$\\
$C(121)=\{121\}$\\
$C(122)=\{122, 124, 130, 148, 202\}$\\
$C(125)=\{125, 133, 157, 203, 229\}$\\
$C(131)=\{131, 149, 151, 205, 211\}$\\
$C(134)=\{134, 160, 206, 230, 238\}$\\
$C(152)=\{152, 158, 212, 214, 232\}$\\
$C(161)=\{161, 215, 233, 239, 241\}$\\
�� There are 50 cosets.

\bigskip
\noindent ${\bf F}_{729}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 3, 9, 27, 81, 243\}$\\
$C(2)=\{2, 6, 18, 54, 162, 486\}$\\
$C(4)=\{4, 12, 36, 108, 244, 324\}$\\
$C(5)=\{5, 15, 45, 135, 405, 487\}$\\
$C(7)=\{7, 21, 63, 189, 245, 567\}$\\
$C(8)=\{8, 24, 72, 216, 488, 648\}$\\
$C(10)=\{10, 30, 82, 90, 246, 270\}$\\
$C(11)=\{11, 33, 99, 163, 297, 489\}$\\
$C(13)=\{13, 39, 117, 247, 325, 351\}$\\
$C(14)=\{14, 42, 126, 378, 406, 490\}$\\
$C(16)=\{16, 48, 144, 248, 432, 568\}$\\
$C(17)=\{17, 51, 153, 459, 491, 649\}$\\
$C(19)=\{19, 57, 83, 171, 249, 513\}$\\
$C(20)=\{20, 60, 164, 180, 492, 540\}$\\
$C(22)=\{22, 66, 198, 250, 326, 594\}$\\
$C(23)=\{23, 69, 207, 407, 493, 621\}$\\
$C(25)=\{25, 75, 225, 251, 569, 675\}$\\
$C(26)=\{26, 78, 234, 494, 650, 702\}$\\
$C(28)=\{28, 84, 252\}$\\
$C(29)=\{29, 55, 87, 165, 261, 495\}$\\
$C(31)=\{31, 93, 109, 253, 279, 327\}$\\
$C(32)=\{32, 96, 136, 288, 408, 496\}$\\
$C(34)=\{34, 102, 190, 254, 306, 570\}$\\
$C(35)=\{35, 105, 217, 315, 497, 651\}$\\
$C(37)=\{37, 85, 111, 255, 271, 333\}$\\
$C(38)=\{38, 114, 166, 298, 342, 498\}$\\
$C(40)=\{40, 120, 256, 328, 352, 360\}$\\
$C(41)=\{41, 123, 369, 379, 409, 499\}$\\
$C(43)=\{43, 129, 257, 387, 433, 571\}$\\
$C(44)=\{44, 132, 396, 460, 500, 652\}$\\
$C(46)=\{46, 86, 138, 258, 414, 514\}$\\
$C(47)=\{47, 141, 167, 423, 501, 541\}$\\
$C(49)=\{49, 147, 259, 329, 441, 595\}$\\
$C(50)=\{50, 150, 410, 450, 502, 622\}$\\
$C(52)=\{52, 156, 260, 468, 572, 676\}$\\
$C(53)=\{53, 159, 477, 503, 653, 703\}$\\
$C(56)=\{56, 168, 504\}$\\
$C(58)=\{58, 110, 174, 262, 330, 522\}$\\
$C(59)=\{59, 137, 177, 411, 505, 531\}$\\
$C(61)=\{61, 183, 191, 263, 549, 573\}$\\
$C(62)=\{62, 186, 218, 506, 558, 654\}$\\
$C(64)=\{64, 88, 192, 264, 272, 576\}$\\
$C(65)=\{65, 169, 195, 299, 507, 585\}$\\
$C(67)=\{67, 201, 265, 331, 353, 603\}$\\
$C(68)=\{68, 204, 380, 412, 508, 612\}$\\
$C(70)=\{70, 210, 266, 434, 574, 630\}$\\
$C(71)=\{71, 213, 461, 509, 639, 655\}$\\
$C(73)=\{73, 89, 219, 267, 515, 657\}$\\
$C(74)=\{74, 170, 222, 510, 542, 666\}$\\
$C(76)=\{76, 228, 268, 332, 596, 684\}$\\
$C(77)=\{77, 231, 413, 511, 623, 693\}$\\
$C(79)=\{79, 237, 269, 575, 677, 711\}$\\
$C(80)=\{80, 240, 512, 656, 704, 720\}$\\
$C(91)=\{91, 273\}$\\
$C(92)=\{92, 100, 172, 276, 300, 516\}$\\
$C(94)=\{94, 118, 274, 282, 334, 354\}$\\
$C(95)=\{95, 127, 285, 381, 415, 517\}$\\
$C(97)=\{97, 145, 275, 291, 435, 577\}$\\
$C(98)=\{98, 154, 294, 462, 518, 658\}$\\
$C(101)=\{101, 173, 181, 303, 519, 543\}$\\
$C(103)=\{103, 199, 277, 309, 335, 597\}$\\
$C(104)=\{104, 208, 312, 416, 520, 624\}$\\
$C(106)=\{106, 226, 278, 318, 578, 678\}$\\
$C(107)=\{107, 235, 321, 521, 659, 705\}$\\
$C(112)=\{112, 280, 336\}$\\
$C(113)=\{113, 139, 289, 339, 417, 523\}$\\
$C(115)=\{115, 193, 281, 307, 345, 579\}$\\
$C(116)=\{116, 220, 316, 348, 524, 660\}$\\
$C(119)=\{119, 175, 301, 343, 357, 525\}$\\
$C(121)=\{121, 283, 337, 355, 361, 363\}$\\
$C(122)=\{122, 366, 370, 382, 418, 526\}$\\
$C(124)=\{124, 284, 372, 388, 436, 580\}$\\
$C(125)=\{125, 375, 397, 463, 527, 661\}$\\
$C(128)=\{128, 176, 384, 424, 528, 544\}$\\
$C(130)=\{130, 286, 338, 390, 442, 598\}$\\
$C(131)=\{131, 393, 419, 451, 529, 625\}$\\
$C(133)=\{133, 287, 399, 469, 581, 679\}$\\
$C(134)=\{134, 402, 478, 530, 662, 706\}$\\
$C(140)=\{140, 420, 532\}$\\
$C(142)=\{142, 194, 290, 426, 550, 582\}$\\
$C(143)=\{143, 221, 429, 533, 559, 663\}$\\
$C(146)=\{146, 178, 302, 438, 534, 586\}$\\
$C(148)=\{148, 292, 340, 356, 444, 604\}$\\
$C(149)=\{149, 383, 421, 447, 535, 613\}$\\
$C(151)=\{151, 293, 437, 453, 583, 631\}$\\
$C(152)=\{152, 456, 464, 536, 640, 664\}$\\
$C(155)=\{155, 179, 465, 537, 545, 667\}$\\
$C(157)=\{157, 295, 341, 471, 599, 685\}$\\
$C(158)=\{158, 422, 474, 538, 626, 694\}$\\
$C(160)=\{160, 296, 480, 584, 680, 712\}$\\
$C(161)=\{161, 483, 539, 665, 707, 721\}$\\
$C(182)=\{182, 546\}$\\
$C(184)=\{184, 200, 304, 344, 552, 600\}$\\
$C(185)=\{185, 209, 425, 547, 555, 627\}$\\
$C(187)=\{187, 227, 305, 561, 587, 681\}$\\
$C(188)=\{188, 236, 548, 564, 668, 708\}$\\
$C(196)=\{196, 308, 588\}$\\
$C(197)=\{197, 223, 317, 551, 591, 669\}$\\
$C(202)=\{202, 310, 346, 358, 362, 606\}$\\
$C(203)=\{203, 371, 385, 427, 553, 609\}$\\
$C(205)=\{205, 311, 389, 439, 589, 615\}$\\
$C(206)=\{206, 398, 466, 554, 618, 670\}$\\
$C(211)=\{211, 313, 347, 443, 601, 633\}$\\
$C(212)=\{212, 428, 452, 556, 628, 636\}$\\
$C(214)=\{214, 314, 470, 590, 642, 682\}$\\
$C(215)=\{215, 479, 557, 645, 671, 709\}$\\
$C(224)=\{224, 560, 672\}$\\
$C(229)=\{229, 319, 349, 359, 605, 687\}$\\
$C(230)=\{230, 386, 430, 562, 614, 690\}$\\
$C(232)=\{232, 320, 440, 592, 632, 696\}$\\
$C(233)=\{233, 467, 563, 641, 673, 699\}$\\
$C(238)=\{238, 322, 350, 602, 686, 714\}$\\
$C(239)=\{239, 431, 565, 629, 695, 717\}$\\
$C(241)=\{241, 323, 593, 683, 713, 723\}$\\
$C(242)=\{242, 566, 674, 710, 722, 726\}$\\
$C(364)=\{364\}$\\
$C(365)=\{365, 367, 373, 391, 445, 607\}$\\
$C(368)=\{368, 376, 400, 472, 608, 688\}$\\
$C(374)=\{374, 394, 446, 454, 610, 634\}$\\
$C(377)=\{377, 403, 481, 611, 689, 715\}$\\
$C(392)=\{392, 448, 616\}$\\
$C(395)=\{395, 457, 473, 617, 643, 691\}$\\
$C(401)=\{401, 449, 475, 619, 635, 697\}$\\
$C(404)=\{404, 484, 620, 692, 716, 724\}$\\
$C(455)=\{455, 637\}$\\
$C(458)=\{458, 482, 638, 646, 698, 718\}$\\
$C(476)=\{476, 644, 700\}$\\
$C(485)=\{485, 647, 701, 719, 725, 727\}$\\
�� There are 129 cosets.
}
\end{multicols}


\begin{multicols}{4}
{\tiny
\noindent ${\bf F}_{25}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 5\}$\\
$C(2)=\{2, 10\}$\\
$C(3)=\{3, 15\}$\\
$C(4)=\{4, 20\}$\\
$C(6)=\{6\}$\\
$C(7)=\{7, 11\}$\\
$C(8)=\{8, 16\}$\\
$C(9)=\{9, 21\}$\\
$C(12)=\{12\}$\\
$C(13)=\{13, 17\}$\\
$C(14)=\{14, 22\}$\\
$C(18)=\{18\}$\\
$C(19)=\{19, 23\}$\\
�� There are 14 cosets.

\bigskip
\noindent ${\bf F}_{125}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 5, 25\}$\\
$C(2)=\{2, 10, 50\}$\\
$C(3)=\{3, 15, 75\}$\\
$C(4)=\{4, 20, 100\}$\\
$C(6)=\{6, 26, 30\}$\\
$C(7)=\{7, 35, 51\}$\\
$C(8)=\{8, 40, 76\}$\\
$C(9)=\{9, 45, 101\}$\\
$C(11)=\{11, 27, 55\}$\\
$C(12)=\{12, 52, 60\}$\\
$C(13)=\{13, 65, 77\}$\\
$C(14)=\{14, 70, 102\}$\\
$C(16)=\{16, 28, 80\}$\\
$C(17)=\{17, 53, 85\}$\\
$C(18)=\{18, 78, 90\}$\\
$C(19)=\{19, 95, 103\}$\\
$C(21)=\{21, 29, 105\}$\\
$C(22)=\{22, 54, 110\}$\\
$C(23)=\{23, 79, 115\}$\\
$C(24)=\{24, 104, 120\}$\\
$C(31)=\{31\}$\\
$C(32)=\{32, 36, 56\}$\\
$C(33)=\{33, 41, 81\}$\\
$C(34)=\{34, 46, 106\}$\\
$C(37)=\{37, 57, 61\}$\\
$C(38)=\{38, 66, 82\}$\\
$C(39)=\{39, 71, 107\}$\\
$C(42)=\{42, 58, 86\}$\\
$C(43)=\{43, 83, 91\}$\\
$C(44)=\{44, 96, 108\}$\\
$C(47)=\{47, 59, 111\}$\\
$C(48)=\{48, 84, 116\}$\\
$C(49)=\{49, 109, 121\}$\\
$C(62)=\{62\}$\\
$C(63)=\{63, 67, 87\}$\\
$C(64)=\{64, 72, 112\}$\\
$C(68)=\{68, 88, 92\}$\\
$C(69)=\{69, 97, 113\}$\\
$C(73)=\{73, 89, 117\}$\\
$C(74)=\{74, 114, 122\}$\\
$C(93)=\{93\}$\\
$C(94)=\{94, 98, 118\}$\\
$C(99)=\{99, 119, 123\}$\\
�� There are 44 cosets.

\bigskip
\noindent ${\bf F}_{625}$

\medskip
\noindent $C(0)=\{0\}$\\
$C(1)=\{1, 5, 25, 125\}$\\
$C(2)=\{2, 10, 50, 250\}$\\
$C(3)=\{3, 15, 75, 375\}$\\
$C(4)=\{4, 20, 100, 500\}$\\
$C(6)=\{6, 30, 126, 150\}$\\
$C(7)=\{7, 35, 175, 251\}$\\
$C(8)=\{8, 40, 200, 376\}$\\
$C(9)=\{9, 45, 225, 501\}$\\
$C(11)=\{11, 55, 127, 275\}$\\
$C(12)=\{12, 60, 252, 300\}$\\
$C(13)=\{13, 65, 325, 377\}$\\
$C(14)=\{14, 70, 350, 502\}$\\
$C(16)=\{16, 80, 128, 400\}$\\
$C(17)=\{17, 85, 253, 425\}$\\
$C(18)=\{18, 90, 378, 450\}$\\
$C(19)=\{19, 95, 475, 503\}$\\
$C(21)=\{21, 105, 129, 525\}$\\
$C(22)=\{22, 110, 254, 550\}$\\
$C(23)=\{23, 115, 379, 575\}$\\
$C(24)=\{24, 120, 504, 600\}$\\
$C(26)=\{26, 130\}$\\
$C(27)=\{27, 51, 135, 255\}$\\
$C(28)=\{28, 76, 140, 380\}$\\
$C(29)=\{29, 101, 145, 505\}$\\
$C(31)=\{31, 131, 151, 155\}$\\
$C(32)=\{32, 160, 176, 256\}$\\
$C(33)=\{33, 165, 201, 381\}$\\
$C(34)=\{34, 170, 226, 506\}$\\
$C(36)=\{36, 132, 180, 276\}$\\
$C(37)=\{37, 185, 257, 301\}$\\
$C(38)=\{38, 190, 326, 382\}$\\
$C(39)=\{39, 195, 351, 507\}$\\
$C(41)=\{41, 133, 205, 401\}$\\
$C(42)=\{42, 210, 258, 426\}$\\
$C(43)=\{43, 215, 383, 451\}$\\
$C(44)=\{44, 220, 476, 508\}$\\
$C(46)=\{46, 134, 230, 526\}$\\
$C(47)=\{47, 235, 259, 551\}$\\
$C(48)=\{48, 240, 384, 576\}$\\
$C(49)=\{49, 245, 509, 601\}$\\
$C(52)=\{52, 260\}$\\
$C(53)=\{53, 77, 265, 385\}$\\
$C(54)=\{54, 102, 270, 510\}$\\
$C(56)=\{56, 136, 152, 280\}$\\
$C(57)=\{57, 177, 261, 285\}$\\
$C(58)=\{58, 202, 290, 386\}$\\
$C(59)=\{59, 227, 295, 511\}$\\
$C(61)=\{61, 137, 277, 305\}$\\
$C(62)=\{62, 262, 302, 310\}$\\
$C(63)=\{63, 315, 327, 387\}$\\
$C(64)=\{64, 320, 352, 512\}$\\
$C(66)=\{66, 138, 330, 402\}$\\
$C(67)=\{67, 263, 335, 427\}$\\
$C(68)=\{68, 340, 388, 452\}$\\
$C(69)=\{69, 345, 477, 513\}$\\
$C(71)=\{71, 139, 355, 527\}$\\
$C(72)=\{72, 264, 360, 552\}$\\
$C(73)=\{73, 365, 389, 577\}$\\
$C(74)=\{74, 370, 514, 602\}$\\
$C(78)=\{78, 390\}$\\
$C(79)=\{79, 103, 395, 515\}$\\
$C(81)=\{81, 141, 153, 405\}$\\
$C(82)=\{82, 178, 266, 410\}$\\
$C(83)=\{83, 203, 391, 415\}$\\
$C(84)=\{84, 228, 420, 516\}$\\
$C(86)=\{86, 142, 278, 430\}$\\
$C(87)=\{87, 267, 303, 435\}$\\
$C(88)=\{88, 328, 392, 440\}$\\
$C(89)=\{89, 353, 445, 517\}$\\
$C(91)=\{91, 143, 403, 455\}$\\
$C(92)=\{92, 268, 428, 460\}$\\
$C(93)=\{93, 393, 453, 465\}$\\
$C(94)=\{94, 470, 478, 518\}$\\
$C(96)=\{96, 144, 480, 528\}$\\
$C(97)=\{97, 269, 485, 553\}$\\
$C(98)=\{98, 394, 490, 578\}$\\
$C(99)=\{99, 495, 519, 603\}$\\
$C(104)=\{104, 520\}$\\
$C(106)=\{106, 146, 154, 530\}$\\
$C(107)=\{107, 179, 271, 535\}$\\
$C(108)=\{108, 204, 396, 540\}$\\
$C(109)=\{109, 229, 521, 545\}$\\
$C(111)=\{111, 147, 279, 555\}$\\
$C(112)=\{112, 272, 304, 560\}$\\
$C(113)=\{113, 329, 397, 565\}$\\
$C(114)=\{114, 354, 522, 570\}$\\
$C(116)=\{116, 148, 404, 580\}$\\
$C(117)=\{117, 273, 429, 585\}$\\
$C(118)=\{118, 398, 454, 590\}$\\
$C(119)=\{119, 479, 523, 595\}$\\
$C(121)=\{121, 149, 529, 605\}$\\
$C(122)=\{122, 274, 554, 610\}$\\
$C(123)=\{123, 399, 579, 615\}$\\
$C(124)=\{124, 524, 604, 620\}$\\
$C(156)=\{156\}$\\
$C(157)=\{157, 161, 181, 281\}$\\
$C(158)=\{158, 166, 206, 406\}$\\
$C(159)=\{159, 171, 231, 531\}$\\
$C(162)=\{162, 186, 282, 306\}$\\
$C(163)=\{163, 191, 331, 407\}$\\
$C(164)=\{164, 196, 356, 532\}$\\
$C(167)=\{167, 211, 283, 431\}$\\
$C(168)=\{168, 216, 408, 456\}$\\
$C(169)=\{169, 221, 481, 533\}$\\
$C(172)=\{172, 236, 284, 556\}$\\
$C(173)=\{173, 241, 409, 581\}$\\
$C(174)=\{174, 246, 534, 606\}$\\
$C(182)=\{182, 286\}$\\
$C(183)=\{183, 207, 291, 411\}$\\
$C(184)=\{184, 232, 296, 536\}$\\
$C(187)=\{187, 287, 307, 311\}$\\
$C(188)=\{188, 316, 332, 412\}$\\
$C(189)=\{189, 321, 357, 537\}$\\
$C(192)=\{192, 288, 336, 432\}$\\
$C(193)=\{193, 341, 413, 457\}$\\
$C(194)=\{194, 346, 482, 538\}$\\
$C(197)=\{197, 289, 361, 557\}$\\
$C(198)=\{198, 366, 414, 582\}$\\
$C(199)=\{199, 371, 539, 607\}$\\
$C(208)=\{208, 416\}$\\
$C(209)=\{209, 233, 421, 541\}$\\
$C(212)=\{212, 292, 308, 436\}$\\
$C(213)=\{213, 333, 417, 441\}$\\
$C(214)=\{214, 358, 446, 542\}$\\
$C(217)=\{217, 293, 433, 461\}$\\
$C(218)=\{218, 418, 458, 466\}$\\
$C(219)=\{219, 471, 483, 543\}$\\
$C(222)=\{222, 294, 486, 558\}$\\
$C(223)=\{223, 419, 491, 583\}$\\
$C(224)=\{224, 496, 544, 608\}$\\
$C(234)=\{234, 546\}$\\
$C(237)=\{237, 297, 309, 561\}$\\
$C(238)=\{238, 334, 422, 566\}$\\
$C(239)=\{239, 359, 547, 571\}$\\
$C(242)=\{242, 298, 434, 586\}$\\
$C(243)=\{243, 423, 459, 591\}$\\
$C(244)=\{244, 484, 548, 596\}$\\
$C(247)=\{247, 299, 559, 611\}$\\
$C(248)=\{248, 424, 584, 616\}$\\
$C(249)=\{249, 549, 609, 621\}$\\
$C(312)=\{312\}$\\
$C(313)=\{313, 317, 337, 437\}$\\
$C(314)=\{314, 322, 362, 562\}$\\
$C(318)=\{318, 342, 438, 462\}$\\
$C(319)=\{319, 347, 487, 563\}$\\
$C(323)=\{323, 367, 439, 587\}$\\
$C(324)=\{324, 372, 564, 612\}$\\
$C(338)=\{338, 442\}$\\
$C(339)=\{339, 363, 447, 567\}$\\
$C(343)=\{343, 443, 463, 467\}$\\
$C(344)=\{344, 472, 488, 568\}$\\
$C(348)=\{348, 444, 492, 588\}$\\
$C(349)=\{349, 497, 569, 613\}$\\
$C(364)=\{364, 572\}$\\
$C(368)=\{368, 448, 464, 592\}$\\
$C(369)=\{369, 489, 573, 597\}$\\
$C(373)=\{373, 449, 589, 617\}$\\
$C(374)=\{374, 574, 614, 622\}$\\
$C(468)=\{468\}$\\
$C(469)=\{469, 473, 493, 593\}$\\
$C(474)=\{474, 498, 594, 618\}$\\
$C(494)=\{494, 598\}$\\
$C(499)=\{499, 599, 619, 623\}$\\
�� There are 164 cosets.
}
\end{multicols}


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\end{thebibliography}
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