Dirichlet Character \(\chi_{89}(29,\cdot)\)

• Conductor: 89
• Order: 88
• Real: no
• Primitive: yes
• Minimal: yes
• Parity: odd
• Orbit label: 89.h

Basic properties

Conductor	=	89
Order	=	88
Real	=	no
Primitive	=	yes
Minimal	=	yes
Parity	=	odd
Orbit label	=	89.h
Orbit index	=	8

Galois orbit

\(\chi_{89}(3,\cdot)\) \(\chi_{89}(6,\cdot)\) \(\chi_{89}(7,\cdot)\) \(\chi_{89}(13,\cdot)\) \(\chi_{89}(14,\cdot)\) \(\chi_{89}(15,\cdot)\) \(\chi_{89}(19,\cdot)\) \(\chi_{89}(23,\cdot)\) \(\chi_{89}(24,\cdot)\) \(\chi_{89}(26,\cdot)\) \(\chi_{89}(27,\cdot)\) \(\chi_{89}(28,\cdot)\) \(\chi_{89}(29,\cdot)\) \(\chi_{89}(30,\cdot)\) \(\chi_{89}(31,\cdot)\) \(\chi_{89}(33,\cdot)\) \(\chi_{89}(35,\cdot)\) \(\chi_{89}(38,\cdot)\) \(\chi_{89}(41,\cdot)\) \(\chi_{89}(43,\cdot)\) \(\chi_{89}(46,\cdot)\) \(\chi_{89}(48,\cdot)\) \(\chi_{89}(51,\cdot)\) \(\chi_{89}(54,\cdot)\) \(\chi_{89}(56,\cdot)\) \(\chi_{89}(58,\cdot)\) \(\chi_{89}(59,\cdot)\) \(\chi_{89}(60,\cdot)\) \(\chi_{89}(61,\cdot)\) \(\chi_{89}(62,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{59}{88}\right)\)

Values

-1	1	2	3	4	5	6	7	8	9	10	11
\(-1\)	\(1\)	\(e\left(\frac{8}{11}\right)\)	\(e\left(\frac{59}{88}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{41}{44}\right)\)	\(e\left(\frac{35}{88}\right)\)	\(e\left(\frac{27}{88}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{15}{44}\right)\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{7}{22}\right)\)

Related number fields

Field of values

\(\Q(\zeta_{88})\)

Gauss sum

\(\displaystyle \tau_{2}(\chi_{89}(29,\cdot)) = \sum_{r\in \Z/89\Z} \chi_{89}(29,r) e\left(\frac{2r}{89}\right) = -8.0234680137+-4.9622536244i \)

Jacobi sum

\( \displaystyle J(\chi_{89}(29,\cdot),\chi_{89}(1,\cdot)) = \sum_{r\in \Z/89\Z} \chi_{89}(29,r) \chi_{89}(1,1-r) = -1 \)

Kloosterman sum

\( \displaystyle K(1,2,\chi_{89}(29,·)) = \sum_{r \in \Z/89\Z} \chi_{89}(29,r) e\left(\frac{1 r + 2 r^{-1}}{89}\right) = -11.1720905083+-9.6806715323i \)