Dirichlet Character \(\chi_{167}(47,\cdot)\)

• Conductor: 167
• Order: 83
• Real: No
• Primitive: Yes
• Parity: Even
• Orbit Label: 167.c

Basic properties

Conductor	=	167
Order	=	83
Real	=	No
Primitive	=	Yes
Parity	=	Even
Orbit label	=	167.c
Orbit index	=	3

Galois orbit

\(\chi_{167}(2,\cdot)\) \(\chi_{167}(3,\cdot)\) \(\chi_{167}(4,\cdot)\) \(\chi_{167}(6,\cdot)\) \(\chi_{167}(7,\cdot)\) \(\chi_{167}(8,\cdot)\) \(\chi_{167}(9,\cdot)\) \(\chi_{167}(11,\cdot)\) \(\chi_{167}(12,\cdot)\) \(\chi_{167}(14,\cdot)\) \(\chi_{167}(16,\cdot)\) \(\chi_{167}(18,\cdot)\) \(\chi_{167}(19,\cdot)\) \(\chi_{167}(21,\cdot)\) \(\chi_{167}(22,\cdot)\) \(\chi_{167}(24,\cdot)\) \(\chi_{167}(25,\cdot)\) \(\chi_{167}(27,\cdot)\) \(\chi_{167}(28,\cdot)\) \(\chi_{167}(29,\cdot)\) \(\chi_{167}(31,\cdot)\) \(\chi_{167}(32,\cdot)\) \(\chi_{167}(33,\cdot)\) \(\chi_{167}(36,\cdot)\) \(\chi_{167}(38,\cdot)\) \(\chi_{167}(42,\cdot)\) \(\chi_{167}(44,\cdot)\) \(\chi_{167}(47,\cdot)\) \(\chi_{167}(48,\cdot)\) \(\chi_{167}(49,\cdot)\) ...

Values on generators

\(5\) → \(e\left(\frac{66}{83}\right)\)

Values

-1	1	2	3	4	5	6	7	8	9	10	11
\(1\)	\(1\)	\(e\left(\frac{67}{83}\right)\)	\(e\left(\frac{62}{83}\right)\)	\(e\left(\frac{51}{83}\right)\)	\(e\left(\frac{66}{83}\right)\)	\(e\left(\frac{46}{83}\right)\)	\(e\left(\frac{69}{83}\right)\)	\(e\left(\frac{35}{83}\right)\)	\(e\left(\frac{41}{83}\right)\)	\(e\left(\frac{50}{83}\right)\)	\(e\left(\frac{22}{83}\right)\)

Related number fields

Field of values

\(\Q(\zeta_{83})\)

Gauss sum

\(\displaystyle \tau_{2}(\chi_{167}(47,\cdot)) = \sum_{r\in \Z/167\Z} \chi_{167}(47,r) e\left(\frac{2r}{167}\right) = -4.7295070225+-12.0262946631i \)

Jacobi sum

\( \displaystyle J(\chi_{167}(47,\cdot),\chi_{167}(1,\cdot)) = \sum_{r\in \Z/167\Z} \chi_{167}(47,r) \chi_{167}(1,1-r) = -1 \)

Kloosterman sum

\( \displaystyle K(1,2,\chi_{167}(47,·)) = \sum_{r \in \Z/167\Z} \chi_{167}(47,r) e\left(\frac{1 r + 2 r^{-1}}{167}\right) = -0.7840508206+0.5428781355i \)