Dirichlet character \(\chi_{83}(44,\cdot)\)

• Conductor: 83
• Order: 41
• Real: no
• Primitive: yes
• Minimal: yes
• Parity: even
• Orbit label: 83.c

Basic properties

Conductor	=	83
Order	=	41
Real	=	no
Primitive	=	yes
Minimal	=	yes
Parity	=	even
Orbit label	=	83.c
Orbit index	=	3

Galois orbit

\(\chi_{83}(3,\cdot)\) \(\chi_{83}(4,\cdot)\) \(\chi_{83}(7,\cdot)\) \(\chi_{83}(9,\cdot)\) \(\chi_{83}(10,\cdot)\) \(\chi_{83}(11,\cdot)\) \(\chi_{83}(12,\cdot)\) \(\chi_{83}(16,\cdot)\) \(\chi_{83}(17,\cdot)\) \(\chi_{83}(21,\cdot)\) \(\chi_{83}(23,\cdot)\) \(\chi_{83}(25,\cdot)\) \(\chi_{83}(26,\cdot)\) \(\chi_{83}(27,\cdot)\) \(\chi_{83}(28,\cdot)\) \(\chi_{83}(29,\cdot)\) \(\chi_{83}(30,\cdot)\) \(\chi_{83}(31,\cdot)\) \(\chi_{83}(33,\cdot)\) \(\chi_{83}(36,\cdot)\) \(\chi_{83}(37,\cdot)\) \(\chi_{83}(38,\cdot)\) \(\chi_{83}(40,\cdot)\) \(\chi_{83}(41,\cdot)\) \(\chi_{83}(44,\cdot)\) \(\chi_{83}(48,\cdot)\) \(\chi_{83}(49,\cdot)\) \(\chi_{83}(51,\cdot)\) \(\chi_{83}(59,\cdot)\) \(\chi_{83}(61,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{13}{41}\right)\)

Values

-1	1	2	3	4	5	6	7	8	9	10	11
\(1\)	\(1\)	\(e\left(\frac{13}{41}\right)\)	\(e\left(\frac{34}{41}\right)\)	\(e\left(\frac{26}{41}\right)\)	\(e\left(\frac{23}{41}\right)\)	\(e\left(\frac{6}{41}\right)\)	\(e\left(\frac{22}{41}\right)\)	\(e\left(\frac{39}{41}\right)\)	\(e\left(\frac{27}{41}\right)\)	\(e\left(\frac{36}{41}\right)\)	\(e\left(\frac{25}{41}\right)\)

Related number fields

Field of values

\(\Q(\zeta_{41})\)

Gauss sum

\(\displaystyle \tau_{2}(\chi_{83}(44,\cdot)) = \sum_{r\in \Z/83\Z} \chi_{83}(44,r) e\left(\frac{2r}{83}\right) = -8.0652803901+4.2368918124i \)

Jacobi sum

\( \displaystyle J(\chi_{83}(44,\cdot),\chi_{83}(1,\cdot)) = \sum_{r\in \Z/83\Z} \chi_{83}(44,r) \chi_{83}(1,1-r) = -1 \)

Kloosterman sum

\( \displaystyle K(1,2,\chi_{83}(44,·)) = \sum_{r \in \Z/83\Z} \chi_{83}(44,r) e\left(\frac{1 r + 2 r^{-1}}{83}\right) = 3.5971082109+5.5545778373i \)