Dirichlet Character \(\chi_{83}(60,\cdot)\)

• Conductor: 83
• Order: 82
• Real: no
• Primitive: yes
• Minimal: yes
• Parity: odd
• Orbit label: 83.d

Basic properties

Conductor	=	83
Order	=	82
Real	=	no
Primitive	=	yes
Minimal	=	yes
Parity	=	odd
Orbit label	=	83.d
Orbit index	=	4

Galois orbit

\(\chi_{83}(2,\cdot)\) \(\chi_{83}(5,\cdot)\) \(\chi_{83}(6,\cdot)\) \(\chi_{83}(8,\cdot)\) \(\chi_{83}(13,\cdot)\) \(\chi_{83}(14,\cdot)\) \(\chi_{83}(15,\cdot)\) \(\chi_{83}(18,\cdot)\) \(\chi_{83}(19,\cdot)\) \(\chi_{83}(20,\cdot)\) \(\chi_{83}(22,\cdot)\) \(\chi_{83}(24,\cdot)\) \(\chi_{83}(32,\cdot)\) \(\chi_{83}(34,\cdot)\) \(\chi_{83}(35,\cdot)\) \(\chi_{83}(39,\cdot)\) \(\chi_{83}(42,\cdot)\) \(\chi_{83}(43,\cdot)\) \(\chi_{83}(45,\cdot)\) \(\chi_{83}(46,\cdot)\) \(\chi_{83}(47,\cdot)\) \(\chi_{83}(50,\cdot)\) \(\chi_{83}(52,\cdot)\) \(\chi_{83}(53,\cdot)\) \(\chi_{83}(54,\cdot)\) \(\chi_{83}(55,\cdot)\) \(\chi_{83}(56,\cdot)\) \(\chi_{83}(57,\cdot)\) \(\chi_{83}(58,\cdot)\) \(\chi_{83}(60,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{19}{82}\right)\)

Values

-1	1	2	3	4	5	6	7	8	9	10	11
\(-1\)	\(1\)	\(e\left(\frac{19}{82}\right)\)	\(e\left(\frac{28}{41}\right)\)	\(e\left(\frac{19}{41}\right)\)	\(e\left(\frac{21}{82}\right)\)	\(e\left(\frac{75}{82}\right)\)	\(e\left(\frac{35}{41}\right)\)	\(e\left(\frac{57}{82}\right)\)	\(e\left(\frac{15}{41}\right)\)	\(e\left(\frac{20}{41}\right)\)	\(e\left(\frac{23}{41}\right)\)

Related number fields

Field of values

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

Gauss sum

\(\displaystyle \tau_{2}(\chi_{83}(60,\cdot)) = \sum_{r\in \Z/83\Z} \chi_{83}(60,r) e\left(\frac{2r}{83}\right) = 8.4236281728+-3.4702288694i \)

Jacobi sum

\( \displaystyle J(\chi_{83}(60,\cdot),\chi_{83}(1,\cdot)) = \sum_{r\in \Z/83\Z} \chi_{83}(60,r) \chi_{83}(1,1-r) = -1 \)

Kloosterman sum

\( \displaystyle K(1,2,\chi_{83}(60,·)) = \sum_{r \in \Z/83\Z} \chi_{83}(60,r) e\left(\frac{1 r + 2 r^{-1}}{83}\right) = -7.0038129965+7.8588864894i \)