Dirichlet Character \(\chi_{59}(38,\cdot)\)

• Conductor: 59
• Order: 58
• Real: No
• Primitive: Yes
• Parity: Odd
• Orbit Label: 59.d

Basic properties

Conductor	=	59
Order	=	58
Real	=	No
Primitive	=	Yes
Parity	=	Odd
Orbit label	=	59.d
Orbit index	=	4

Galois orbit

\(\chi_{59}(2,\cdot)\) \(\chi_{59}(6,\cdot)\) \(\chi_{59}(8,\cdot)\) \(\chi_{59}(10,\cdot)\) \(\chi_{59}(11,\cdot)\) \(\chi_{59}(13,\cdot)\) \(\chi_{59}(14,\cdot)\) \(\chi_{59}(18,\cdot)\) \(\chi_{59}(23,\cdot)\) \(\chi_{59}(24,\cdot)\) \(\chi_{59}(30,\cdot)\) \(\chi_{59}(31,\cdot)\) \(\chi_{59}(32,\cdot)\) \(\chi_{59}(33,\cdot)\) \(\chi_{59}(34,\cdot)\) \(\chi_{59}(37,\cdot)\) \(\chi_{59}(38,\cdot)\) \(\chi_{59}(39,\cdot)\) \(\chi_{59}(40,\cdot)\) \(\chi_{59}(42,\cdot)\) \(\chi_{59}(43,\cdot)\) \(\chi_{59}(44,\cdot)\) \(\chi_{59}(47,\cdot)\) \(\chi_{59}(50,\cdot)\) \(\chi_{59}(52,\cdot)\) \(\chi_{59}(54,\cdot)\) \(\chi_{59}(55,\cdot)\) \(\chi_{59}(56,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{39}{58}\right)\)

Values

-1	1	2	3	4	5	6	7	8	9	10	11
\(-1\)	\(1\)	\(e\left(\frac{39}{58}\right)\)	\(e\left(\frac{18}{29}\right)\)	\(e\left(\frac{10}{29}\right)\)	\(e\left(\frac{1}{29}\right)\)	\(e\left(\frac{17}{58}\right)\)	\(e\left(\frac{3}{29}\right)\)	\(e\left(\frac{1}{58}\right)\)	\(e\left(\frac{7}{29}\right)\)	\(e\left(\frac{41}{58}\right)\)	\(e\left(\frac{47}{58}\right)\)

Related number fields

Field of values

\(\Q(\zeta_{29})\)

Gauss sum

\(\displaystyle \tau_{2}(\chi_{59}(38,\cdot)) = \sum_{r\in \Z/59\Z} \chi_{59}(38,r) e\left(\frac{2r}{59}\right) = -6.2210891581+4.5053356908i \)

Jacobi sum

\( \displaystyle J(\chi_{59}(38,\cdot),\chi_{59}(1,\cdot)) = \sum_{r\in \Z/59\Z} \chi_{59}(38,r) \chi_{59}(1,1-r) = -1 \)

Kloosterman sum

\( \displaystyle K(1,2,\chi_{59}(38,·)) = \sum_{r \in \Z/59\Z} \chi_{59}(38,r) e\left(\frac{1 r + 2 r^{-1}}{59}\right) = 1.3315876605+0.8011897124i \)