Dirichlet character \(\chi_{199}(25,\cdot)\)

• Label: 199.25
• Modulus: $199$
• Conductor: $199$
• Order: $33$
• Real: no
• Primitive: yes
• Minimal: yes
• Parity: even

Basic properties

Modulus:	\(199\)
Conductor:	\(199\)
Order:	\(33\)
Real:	no
Primitive:	yes
Minimal:	yes
Parity:	even

Galois orbit 199.i

\(\chi_{199}(5,\cdot)\) \(\chi_{199}(8,\cdot)\) \(\chi_{199}(25,\cdot)\) \(\chi_{199}(28,\cdot)\) \(\chi_{199}(40,\cdot)\) \(\chi_{199}(52,\cdot)\) \(\chi_{199}(64,\cdot)\) \(\chi_{199}(90,\cdot)\) \(\chi_{199}(98,\cdot)\) \(\chi_{199}(111,\cdot)\) \(\chi_{199}(116,\cdot)\) \(\chi_{199}(117,\cdot)\) \(\chi_{199}(123,\cdot)\) \(\chi_{199}(132,\cdot)\) \(\chi_{199}(140,\cdot)\) \(\chi_{199}(144,\cdot)\) \(\chi_{199}(157,\cdot)\) \(\chi_{199}(172,\cdot)\) \(\chi_{199}(182,\cdot)\) \(\chi_{199}(187,\cdot)\)

Related number fields

Field of values:	\(\Q(\zeta_{33})\)
Fixed field:	33.33.36584611296554742180833097810429342639777502523008874222975105176339833601.1

Values on generators

\(3\) → \(e\left(\frac{13}{33}\right)\)

Values

\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(1\)	\(1\)	\(e\left(\frac{25}{33}\right)\)	\(e\left(\frac{13}{33}\right)\)	\(e\left(\frac{17}{33}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{5}{33}\right)\)	\(e\left(\frac{31}{33}\right)\)	\(e\left(\frac{3}{11}\right)\)	\(e\left(\frac{26}{33}\right)\)	\(e\left(\frac{4}{33}\right)\)	\(e\left(\frac{5}{11}\right)\)

value at

e.g. 2

Gauss sum

\(\displaystyle \tau_{2}(\chi_{199}(25,\cdot)) = \sum_{r\in \Z/199\Z} \chi_{199}(25,r) e\left(\frac{2r}{199}\right) = 0.2156556218+-14.1050874741i \)

Jacobi sum

\( \displaystyle J(\chi_{199}(25,\cdot),\chi_{199}(1,\cdot)) = \sum_{r\in \Z/199\Z} \chi_{199}(25,r) \chi_{199}(1,1-r) = -1 \)

Kloosterman sum

\( \displaystyle K(1,2,\chi_{199}(25,·)) = \sum_{r \in \Z/199\Z} \chi_{199}(25,r) e\left(\frac{1 r + 2 r^{-1}}{199}\right) = 7.2605813318+-6.9229503166i \)