Dirichlet character \(\chi_{47}(12,\cdot)\)

• Label: 47.12
• Modulus: $47$
• Conductor: $47$
• Order: $23$
• Real: no
• Primitive: yes
• Minimal: yes
• Parity: even

Basic properties

Modulus:	\(47\)
Conductor:	\(47\)
Order:	\(23\)
Real:	no
Primitive:	yes
Minimal:	yes
Parity:	even

Galois orbit 47.c

\(\chi_{47}(2,\cdot)\) \(\chi_{47}(3,\cdot)\) \(\chi_{47}(4,\cdot)\) \(\chi_{47}(6,\cdot)\) \(\chi_{47}(7,\cdot)\) \(\chi_{47}(8,\cdot)\) \(\chi_{47}(9,\cdot)\) \(\chi_{47}(12,\cdot)\) \(\chi_{47}(14,\cdot)\) \(\chi_{47}(16,\cdot)\) \(\chi_{47}(17,\cdot)\) \(\chi_{47}(18,\cdot)\) \(\chi_{47}(21,\cdot)\) \(\chi_{47}(24,\cdot)\) \(\chi_{47}(25,\cdot)\) \(\chi_{47}(27,\cdot)\) \(\chi_{47}(28,\cdot)\) \(\chi_{47}(32,\cdot)\) \(\chi_{47}(34,\cdot)\) \(\chi_{47}(36,\cdot)\) \(\chi_{47}(37,\cdot)\) \(\chi_{47}(42,\cdot)\)

Related number fields

Field of values:	\(\Q(\zeta_{23})\)
Fixed field:	\(\Q(\zeta_{47})^+\)

Values on generators

\(5\) → \(e\left(\frac{5}{23}\right)\)

Values

\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(1\)	\(1\)	\(e\left(\frac{21}{23}\right)\)	\(e\left(\frac{8}{23}\right)\)	\(e\left(\frac{19}{23}\right)\)	\(e\left(\frac{5}{23}\right)\)	\(e\left(\frac{6}{23}\right)\)	\(e\left(\frac{22}{23}\right)\)	\(e\left(\frac{17}{23}\right)\)	\(e\left(\frac{16}{23}\right)\)	\(e\left(\frac{3}{23}\right)\)	\(e\left(\frac{12}{23}\right)\)

value at

e.g. 2

Gauss sum

\(\displaystyle \tau_{2}(\chi_{47}(12,\cdot)) = \sum_{r\in \Z/47\Z} \chi_{47}(12,r) e\left(\frac{2r}{47}\right) = 5.8952695944+3.4993994355i \)

Jacobi sum

\( \displaystyle J(\chi_{47}(12,\cdot),\chi_{47}(1,\cdot)) = \sum_{r\in \Z/47\Z} \chi_{47}(12,r) \chi_{47}(1,1-r) = -1 \)

Kloosterman sum

\( \displaystyle K(1,2,\chi_{47}(12,·)) = \sum_{r \in \Z/47\Z} \chi_{47}(12,r) e\left(\frac{1 r + 2 r^{-1}}{47}\right) = 1.5512253922+-0.4346329717i \)