Dirichlet character \(\chi_{690}(77,\cdot)\)

• Label: 690.77
• Modulus: $690$
• Conductor: $345$
• Order: $44$
• Real: no
• Primitive: no
• Minimal: yes
• Parity: even

Basic properties

Modulus:	\(690\)
Conductor:	\(345\)
Order:	\(44\)
Real:	no
Primitive:	no, induced from \(\chi_{345}(77,\cdot)\)
Minimal:	yes
Parity:	even

Galois orbit 690.x

\(\chi_{690}(77,\cdot)\) \(\chi_{690}(167,\cdot)\) \(\chi_{690}(173,\cdot)\) \(\chi_{690}(197,\cdot)\) \(\chi_{690}(233,\cdot)\) \(\chi_{690}(257,\cdot)\) \(\chi_{690}(317,\cdot)\) \(\chi_{690}(347,\cdot)\) \(\chi_{690}(353,\cdot)\) \(\chi_{690}(377,\cdot)\) \(\chi_{690}(407,\cdot)\) \(\chi_{690}(443,\cdot)\) \(\chi_{690}(473,\cdot)\) \(\chi_{690}(533,\cdot)\) \(\chi_{690}(587,\cdot)\) \(\chi_{690}(593,\cdot)\) \(\chi_{690}(623,\cdot)\) \(\chi_{690}(647,\cdot)\) \(\chi_{690}(653,\cdot)\) \(\chi_{690}(683,\cdot)\)

Values on generators

\((461,277,511)\) → \((-1,i,e\left(\frac{3}{11}\right))\)

Values

\(-1\)	\(1\)	\(7\)	\(11\)	\(13\)	\(17\)	\(19\)	\(29\)	\(31\)	\(37\)	\(41\)	\(43\)
\(1\)	\(1\)	\(e\left(\frac{19}{44}\right)\)	\(e\left(\frac{21}{22}\right)\)	\(e\left(\frac{25}{44}\right)\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{13}{22}\right)\)	\(e\left(\frac{10}{11}\right)\)	\(e\left(\frac{7}{11}\right)\)	\(e\left(\frac{43}{44}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{5}{44}\right)\)

value at

e.g. 2

Related number fields

Field of values:	\(\Q(\zeta_{44})\)
Fixed field:	44.44.10759477646680772288556262812807675479811013525016604751482416550745256245136260986328125.1

Gauss sum

\(\displaystyle \tau_{2}(\chi_{690}(77,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(77,r) e\left(\frac{r}{345}\right) = 15.815037662+-9.7408718167i \)

Jacobi sum

\( \displaystyle J(\chi_{690}(77,\cdot),\chi_{690}(1,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(77,r) \chi_{690}(1,1-r) = 0 \)

Kloosterman sum

\( \displaystyle K(1,2,\chi_{690}(77,·)) = \sum_{r \in \Z/690\Z} \chi_{690}(77,r) e\left(\frac{1 r + 2 r^{-1}}{690}\right) = -0.0 \)