Dirichlet character \(\chi_{690}(7,\cdot)\)

• Label: 690.7
• Modulus: $690$
• Conductor: $115$
• Order: $44$
• Real: no
• Primitive: no
• Minimal: yes
• Parity: even

Basic properties

Modulus:	\(690\)
Conductor:	\(115\)
Order:	\(44\)
Real:	no
Primitive:	no, induced from \(\chi_{115}(7,\cdot)\)
Minimal:	yes
Parity:	even

Galois orbit 690.w

\(\chi_{690}(7,\cdot)\) \(\chi_{690}(37,\cdot)\) \(\chi_{690}(43,\cdot)\) \(\chi_{690}(67,\cdot)\) \(\chi_{690}(97,\cdot)\) \(\chi_{690}(103,\cdot)\) \(\chi_{690}(157,\cdot)\) \(\chi_{690}(217,\cdot)\) \(\chi_{690}(247,\cdot)\) \(\chi_{690}(283,\cdot)\) \(\chi_{690}(313,\cdot)\) \(\chi_{690}(337,\cdot)\) \(\chi_{690}(343,\cdot)\) \(\chi_{690}(373,\cdot)\) \(\chi_{690}(433,\cdot)\) \(\chi_{690}(457,\cdot)\) \(\chi_{690}(493,\cdot)\) \(\chi_{690}(517,\cdot)\) \(\chi_{690}(523,\cdot)\) \(\chi_{690}(613,\cdot)\)

Values on generators

\((461,277,511)\) → \((1,i,e\left(\frac{19}{22}\right))\)

Values

\(-1\)	\(1\)	\(7\)	\(11\)	\(13\)	\(17\)	\(19\)	\(29\)	\(31\)	\(37\)	\(41\)	\(43\)
\(1\)	\(1\)	\(e\left(\frac{29}{44}\right)\)	\(e\left(\frac{17}{22}\right)\)	\(e\left(\frac{37}{44}\right)\)	\(e\left(\frac{13}{44}\right)\)	\(e\left(\frac{5}{11}\right)\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{2}{11}\right)\)	\(e\left(\frac{17}{44}\right)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{3}{44}\right)\)

value at

e.g. 2

Related number fields

Field of values:	\(\Q(\zeta_{44})\)
Fixed field:	\(\Q(\zeta_{115})^+\)

Gauss sum

\(\displaystyle \tau_{2}(\chi_{690}(7,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(7,r) e\left(\frac{r}{345}\right) = 7.7979078568+-7.3615645794i \)

Jacobi sum

\( \displaystyle J(\chi_{690}(7,\cdot),\chi_{690}(1,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(7,r) \chi_{690}(1,1-r) = 0 \)

Kloosterman sum

\( \displaystyle K(1,2,\chi_{690}(7,·)) = \sum_{r \in \Z/690\Z} \chi_{690}(7,r) e\left(\frac{1 r + 2 r^{-1}}{690}\right) = 41.5825976865+-2.9740460828i \)