LMFDB - Group of Dirichlet characters of modulus 17

Show commands: PariGP / SageMath

sage: H = DirichletGroup(17)

pari: g = idealstar(,17,2)

Character group

sage: G.order() pari: g.no
Order	=	16
sage: H.invariants() pari: g.cyc
Structure	=	$C_{16}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{17}(3,\cdot)$

Characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character	Orbit	Order	Primitive	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{17}(1,\cdot)$	17.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{17}(2,\cdot)$	17.d	8	yes	$1$	$1$	$i$	$e\left(\frac{7}{8}\right)$	$-1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{5}{8}\right)$	$-i$	$-i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{1}{8}\right)$
$\chi_{17}(3,\cdot)$	17.e	16	yes	$-1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{16}\right)$	$-i$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{15}{16}\right)$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{7}{16}\right)$
$\chi_{17}(4,\cdot)$	17.c	4	yes	$1$	$1$	$-1$	$-i$	$1$	$-i$	$i$	$i$	$-1$	$-1$	$i$	$i$
$\chi_{17}(5,\cdot)$	17.e	16	yes	$-1$	$1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{5}{16}\right)$	$-i$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{7}{16}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{15}{16}\right)$	$e\left(\frac{3}{16}\right)$
$\chi_{17}(6,\cdot)$	17.e	16	yes	$-1$	$1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{15}{16}\right)$	$i$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{1}{16}\right)$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{9}{16}\right)$
$\chi_{17}(7,\cdot)$	17.e	16	yes	$-1$	$1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{11}{16}\right)$	$i$	$e\left(\frac{7}{16}\right)$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{1}{16}\right)$	$e\left(\frac{13}{16}\right)$
$\chi_{17}(8,\cdot)$	17.d	8	yes	$1$	$1$	$-i$	$e\left(\frac{5}{8}\right)$	$-1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{7}{8}\right)$	$i$	$i$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{3}{8}\right)$
$\chi_{17}(9,\cdot)$	17.d	8	yes	$1$	$1$	$-i$	$e\left(\frac{1}{8}\right)$	$-1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{3}{8}\right)$	$i$	$i$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{7}{8}\right)$
$\chi_{17}(10,\cdot)$	17.e	16	yes	$-1$	$1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{16}\right)$	$i$	$e\left(\frac{15}{16}\right)$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{1}{16}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{5}{16}\right)$
$\chi_{17}(11,\cdot)$	17.e	16	yes	$-1$	$1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{7}{16}\right)$	$i$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{16}\right)$	$e\left(\frac{1}{16}\right)$
$\chi_{17}(12,\cdot)$	17.e	16	yes	$-1$	$1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{13}{16}\right)$	$-i$	$e\left(\frac{1}{16}\right)$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{15}{16}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{7}{16}\right)$	$e\left(\frac{11}{16}\right)$
$\chi_{17}(13,\cdot)$	17.c	4	yes	$1$	$1$	$-1$	$i$	$1$	$i$	$-i$	$-i$	$-1$	$-1$	$-i$	$-i$
$\chi_{17}(14,\cdot)$	17.e	16	yes	$-1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{9}{16}\right)$	$-i$	$e\left(\frac{13}{16}\right)$	$e\left(\frac{7}{16}\right)$	$e\left(\frac{3}{16}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{11}{16}\right)$	$e\left(\frac{15}{16}\right)$
$\chi_{17}(15,\cdot)$	17.d	8	yes	$1$	$1$	$i$	$e\left(\frac{3}{8}\right)$	$-1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{1}{8}\right)$	$-i$	$-i$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{5}{8}\right)$
$\chi_{17}(16,\cdot)$	17.b	2	yes	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Modulus	$17$
Structure	\(C_{16}\)
Order	$16$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)
\(\chi_{17}(1,\cdot)\)	17.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{17}(2,\cdot)\)	17.d	8	yes	\(1\)	\(1\)	\(i\)	\(e\left(\frac{7}{8}\right)\)	\(-1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(-i\)	\(-i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)
\(\chi_{17}(3,\cdot)\)	17.e	16	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{16}\right)\)	\(-i\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{15}{16}\right)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{7}{16}\right)\)
\(\chi_{17}(4,\cdot)\)	17.c	4	yes	\(1\)	\(1\)	\(-1\)	\(-i\)	\(1\)	\(-i\)	\(i\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(i\)
\(\chi_{17}(5,\cdot)\)	17.e	16	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{5}{16}\right)\)	\(-i\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{7}{16}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{15}{16}\right)\)	\(e\left(\frac{3}{16}\right)\)
\(\chi_{17}(6,\cdot)\)	17.e	16	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{15}{16}\right)\)	\(i\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{1}{16}\right)\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{9}{16}\right)\)
\(\chi_{17}(7,\cdot)\)	17.e	16	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{11}{16}\right)\)	\(i\)	\(e\left(\frac{7}{16}\right)\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{1}{16}\right)\)	\(e\left(\frac{13}{16}\right)\)
\(\chi_{17}(8,\cdot)\)	17.d	8	yes	\(1\)	\(1\)	\(-i\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(i\)	\(i\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)
\(\chi_{17}(9,\cdot)\)	17.d	8	yes	\(1\)	\(1\)	\(-i\)	\(e\left(\frac{1}{8}\right)\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(i\)	\(i\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)
\(\chi_{17}(10,\cdot)\)	17.e	16	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{16}\right)\)	\(i\)	\(e\left(\frac{15}{16}\right)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{1}{16}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{5}{16}\right)\)
\(\chi_{17}(11,\cdot)\)	17.e	16	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{7}{16}\right)\)	\(i\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{1}{16}\right)\)
\(\chi_{17}(12,\cdot)\)	17.e	16	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{13}{16}\right)\)	\(-i\)	\(e\left(\frac{1}{16}\right)\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{15}{16}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{7}{16}\right)\)	\(e\left(\frac{11}{16}\right)\)
\(\chi_{17}(13,\cdot)\)	17.c	4	yes	\(1\)	\(1\)	\(-1\)	\(i\)	\(1\)	\(i\)	\(-i\)	\(-i\)	\(-1\)	\(-1\)	\(-i\)	\(-i\)
\(\chi_{17}(14,\cdot)\)	17.e	16	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{9}{16}\right)\)	\(-i\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{7}{16}\right)\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{15}{16}\right)\)
\(\chi_{17}(15,\cdot)\)	17.d	8	yes	\(1\)	\(1\)	\(i\)	\(e\left(\frac{3}{8}\right)\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(-i\)	\(-i\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)
\(\chi_{17}(16,\cdot)\)	17.b	2	yes	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.