LMFDB - Group of Dirichlet characters of modulus 13

Show commands: Magma / Pari/GP / SageMath

comment:Define the Dirichlet group

sage:G = DirichletGroup(13)

gp:g = idealstar(,13,2)

magma:G = FullDirichletGroup(13);

Character group

Order	=	12	comment:Order sage:G.order() gp:g.no magma:Order(G);
Structure	=	$C_{12}$	comment:Group structure sage:sorted(g.order() for g in G.gens()) gp:g.cyc magma:PrimaryInvariants(G);
Generators	=	$\chi_{13}(2,\cdot)$	comment:Generators sage:G.gens() gp:g.gen magma:Generators(G);

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	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$\chi_{13}(1,\cdot)$	13.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{13}(2,\cdot)$	13.f	12	yes	$-1$	$1$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$-i$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{11}{12}\right)$	$i$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{7}{12}\right)$
$\chi_{13}(3,\cdot)$	13.c	3	yes	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$
$\chi_{13}(4,\cdot)$	13.e	6	yes	$1$	$1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{6}\right)$
$\chi_{13}(5,\cdot)$	13.d	4	yes	$-1$	$1$	$-i$	$1$	$-1$	$-i$	$-i$	$i$	$i$	$1$	$-1$	$i$
$\chi_{13}(6,\cdot)$	13.f	12	yes	$-1$	$1$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$-i$	$e\left(\frac{1}{12}\right)$	$e\left(\frac{7}{12}\right)$	$i$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{11}{12}\right)$
$\chi_{13}(7,\cdot)$	13.f	12	yes	$-1$	$1$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$i$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{12}\right)$	$-i$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{5}{12}\right)$
$\chi_{13}(8,\cdot)$	13.d	4	yes	$-1$	$1$	$i$	$1$	$-1$	$i$	$i$	$-i$	$-i$	$1$	$-1$	$-i$
$\chi_{13}(9,\cdot)$	13.c	3	yes	$1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$
$\chi_{13}(10,\cdot)$	13.e	6	yes	$1$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{6}\right)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$
$\chi_{13}(11,\cdot)$	13.f	12	yes	$-1$	$1$	$e\left(\frac{7}{12}\right)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$i$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{5}{12}\right)$	$-i$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{12}\right)$
$\chi_{13}(12,\cdot)$	13.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 $\F_{q}$
Belyi maps

Modulus	$13$
Structure	\(C_{12}\)
Order	$12$

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