LMFDB - Group of Dirichlet characters of modulus 32

Show commands: Magma / Pari/GP / SageMath

comment:Define the Dirichlet group

sage:G = DirichletGroup(32)

gp:g = idealstar(,32,2)

magma:G = FullDirichletGroup(32);

Character group

Order	=	16	comment:Order sage:G.order() gp:g.no magma:Order(G);
Structure	=	$C_{2}\times C_{8}$	comment:Group structure sage:sorted(g.order() for g in G.gens()) gp:g.cyc magma:PrimaryInvariants(G);
Generators	=	$\chi_{32}(31,\cdot)$, $\chi_{32}(5,\cdot)$	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	$-1$	$1$	$3$	$5$	$7$	$9$	$11$	$13$	$15$	$17$	$19$	$21$
$\chi_{32}(1,\cdot)$	32.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{32}(3,\cdot)$	32.h	8	yes	$-1$	$1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$	$i$	$i$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{5}{8}\right)$	$1$	$-1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{7}{8}\right)$
$\chi_{32}(5,\cdot)$	32.g	8	yes	$1$	$1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{1}{8}\right)$	$i$	$-i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{7}{8}\right)$	$-1$	$-1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{8}\right)$
$\chi_{32}(7,\cdot)$	32.f	4	no	$-1$	$1$	$i$	$i$	$1$	$-1$	$-i$	$-i$	$-1$	$1$	$i$	$i$
$\chi_{32}(9,\cdot)$	32.e	4	no	$1$	$1$	$i$	$-i$	$-1$	$-1$	$-i$	$i$	$1$	$1$	$i$	$-i$
$\chi_{32}(11,\cdot)$	32.h	8	yes	$-1$	$1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{5}{8}\right)$	$-i$	$-i$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$	$1$	$-1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$
$\chi_{32}(13,\cdot)$	32.g	8	yes	$1$	$1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{7}{8}\right)$	$-i$	$i$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{1}{8}\right)$	$-1$	$-1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{3}{8}\right)$
$\chi_{32}(15,\cdot)$	32.d	2	no	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
$\chi_{32}(17,\cdot)$	32.b	2	no	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
$\chi_{32}(19,\cdot)$	32.h	8	yes	$-1$	$1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{7}{8}\right)$	$i$	$i$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$	$1$	$-1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{3}{8}\right)$
$\chi_{32}(21,\cdot)$	32.g	8	yes	$1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{8}\right)$	$i$	$-i$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{3}{8}\right)$	$-1$	$-1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{1}{8}\right)$
$\chi_{32}(23,\cdot)$	32.f	4	no	$-1$	$1$	$-i$	$-i$	$1$	$-1$	$i$	$i$	$-1$	$1$	$-i$	$-i$
$\chi_{32}(25,\cdot)$	32.e	4	no	$1$	$1$	$-i$	$i$	$-1$	$-1$	$i$	$-i$	$1$	$1$	$-i$	$i$
$\chi_{32}(27,\cdot)$	32.h	8	yes	$-1$	$1$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{1}{8}\right)$	$-i$	$-i$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{7}{8}\right)$	$1$	$-1$	$e\left(\frac{3}{8}\right)$	$e\left(\frac{5}{8}\right)$
$\chi_{32}(29,\cdot)$	32.g	8	yes	$1$	$1$	$e\left(\frac{1}{8}\right)$	$e\left(\frac{3}{8}\right)$	$-i$	$i$	$e\left(\frac{7}{8}\right)$	$e\left(\frac{5}{8}\right)$	$-1$	$-1$	$e\left(\frac{5}{8}\right)$	$e\left(\frac{7}{8}\right)$
$\chi_{32}(31,\cdot)$	32.c	2	no	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$

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	$32$
Structure	\(C_{2}\times C_{8}\)
Order	$16$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(3\)	\(5\)	\(7\)	\(9\)	\(11\)	\(13\)	\(15\)	\(17\)	\(19\)	\(21\)
\(\chi_{32}(1,\cdot)\)	32.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{32}(3,\cdot)\)	32.h	8	yes	\(-1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(i\)	\(i\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)
\(\chi_{32}(5,\cdot)\)	32.g	8	yes	\(1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(i\)	\(-i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(-1\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)
\(\chi_{32}(7,\cdot)\)	32.f	4	no	\(-1\)	\(1\)	\(i\)	\(i\)	\(1\)	\(-1\)	\(-i\)	\(-i\)	\(-1\)	\(1\)	\(i\)	\(i\)
\(\chi_{32}(9,\cdot)\)	32.e	4	no	\(1\)	\(1\)	\(i\)	\(-i\)	\(-1\)	\(-1\)	\(-i\)	\(i\)	\(1\)	\(1\)	\(i\)	\(-i\)
\(\chi_{32}(11,\cdot)\)	32.h	8	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(-i\)	\(-i\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)
\(\chi_{32}(13,\cdot)\)	32.g	8	yes	\(1\)	\(1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(-i\)	\(i\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)
\(\chi_{32}(15,\cdot)\)	32.d	2	no	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)
\(\chi_{32}(17,\cdot)\)	32.b	2	no	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)
\(\chi_{32}(19,\cdot)\)	32.h	8	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(i\)	\(i\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)
\(\chi_{32}(21,\cdot)\)	32.g	8	yes	\(1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(i\)	\(-i\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(-1\)	\(-1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)
\(\chi_{32}(23,\cdot)\)	32.f	4	no	\(-1\)	\(1\)	\(-i\)	\(-i\)	\(1\)	\(-1\)	\(i\)	\(i\)	\(-1\)	\(1\)	\(-i\)	\(-i\)
\(\chi_{32}(25,\cdot)\)	32.e	4	no	\(1\)	\(1\)	\(-i\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(-i\)	\(1\)	\(1\)	\(-i\)	\(i\)
\(\chi_{32}(27,\cdot)\)	32.h	8	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{8}\right)\)	\(-i\)	\(-i\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)	\(1\)	\(-1\)	\(e\left(\frac{3}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)
\(\chi_{32}(29,\cdot)\)	32.g	8	yes	\(1\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{3}{8}\right)\)	\(-i\)	\(i\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{7}{8}\right)\)
\(\chi_{32}(31,\cdot)\)	32.c	2	no	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)

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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.