LMFDB - Group of Dirichlet characters of modulus 48

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comment:Define the Dirichlet group

sage:G = DirichletGroup(48)

gp:g = idealstar(,48,2)

magma:G = FullDirichletGroup(48);

Character group

Order	=	16	comment:Order sage:G.order() gp:g.no magma:Order(G);
Structure	=	$C_{2}\times C_{2}\times C_{4}$	comment:Group structure sage:sorted(g.order() for g in G.gens()) gp:g.cyc magma:PrimaryInvariants(G);
Generators	=	$\chi_{48}(31,\cdot)$, $\chi_{48}(37,\cdot)$, $\chi_{48}(17,\cdot)$	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	$-1$	$1$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$25$	$29$	$31$
$\chi_{48}(1,\cdot)$	48.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{48}(5,\cdot)$	48.i	4	yes	$-1$	$1$	$-i$	$-1$	$-i$	$-i$	$-1$	$-i$	$1$	$-1$	$i$	$1$
$\chi_{48}(7,\cdot)$	48.b	2	no	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
$\chi_{48}(11,\cdot)$	48.k	4	yes	$1$	$1$	$-i$	$1$	$i$	$-i$	$-1$	$i$	$-1$	$-1$	$i$	$-1$
$\chi_{48}(13,\cdot)$	48.j	4	no	$1$	$1$	$-i$	$-1$	$-i$	$i$	$1$	$i$	$-1$	$-1$	$i$	$1$
$\chi_{48}(17,\cdot)$	48.e	2	no	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$
$\chi_{48}(19,\cdot)$	48.l	4	no	$-1$	$1$	$-i$	$1$	$i$	$i$	$1$	$-i$	$1$	$-1$	$i$	$-1$
$\chi_{48}(23,\cdot)$	48.f	2	no	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$
$\chi_{48}(25,\cdot)$	48.d	2	no	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$
$\chi_{48}(29,\cdot)$	48.i	4	yes	$-1$	$1$	$i$	$-1$	$i$	$i$	$-1$	$i$	$1$	$-1$	$-i$	$1$
$\chi_{48}(31,\cdot)$	48.g	2	no	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
$\chi_{48}(35,\cdot)$	48.k	4	yes	$1$	$1$	$i$	$1$	$-i$	$i$	$-1$	$-i$	$-1$	$-1$	$-i$	$-1$
$\chi_{48}(37,\cdot)$	48.j	4	no	$1$	$1$	$i$	$-1$	$i$	$-i$	$1$	$-i$	$-1$	$-1$	$-i$	$1$
$\chi_{48}(41,\cdot)$	48.h	2	no	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
$\chi_{48}(43,\cdot)$	48.l	4	no	$-1$	$1$	$i$	$1$	$-i$	$-i$	$1$	$i$	$1$	$-1$	$-i$	$-1$
$\chi_{48}(47,\cdot)$	48.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	$48$
Structure	\(C_{2}\times C_{2}\times C_{4}\)
Order	$16$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(5\)	\(7\)	\(11\)	\(13\)	\(17\)	\(19\)	\(23\)	\(25\)	\(29\)	\(31\)
\(\chi_{48}(1,\cdot)\)	48.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{48}(5,\cdot)\)	48.i	4	yes	\(-1\)	\(1\)	\(-i\)	\(-1\)	\(-i\)	\(-i\)	\(-1\)	\(-i\)	\(1\)	\(-1\)	\(i\)	\(1\)
\(\chi_{48}(7,\cdot)\)	48.b	2	no	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)
\(\chi_{48}(11,\cdot)\)	48.k	4	yes	\(1\)	\(1\)	\(-i\)	\(1\)	\(i\)	\(-i\)	\(-1\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(-1\)
\(\chi_{48}(13,\cdot)\)	48.j	4	no	\(1\)	\(1\)	\(-i\)	\(-1\)	\(-i\)	\(i\)	\(1\)	\(i\)	\(-1\)	\(-1\)	\(i\)	\(1\)
\(\chi_{48}(17,\cdot)\)	48.e	2	no	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)
\(\chi_{48}(19,\cdot)\)	48.l	4	no	\(-1\)	\(1\)	\(-i\)	\(1\)	\(i\)	\(i\)	\(1\)	\(-i\)	\(1\)	\(-1\)	\(i\)	\(-1\)
\(\chi_{48}(23,\cdot)\)	48.f	2	no	\(1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)
\(\chi_{48}(25,\cdot)\)	48.d	2	no	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(1\)
\(\chi_{48}(29,\cdot)\)	48.i	4	yes	\(-1\)	\(1\)	\(i\)	\(-1\)	\(i\)	\(i\)	\(-1\)	\(i\)	\(1\)	\(-1\)	\(-i\)	\(1\)
\(\chi_{48}(31,\cdot)\)	48.g	2	no	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)
\(\chi_{48}(35,\cdot)\)	48.k	4	yes	\(1\)	\(1\)	\(i\)	\(1\)	\(-i\)	\(i\)	\(-1\)	\(-i\)	\(-1\)	\(-1\)	\(-i\)	\(-1\)
\(\chi_{48}(37,\cdot)\)	48.j	4	no	\(1\)	\(1\)	\(i\)	\(-1\)	\(i\)	\(-i\)	\(1\)	\(-i\)	\(-1\)	\(-1\)	\(-i\)	\(1\)
\(\chi_{48}(41,\cdot)\)	48.h	2	no	\(-1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(1\)
\(\chi_{48}(43,\cdot)\)	48.l	4	no	\(-1\)	\(1\)	\(i\)	\(1\)	\(-i\)	\(-i\)	\(1\)	\(i\)	\(1\)	\(-1\)	\(-i\)	\(-1\)
\(\chi_{48}(47,\cdot)\)	48.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.