LMFDB - Group of Dirichlet characters of modulus 33

Show commands: PariGP / SageMath

sage: H = DirichletGroup(33)

pari: g = idealstar(,33,2)

Character group

sage: G.order() pari: g.no
Order	=	20
sage: H.invariants() pari: g.cyc
Structure	=	$C_{2}\times C_{10}$
sage: H.gens() pari: g.gen
Generators	=	$\chi_{33}(23,\cdot)$, $\chi_{33}(13,\cdot)$

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	$-1$	$1$	$2$	$4$	$5$	$7$	$8$	$10$	$13$	$14$	$16$	$17$
$\chi_{33}(1,\cdot)$	33.a	1	no	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{33}(2,\cdot)$	33.f	10	yes	$1$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{4}{5}\right)$	$-1$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{2}{5}\right)$
$\chi_{33}(4,\cdot)$	33.e	5	no	$1$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{3}{5}\right)$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{4}{5}\right)$
$\chi_{33}(5,\cdot)$	33.h	10	yes	$-1$	$1$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{7}{10}\right)$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{10}\right)$
$\chi_{33}(7,\cdot)$	33.g	10	no	$-1$	$1$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{1}{10}\right)$	$-1$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{10}\right)$
$\chi_{33}(8,\cdot)$	33.f	10	yes	$1$	$1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{2}{5}\right)$	$-1$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{5}\right)$
$\chi_{33}(10,\cdot)$	33.c	2	no	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$
$\chi_{33}(13,\cdot)$	33.g	10	no	$-1$	$1$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{3}{10}\right)$	$-1$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{9}{10}\right)$
$\chi_{33}(14,\cdot)$	33.h	10	yes	$-1$	$1$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{9}{10}\right)$	$1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{7}{10}\right)$
$\chi_{33}(16,\cdot)$	33.e	5	no	$1$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{5}\right)$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{3}{5}\right)$
$\chi_{33}(17,\cdot)$	33.f	10	yes	$1$	$1$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{1}{5}\right)$	$-1$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{3}{5}\right)$
$\chi_{33}(19,\cdot)$	33.g	10	no	$-1$	$1$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{9}{10}\right)$	$-1$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{7}{10}\right)$
$\chi_{33}(20,\cdot)$	33.h	10	yes	$-1$	$1$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{10}\right)$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{9}{10}\right)$
$\chi_{33}(23,\cdot)$	33.b	2	no	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
$\chi_{33}(25,\cdot)$	33.e	5	no	$1$	$1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{2}{5}\right)$	$1$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{5}\right)$
$\chi_{33}(26,\cdot)$	33.h	10	yes	$-1$	$1$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{10}\right)$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{10}\right)$
$\chi_{33}(28,\cdot)$	33.g	10	no	$-1$	$1$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{7}{10}\right)$	$-1$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{10}\right)$
$\chi_{33}(29,\cdot)$	33.f	10	yes	$1$	$1$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{3}{10}\right)$	$e\left(\frac{9}{10}\right)$	$e\left(\frac{3}{5}\right)$	$-1$	$e\left(\frac{7}{10}\right)$	$e\left(\frac{1}{10}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{4}{5}\right)$
$\chi_{33}(31,\cdot)$	33.e	5	no	$1$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{1}{5}\right)$	$e\left(\frac{4}{5}\right)$	$1$	$e\left(\frac{3}{5}\right)$	$e\left(\frac{4}{5}\right)$	$e\left(\frac{2}{5}\right)$	$e\left(\frac{2}{5}\right)$
$\chi_{33}(32,\cdot)$	33.d	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	$33$
Structure	\(C_{2}\times C_{10}\)
Order	$20$

Character	Orbit	Order	Primitive	\(-1\)	\(1\)	\(2\)	\(4\)	\(5\)	\(7\)	\(8\)	\(10\)	\(13\)	\(14\)	\(16\)	\(17\)
\(\chi_{33}(1,\cdot)\)	33.a	1	no	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{33}(2,\cdot)\)	33.f	10	yes	\(1\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(-1\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)
\(\chi_{33}(4,\cdot)\)	33.e	5	no	\(1\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)
\(\chi_{33}(5,\cdot)\)	33.h	10	yes	\(-1\)	\(1\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)
\(\chi_{33}(7,\cdot)\)	33.g	10	no	\(-1\)	\(1\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(-1\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)
\(\chi_{33}(8,\cdot)\)	33.f	10	yes	\(1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(-1\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)
\(\chi_{33}(10,\cdot)\)	33.c	2	no	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)	\(1\)	\(1\)	\(-1\)
\(\chi_{33}(13,\cdot)\)	33.g	10	no	\(-1\)	\(1\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(-1\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{9}{10}\right)\)
\(\chi_{33}(14,\cdot)\)	33.h	10	yes	\(-1\)	\(1\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)
\(\chi_{33}(16,\cdot)\)	33.e	5	no	\(1\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)
\(\chi_{33}(17,\cdot)\)	33.f	10	yes	\(1\)	\(1\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(-1\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)
\(\chi_{33}(19,\cdot)\)	33.g	10	no	\(-1\)	\(1\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{7}{10}\right)\)
\(\chi_{33}(20,\cdot)\)	33.h	10	yes	\(-1\)	\(1\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{9}{10}\right)\)
\(\chi_{33}(23,\cdot)\)	33.b	2	no	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)
\(\chi_{33}(25,\cdot)\)	33.e	5	no	\(1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)
\(\chi_{33}(26,\cdot)\)	33.h	10	yes	\(-1\)	\(1\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)
\(\chi_{33}(28,\cdot)\)	33.g	10	no	\(-1\)	\(1\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{7}{10}\right)\)	\(-1\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{10}\right)\)
\(\chi_{33}(29,\cdot)\)	33.f	10	yes	\(1\)	\(1\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{3}{10}\right)\)	\(e\left(\frac{9}{10}\right)\)	\(e\left(\frac{3}{5}\right)\)	\(-1\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{1}{10}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)
\(\chi_{33}(31,\cdot)\)	33.e	5	no	\(1\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(1\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{2}{5}\right)\)
\(\chi_{33}(32,\cdot)\)	33.d	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.